Being HOD-of-a-set is invariant throughout the generic multiverse

Posted on September 23, 2015 by Joel David Hamkins

The axiom $𝑉 = H O D$ , introduced by Gödel, asserts that every set is ordinal definable. This axiom has a subtler foundational aspect than might at first be expected. The reason is that the general concept of “object $𝑥$ is definable using parameter $𝑝$ ” is not in general first-order expressible in set theory; it is of course a second-order property, which makes sense only relative to a truth predicate, and by Tarski’s theorem, we can have no first-order definable truth predicate. Thus, the phrase “definable using ordinal parameters” is not directly meaningful in the first-order language of set theory without further qualification or explanation. Fortunately, however, it is a remarkable fact that when we allow definitions to use arbitrary ordinal parameters, as we do with $H O D$ , then we can in fact make such qualifications in such a way that the axiom becomes first-order expressible in set theory. Specifically, we say officially that $𝑉 = H O D$ holds, if for every set $𝑥$ , there is an ordinal $𝜃$ with $𝑥 \in 𝑉 𝜃$ , for which which $𝑥$ is definable by some formula $𝜓 (𝑥)$ in the structure $⟨ 𝑉 𝜃, \in ⟩$ using ordinal parameters. Since $𝑉 𝜃$ is a set, we may freely make reference to first-order truth in $𝑉 𝜃$ without requiring any truth predicate in $𝑉$ . Certainly any such $𝑥$ as this is also ordinal-definable in $𝑉$ , since we may use $𝜃$ and the Gödel-code of $𝜓$ also as parameters, and note that $𝑥$ is the unique object such that it is in $𝑉 𝜃$ and satisfies $𝜓$ in $𝑉 𝜃$ . (Note that inside an $𝜔$ -nonstandard model of set theory, we may really need to use $𝜓$ as a parameter, since it may be nonstandard, and $𝑥$ may not be definable in $𝑉 𝜃$ using a meta-theoretically standard natural number; but fortunately, the Gödel code of a formula is an integer, which is still an ordinal, and this issue is the key to the issue.) Conversely, if $𝑥$ is definable in $𝑉$ using formula $𝜑 (𝑥, ⃗ 𝛼)$ with ordinal parameters $⃗ 𝛼$ , then it follows by the reflection theorem that $𝑥$ is defined by $𝜑 (𝑥, ⃗ 𝛼)$ inside some $𝑉 𝜃$ . So this formulation of $𝑉 = 𝐻 𝑂 𝐷$ is expressible and exactly captures the desired second-order property that every set is ordinal-definable.

Consider next the axiom $𝑉 = H O D (𝑏)$ , asserting that every set is definable from ordinal parameters and parameter $𝑏$ . Officially, as before, $𝑉 = H O D (𝑏)$ asserts that for every $𝑥$ , there is an ordinal $𝜃$ , formula $𝜓$ and ordinals $⃗ 𝛼 < 𝜃$ , such that $𝑥$ is the unique object in $𝑉 𝜃$ for which $⟨ 𝑉 𝜃, \in ⟩ ⊧ 𝜓 (𝑥, ⃗ 𝛼, 𝑏)$ , and the reflection argument shows again that this way of defining the axiom exactly captures the intended idea.

The axiom I actually want to focus on is $\exists 𝑏 (𝑉 = H O D (𝑏))$ , asserting that the universe is $H O D$ of a set. (I assume ZFC in the background theory.) It turns out that this axiom is constant throughout the generic multiverse.

Theorem. The assertion $\exists 𝑏 (𝑉 = H O D (𝑏))$ is forcing invariant.

If it holds in $𝑉$ , then it continues to hold in every set forcing extension of $𝑉$ .
If it holds in $𝑉$ , then it holds in every ground of $𝑉$ .

Thus, the truth of this axiom is invariant throughout the generic multiverse.

Proof. Suppose that $Z F C + 𝑉 = H O D (𝑏)$ , and $𝑉 [𝐺]$ is a forcing extension of $𝑉$ by generic filter $𝐺 \subset ℙ \in 𝑉$ . By the ground-model definability theorem, it follows that $𝑉$ is definable in $𝑉 [𝐺]$ from parameter $𝑃 (ℙ) 𝑉$ . Thus, using this parameter, as well as $𝑏$ and additional ordinal parameters, we can define in $𝑉 [𝐺]$ any particular object in $𝑉$ . Since this includes all the $ℙ$ -names used to form $𝑉 [𝐺]$ , it follows that $𝑉 [𝐺] = H O D (𝑏, 𝑃 (ℙ) 𝑉, 𝐺)$ , and so $𝑉 [𝐺]$ is $H O D$ of a set, as desired.

Conversely, suppose that $𝑊$ is a ground of $𝑉$ , so that $𝑉 = 𝑊 [𝐺]$ for some $𝑊$ -generic filter $𝐺 \subset ℙ \in 𝑊$ , and $𝑉 = H O D (𝑏)$ for some set $𝑏$ . Let $˙ 𝑏$ be a name for which $˙ 𝑏 𝐺 = 𝑏$ . Every object $𝑥 \in 𝑊$ is definable in $𝑊 [𝐺]$ from $𝑏$ and ordinal parameters $⃗ 𝛼$ , so there is some formula $𝜓$ for which $𝑥$ is unique such that $𝜓 (𝑥, 𝑏, ⃗ 𝛼)$ . Thus, there is some condition $𝑝 \in ℙ$ such that $𝑥$ is unique such that $𝑝 ⊩ 𝜓 (ˇ 𝑥, ˙ 𝑏, ˇ ⃗ 𝛼)$ . If $⟨ 𝑝 𝛽 ∣ 𝛽 < | ℙ | ⟩$ is a fixed enumeration of $ℙ$ in $𝑊$ , then $𝑝 = 𝑝 𝛽$ for some ordinal $𝛽$ , and we may therefore define $𝑥$ in $𝑊$ using ordinal parameters, along with $˙ 𝑏$ and the fixed enumeration of $ℙ$ . So $𝑊$ thinks the universe is $H O D$ of a set, as desired.

Since the generic multiverse is obtained by iteratively moving to forcing extensions to grounds, and each such movement preserves the axiom, it follows that $\exists 𝑏 (𝑉 = H O D (𝑏))$ is constant throughout the generic multiverse. QED

Theorem. If $𝑉 = H O D (𝑏)$ , then there is a forcing extension $𝑉 [𝐺]$ in which $𝑉 = H O D$ holds.

Proof. We are working in ZFC. Suppose that $𝑉 = H O D (𝑏)$ . We may assume $𝑏$ is a set of ordinals, since such sets can code any given set. Consider the following forcing iteration: first add a Cohen real $𝑐$ , and then perform forcing $𝐺$ that codes $𝑐$ , $𝑃 (𝜔) 𝑉$ and $𝑏$ into the GCH pattern at uncountable cardinals, and then perform self-encoding forcing $𝐻$ above that coding, coding also $𝐺$ (see my paper on Set-theoretic geology for further details on self-encoding forcing). In the final model $𝑉 [𝑐] [𝐺] [𝐻]$ , therefore, the objects $𝑐$ , $𝑏$ , $𝑃 (𝜔) 𝑉$ , $𝐺$ and $𝐻$ are all definable without parameters. Since $𝑉 \subset 𝑉 [𝑐] [𝐺] [𝐻]$ has a closure point at $𝜔$ , it satisfies the $𝜔 1$ -approximation and cover properties, and therefore the class $𝑉$ is definable in $𝑉 [𝑐] [𝐺] [𝐻]$ using $𝑃 (𝜔) 𝑉$ as a parameter. Since this parameter is itself definable without parameters, it follows that $𝑉$ is parameter-free definable in $𝑉 [𝑐] [𝐺] [𝐻]$ . Since $𝑏$ is also definable there, it follows that every element of $H O D (𝑏) 𝑉 = 𝑉$ is ordinal-definable in $𝑉 [𝑐] [𝐺] [𝐻]$ . And since $𝑐$ , $𝐺$ and $𝐻$ are also definable without parameters, we have $𝑉 [𝑐] [𝐺] [𝐻] ⊧ 𝑉 = H O D$ , as desired. QED

Corollary. The following are equivalent.

The universe is $H O D$ of a set: $\exists 𝑏 (𝑉 = H O D (𝑏))$ .
Somewhere in the generic multiverse, the universe is $H O D$ of a set.
Somewhere in the generic multiverse, the axiom $𝑉 = H O D$ holds.
The axiom $𝑉 = H O D$ is forceable.

Proof. This is an immediate consequence of the previous theorems. $1 \to 4 \to 3 \to 2 \to 1$ . QED

Corollary. The axiom $𝑉 = H O D$ , if true, even if true anywhere in the generic multiverse, is a switch.

Proof. A switch is a statement such that both it and its negation are necessarily possible by forcing; that is, in every set forcing extension, one can force the statement to be true and also force it to be false. We can always force $𝑉 = H O D$ to fail, simply by adding a Cohen real. If $𝑉 = H O D$ is true, then by the first theorem, every forcing extension has $𝑉 = H O D (𝑏)$ for some $𝑏$ , in which case $𝑉 = H O D$ remains forceable, by the second theorem. QED

Quoted in New York magazine: The Chalk for Math Professors

Posted on September 22, 2015 by Joel David Hamkins

The Chalk for Math Professors: Hagoromo Fulltouch, by Alex Ronan for the current issue of New York magazine, part of the Status Survey on various items for professionals.

The smooth texture flows so easily across the chalkboard like a fountain pen … One puts up mathematics on the chalkboard as if tracing out an idea in the air.

Meanwhile, I happened to be at a conference at the Research Institute for Mathematical Sciences in Kyoto last week, and I gave my talk using the chalkboard and the plentiful supply of Hagoromo chalk provided there.

Different models of set theory with the same subset relation

Posted on September 9, 2015 by Joel David Hamkins

Recently Makoto Kikuchi (Kobe University) asked me the following interesting question, which arises very naturally if one should adopt a mereological perspective in the foundations of mathematics, placing a focus on the parthood relation rather than the element-of relation. In set theory, this perspective would lead one to view the subset or inclusion relation $\subseteq$ as the primary fundamental relation, rather than the membership $\in$ relation.

Question. Can there be two different models of set theory, with the same inclusion relation?

We spent an evening discussing it, over delicious (Rokko-michi-style) okonomiyaki and bi-ru, just like old times, except that we are in Tokyo at the CTFM 2015, and I’d like to explain the answer, which is yes, this always happens in every model of set theory.

Theorem. In any universe of set theory $⟨ 𝑉, \in ⟩$ , there is a definable relation $\in *$ , different from $\in$ , such that $⟨ 𝑉, \in * ⟩$ is a model of set theory, in fact isomorphic to the original universe $⟨ 𝑉, \in ⟩$ , for which the corresponding inclusion relation $𝑢 \subseteq * 𝑣 ⟺ \forall 𝑎 (𝑎 \in * 𝑢 \to 𝑎 \in * 𝑣)$ is identical to the usual inclusion relation $𝑢 \subseteq 𝑣$ .

Proof. Let $𝜃 : 𝑉 \to 𝑉$ be any definable non-identity permutation of the universe, and let $𝜏 : 𝑢 \mapsto 𝜃 [𝑢] = {𝜃 (𝑎) ∣ 𝑎 \in 𝑢}$ be the function determined by pointwise image under $𝜃$ . Since $𝜃$ is bijective, it follows that $𝜏$ is also a bijection of $𝑉$ to $𝑉$ , since every set is the $𝜃$ -image of a unique set. Furthermore, $𝜏$ is an automorphism of $⟨ 𝑉, \subseteq ⟩$ , since $𝑢 \subseteq 𝑣 ⟺ 𝜃 [𝑢] \subseteq 𝜃 [𝑣] ⟺ 𝜏 (𝑢) \subseteq 𝜏 (𝑣) .$ I had used this idea a few years ago in my answer to the MathOverflow question, Is the inclusion version of Kunen inconsistency theorem true?, which shows that there are nontrivial $\subseteq$ automorphisms of the universe. Note that since $𝜏 ({𝑎}) = {𝜃 (𝑎)}$ , it follows that any instance of nontriviality $𝜃 (𝑎) \neq 𝑎$ in $𝜃$ leads immediately to an instance of nontriviality in $𝜏$ .

Using the map $𝜏$ , define $𝑎 \in * 𝑏 ⟺ 𝜏 (𝑎) \in 𝜏 (𝑏)$ . By definition, therefore, $𝜏$ is an isomorphism of $⟨ 𝑉, \in * ⟩ ≅ ⟨ 𝑉, \in ⟩$ . Let us show that $\in * \neq \in$ . Since $𝜃$ is nontrivial, there is an $\in$ -minimal set $𝑎$ with $𝜃 (𝑎) \neq 𝑎$ . By minimality, $𝜃 [𝑎] = 𝑎$ and so $𝜏 (𝑎) = 𝑎$ . But as mentioned, $𝜏 ({𝑎}) = {𝜃 (𝑎)} \neq {𝑎}$ . So we have $𝑎 \in {𝑎}$ , but $𝜏 (𝑎) = 𝑎 \notin {𝜃 (𝑎)} = 𝜏 ({𝑎})$ and hence $𝑎 \notin * {𝑎}$ . So the two relations are different.

Meanwhile, consider the corresponding subset relation. Specifically, $𝑢 \subseteq * 𝑣$ is defined to mean $\forall 𝑎 (𝑎 \in * 𝑢 \to 𝑎 \in * 𝑣)$ , which holds if and only if $\forall 𝑎 (𝜏 (𝑎) \in 𝜏 (𝑢) \to 𝜏 (𝑎) \in 𝜏 (𝑣))$ ; but since $𝜏$ is surjective, this holds if and only if $𝜏 (𝑢) \subseteq 𝜏 (𝑣)$ , which as we observed at the beginning of the proof, holds if and only if $𝑢 \subseteq 𝑣$ . So the corresponding subset relations $\subseteq *$ and $\subseteq$ are identical, as desired.

Another way to express what is going on is that $𝜏$ is an isomorphism of the structure $⟨ 𝑉, \in *, \subseteq ⟩$ with $⟨ 𝑉, \in, \subseteq ⟩$ , and so $\subseteq$ is in fact that same as the corresponding inclusion relation $\subseteq *$ that one would define from $\in *$ . QED

Corollary. One cannot define $\in$ from $\subseteq$ in a model of set theory.

Proof. The map $𝜏$ is a $\subseteq$ -automorphism, and so it preserves every relation definable from $\subseteq$ , but it does not preserve $\in$ . QED

Nevertheless, I claim that the isomorphism type of $⟨ 𝑉, \in ⟩$ is implicit in the inclusion relation $\subseteq$ , in the sense that any other class relation $\in *$ having that same inclusion relation is isomorphic to the $\in$ relation.

Theorem. Assume ZFC in the universe $⟨ 𝑉, \in ⟩$ . Suppose that $\in *$ is a class relation for which $⟨ 𝑉, \in * ⟩$ is a model of set theory (a weak set theory suffices), such that the corresponding inclusion relation $𝑢 \subseteq * 𝑣 ⟺ \forall 𝑎 (𝑎 \in * 𝑢 \to 𝑎 \in * 𝑣)$ is the same as the usual inclusion relation $𝑢 \subseteq 𝑣$ . Then the two membership relations are isomorphic $⟨ 𝑉, \in ⟩ ≅ ⟨ 𝑉, \in * ⟩ .$

Proof. Since the singleton set ${𝑎}$ has exactly two subsets with respect to the usual $\subseteq$ relation — the empty set and itself — this must also be true with respect to the inclusion relation $\subseteq *$ defined via $\in *$ , since we have assumed $\subseteq * = \subseteq$ . Thus, the object ${𝑎}$ is also a singleton with respect to $\in *$ , and so there is a unique object $𝜂 (𝑎)$ such that $𝑥 \in * 𝑎 ⟺ 𝑥 = 𝜂 (𝑎)$ . By extensionality and since every object has its singleton, it follows that $𝜂 : 𝑉 \to 𝑉$ is both one-to-one and onto. Let $𝜃 = 𝜂 - 1$ be the inverse permutation.

Observe that $𝑎 \in 𝑢 ⟺ {𝑎} \subseteq 𝑢 ⟺ {𝑎} \subseteq * 𝑢 ⟺ 𝜂 (𝑎) \in * 𝑢$ . Thus, $𝑏 \in * 𝑢 ⟺ 𝜃 (𝑏) \in 𝑢 .$

Using $\in$ -recursion, define $𝑏 * = {𝜃 (𝑎 *) ∣ 𝑎 \in 𝑏}$ . The map $𝑏 \mapsto 𝑏 *$ is one-to-one by $\in$ -recursion, since if there is no violation of this for the elements of $𝑏$ , then we may recover $𝑏$ from $𝑏 *$ by applying $𝜃 - 1$ to the elements of $𝑏 *$ and then using the induction assumption to find the unique $𝑎$ from $𝑎 *$ for each $𝜃 (𝑎 *) \in 𝑏 *$ , thereby recovering $𝑏$ . So $𝑏 \mapsto 𝑏 *$ is injective.

I claim that this map is also surjective. If $𝑦 0 \neq 𝑏 *$ for any $𝑏$ , then there must be an element of $𝑦 0$ that is not of the form $𝜃 (𝑏 *)$ for any $𝑏$ . Since $𝜃$ is surjective, this means there is $𝜃 (𝑦 1) \in 𝑦 0$ with $𝑦 1 \neq 𝑏 *$ for any $𝑏$ . Continuing, there is $𝑦 𝑛 + 1$ with $𝜃 (𝑦 𝑛 + 1) \in 𝑦 𝑛$ and $𝑦 𝑛 + 1 \neq 𝑏 *$ for any $𝑏$ . Let $𝑧 = {𝜃 (𝑦 𝑛) ∣ 𝑛 \in 𝜔}$ . Since $𝑥 \in * 𝑢 ⟺ 𝜃 (𝑥) \in 𝑢$ , it follows that the $\in *$ -elements of $𝑧$ are precisely the $𝑦 𝑛$ ’s. But $𝜃 (𝑦 𝑛 + 1) \in 𝑦 𝑛$ , and so $𝑦 𝑛 + 1 \in * 𝑦 𝑛$ . So $𝑧$ has no $\in *$ -minimal element, violating the axiom of foundation for $\in *$ , a contradiction. So the map $𝑏 \mapsto 𝑏 *$ is a bijection of $𝑉$ with $𝑉$ .

Finally, we observe that because $𝑎 \in 𝑏 ⟺ 𝜃 (𝑎 *) \in 𝑏 * ⟺ 𝑎 * \in * 𝑏 *,$ it follows that the map $𝑏 \mapsto 𝑏 *$ is an isomorphism of $⟨ 𝑉, \in ⟩$ with $⟨ 𝑉, \in * ⟩$ , as desired. QED

The conclusion is that although $\in$ is not definable from $\subseteq$ , nevertheless, the isomorphism type of $\in$ is implicit in $\subseteq$ , in the sense that any other class relation $\in *$ giving rise to the same inclusion relation $\subseteq * = \subseteq$ is isomorphic to $\in$ .

Meanwhile, I do not yet know what the situation is when one drops the assumption that $\in *$ is a class with respect to the $⟨ 𝑉, \in ⟩$ universe.

Question. Can there be two models of set theory $⟨ 𝑀, \in ⟩$ and $⟨ 𝑀, \in * ⟩$ , not necessarily classes with respect to each other, which have the same inclusion relation $\subseteq = \subseteq *$ , but which are not isomorphic?

(This question is now answered! See my joint paper with Kikuchi at Set-theoretic mereology.)

Upward countable closure in the generic multiverse of forcing to add a Cohen real

Posted on September 4, 2015 by Joel David Hamkins

I’d like to discuss my theorem that the collection of models $𝑀 [𝑐]$ obtained by adding an $𝑀$ -generic Cohen real $𝑐$ over a fixed countable transitive model of set theory $𝑀$ is upwardly countably closed, in the sense that every increasing countable chain has an upper bound.

I proved this theorem back in 2011, while at the Young Set Theory Workshop in Bonn and continuing at the London summer school on set theory, in a series of conversations with Giorgio Venturi. The argument has recently come up again in various discussions, and so let me give an account of it.

We consider the collection of all forcing extensions of a fixed countable transitive model $𝑀$ of ZFC by the forcing to add a Cohen real, models of the form $𝑀 [𝑐]$ , and consider the question of whether every countable increasing chain of these models has an upper bound. The answer is yes! (Actually, Giorgio wants to undertake forcing constructions by forcing over this collection of models to add a generic upward directed system of models; it follows from this theorem that this forcing is countably closed.) This theorem fits into the theme of my earlier post, Upward closure in the toy multiverse of all countable models of set theory, where similar theorems are proved, but not this one exactly.

Theorem. For any countable transitive model $𝑀 ⊧ Z F C$ , the collection of all forcing extensions $𝑀 [𝑐]$ by adding an $𝑀$ -generic Cohen real is upward-countably closed. That is, for any countable tower of such forcing extensions
$𝑀 [𝑐 0] \subset 𝑀 [𝑐 1] \subset \dots \subset 𝑀 [𝑐 𝑛] \subset \dots,$
we may find an $𝑀$ -generic Cohen real $𝑑$ such that $𝑀 [𝑐 𝑛] \subset 𝑀 [𝑑]$ for every natural number $𝑛$ .

Proof. Suppose that we have such a tower of forcing extensions $𝑀 [𝑐 0] \subset 𝑀 [𝑐 1] \subset \dots$ , and so on. Note that if $𝑀 [𝑏] \subset 𝑀 [𝑐]$ for $𝑀$ -generic Cohen reals $𝑏$ and $𝑐$ , then $𝑀 [𝑐]$ is a forcing extension of $𝑀 [𝑏]$ by a quotient of the Cohen-real forcing. But since the Cohen forcing itself has a countable dense set, it follows that all such quotients also have a countable dense set, and so $𝑀 [𝑐]$ is actually $𝑀 [𝑏] [𝑏 1]$ for some $𝑀 [𝑏]$ -generic Cohen real $𝑏 1$ . Thus, we may view the tower as having the form:
$𝑀 [𝑏 0] \subset 𝑀 [𝑏 0 \times 𝑏 1] \subset \dots \subset 𝑀 [𝑏 0 \times 𝑏 1 \times \dots \times 𝑏 𝑛] \subset \dots,$
where now it follows that any finite collection of the reals $𝑏 𝑖$ are mutually $𝑀$ -generic.

Of course, we cannot expect in general that the real $⟨ 𝑏 𝑛 ∣ 𝑛 < 𝜔 ⟩$ is $𝑀$ -generic for $A d d (𝜔, 𝜔)$ , since this real may be very badly behaved. For example, the sequence of first-bits of the $𝑏 𝑛$ ’s may code a very naughty real $𝑧$ , which cannot be added by forcing over $𝑀$ at all. So in general, we cannot allow that this sequence is added to the limit model $𝑀 [𝑑]$ . (See further discussion in my post Upward closure in the toy multiverse of all countable models of set theory.)

We shall instead undertake a construction by making finitely many changes to each real $𝑏 𝑛$ , resulting in a real $𝑑 𝑛$ , in such a way that the resulting combined real $𝑑 = \oplus 𝑛 𝑑 𝑛$ is $𝑀$ -generic for the forcing to add $𝜔$ -many Cohen reals, which is of course isomorphic to adding just one. To do this, let’s get a little more clear with our notation. We regard each $𝑏 𝑛$ as an element of Cantor space $2 𝜔$ , that is, an infinite binary sequence, and the corresponding filter associated with this real is the collection of finite initial segments of $𝑏 𝑛$ , which will be an $𝑀$ -generic filter through the partial order of finite binary sequences $2 < 𝜔$ , which is one of the standard isomorphic copies of Cohen forcing. We will think of $𝑑$ as a binary function on the plane $𝑑 : 𝜔 \times 𝜔 \to 2$ , where the $𝑛 𝑡 ℎ$ slice $𝑑 𝑛$ is the corresponding function $𝜔 \to 2$ obtained by fixing the first coordinate to be $𝑛$ .

Now, we enumerate the countably many open dense subsets for the forcing to add a Cohen real $𝜔 \times 𝜔 \to 2$ as $𝐷 0$ , $𝐷 1$ , and so on. There are only countably many such dense sets, because $𝑀$ is countable. Now, we construct $𝑑$ in stages. Before stage $𝑛$ , we will have completely specified $𝑑 𝑘$ for $𝑘 < 𝑛$ , and we also may be committed to a finite condition $𝑝 𝑛 - 1$ in the forcing to add $𝜔$ many Cohen reals. We consider the dense set $𝐷 𝑛$ . We may factor $A d d (𝜔, 𝜔)$ as $A d d (𝜔, 𝑛) \times A d d (𝜔, [𝑛, 𝜔))$ . Since $𝑑 0 \times \dots \times 𝑑 𝑛 - 1$ is actually $𝑀$ -generic (since these are finite modifications of the corresponding $𝑏 𝑘$ ’s, which are mutually $𝑀$ -generic, it follows that there is some finite extension of our condition $𝑝 𝑛 - 1$ to a condition $𝑝 𝑛 \in 𝐷 𝑛$ , which is compatible with $𝑑 0 \times \dots \times 𝑑 𝑛 - 1$ . Let $𝑑 𝑛$ be the same as $𝑏 𝑛$ , except finitely modified to be compatible with $𝑝 𝑛$ . In this way, our final real $\oplus 𝑛 𝑑 𝑛$ will contain all the conditions $𝑝 𝑛$ , and therefore be $𝑀$ -generic for $A d d (𝜔, 𝜔)$ , yet every $𝑏 𝑛$ will differ only finitely from $𝑑 𝑛$ and hence be an element of $𝑀 [𝑑]$ . So we have $𝑀 [𝑏 0] \dots [𝑏 𝑛] \subset 𝑀 [𝑑]$ , and we have found our upper bound. QED

Notice that the real $𝑑$ we construct is not only $𝑀$ -generic, but also $𝑀 [𝑐 𝑛]$ -generic for every $𝑛$ .

My related post, Upward closure in the toy multiverse of all countable models of set theory, which is based on material in my paper Set-theoretic geology, discusses some similar results.

Open determinacy for class games

Posted on September 4, 2015 by Joel David Hamkins

[bibtex key=GitmanHamkins2016:OpenDeterminacyForClassGames]

Abstract. The principle of open determinacy for class games — two-player games of perfect information with plays of length $𝜔$ , where the moves are chosen from a possibly proper class, such as games on the ordinals — is not provable in Zermelo-Fraenkel set theory ZFC or Godel-Bernays set theory GBC, if these theories are consistent, because provably in ZFC there is a definable open proper class game with no definable winning strategy. In fact, the principle of open determinacy and even merely clopen determinacy for class games implies Con(ZFC) and iterated instances Con(Con(ZFC)) and more, because it implies that there is a satisfaction class for first-order truth, and indeed a transfinite tower of truth predicates $T r 𝛼$ for iterated truth-about-truth, relative to any class parameter. This is perhaps explained, in light of the Tarskian recursive definition of truth, by the more general fact that the principle of clopen determinacy is exactly equivalent over GBC to the principle of transfinite recursion over well-founded class relations. Meanwhile, the principle of open determinacy for class games is provable in the stronger theory GBC $+ Π 11$ -comprehension, a proper fragment of Kelley-Morse set theory KM.

See my earlier posts on part of this material:

Upward closure in the generic multiverse of a countable model of set theory, RIMS 2015, Kyoto, Japan

Posted on August 15, 2015 by Joel David Hamkins

This will be a talk for the conference Recent Developments in Axiomatic Set Theory at the Research Institute for Mathematical Sciences (RIMS) in Kyoto, Japan, September 16-18, 2015.

Abstract. Consider a countable model of set theory amongst its forcing extensions, the ground models of those extensions, the extensions of those models and so on, closing under the operations of forcing extension and ground model. This collection is known as the generic multiverse of the original model. I shall present a number of upward-oriented closure results in this context. For example, for a long-known negative result, it is a fun exercise to construct forcing extensions $𝑀 [𝑐]$ and $𝑀 [𝑑]$ of a given countable model of set theory $𝑀$ , each by adding an $𝑀$ -generic Cohen real, which cannot be amalgamated, in the sense that there is no common extension model $𝑁$ that contains both $𝑀 [𝑐]$ and $𝑀 [𝑑]$ and has the same ordinals as $𝑀$ . On the positive side, however, any increasing sequence of extensions $𝑀 [𝐺 0] \subset 𝑀 [𝐺 1] \subset 𝑀 [𝐺 2] \subset \dots$ , by forcing of uniformly bounded size in $𝑀$ , has an upper bound in a single forcing extension $𝑀 [𝐺]$ . (Note that one cannot generally have the sequence $⟨ 𝐺 𝑛 ∣ 𝑛 < 𝜔 ⟩$ in $𝑀 [𝐺]$ , so a naive approach to this will fail.) I shall discuss these and related results, many of which appear in the “brief upward glance” section of my recent paper: G. Fuchs, J. D. Hamkins and J. Reitz, Set-theoretic geology.

Set-theoretic geology | Upward closure in the toy multiverse | RIMS set theory workshop | Program

Universality and embeddability amongst the models of set theory, CTFM 2015, Tokyo, Japan

Posted on August 15, 2015 by Joel David Hamkins

This will be a talk for the Computability Theory and Foundations of Mathematics conference at the Tokyo Institute of Technology, September 7-11, 2015. The conference is held in celebration of Professor Kazuyuki Tanaka’s 60th birthday.

Abstract. Recent results on the embeddability phenomenon and universality amongst the models of set theory are an appealing blend of ideas from set theory, model theory and computability theory. Central questions remain open.

A surprisingly vigorous embeddability phenomenon has recently been uncovered amongst the countable models of set theory. It turns out, for instance, that among these models embeddability is linear: for any two countable models of set theory, one of them embeds into the other. Indeed, one countable model of set theory $𝑀$ embeds into another $𝑁$ just in case the ordinals of $𝑀$ order-embed into the ordinals of $𝑁$ . This leads to many surprising instances of embeddability: every forcing extension of a countable model of set theory, for example, embeds into its ground model, and every countable model of set theory, including every well-founded model, embeds into its own constructible universe.

Although the embedding concept here is the usual model-theoretic embedding concept for relational structures, namely, a map $𝑗 : 𝑀 \to 𝑁$ for which $𝑥 \in 𝑀 𝑦$ if and only if $𝑗 (𝑥) \in 𝑁 𝑗 (𝑦)$ , it is a weaker embedding concept than is usually considered in set theory, where embeddings are often elementary and typically at least $Δ 0$ -elementary. Indeed, the embeddability result is surprising precisely because we can easily prove that in many of these instances, there can be no $Δ 0$ -elementary embedding.

The proof of the embedding theorem makes use of universality ideas in digraph combinatorics, including an acyclic version of the countable random digraph, the countable random $ℚ$ -graded digraph, and higher analogues arising as uncountable Fraïssé limits, leading to the hypnagogic digraph, a universal homogeneous graded acyclic class digraph, closely connected with the surreal numbers. Thus, the methods are a blend of ideas from set theory, model theory and computability theory.

Results from Incomparable $𝜔 1$ -like models of set theory show that the embedding phenomenon does not generally extend to uncountable models. Current joint work of myself, Aspero, Hayut, Magidor and Woodin is concerned with questions on the extent to which the embeddings arising in the embedding theorem can exist as classes inside the models in question. Since the embeddings of the theorem are constructed externally to the model, by means of a back-and-forth-style construction, there is little reason to expect, for example, that the resulting embedding $𝑗 : 𝑀 \to 𝐿 𝑀$ should be a class in $𝑀$ . Yet, it has not yet known how to refute in ZFC the existence of a class embedding $𝑗 : 𝑉 \to 𝐿$ when $𝑉 \neq 𝐿$ . However, many partial results are known. For example, if the GCH fails at an uncountable cardinal, if $0 ♯$ exists, or if the universe is a nontrivial forcing extension of some ground model, then there is no embedding $𝑗 : 𝑉 \to 𝐿$ . Meanwhile, it is consistent that there are non-constructible reals, yet $⟨ 𝑃 (𝜔), \in ⟩$ embeds into $⟨ 𝑃 (𝜔) 𝐿, \in ⟩$ .

CFTM 2015 extended abstract | Article | CFTM | Slides

Every ordinal has only finitely many order-types for its final segments

Posted on August 1, 2015 by Joel David Hamkins

droste_effect_clock_by_kiluncle-d4ya2gn I was recently asked an interesting elementary question about the number of possible order types of the final segments of an ordinal, and in particular, whether there could be an ordinal realizing infinitely many different such order types as final segments. Since I found it interesting, let me write here how I replied.

The person asking me had noted that every nonempty final segment of the first infinite ordinal $𝜔$ is isomorphic to $𝜔$ again, since if you start counting from $5$ or from a million, you have just as far to go in the natural numbers. Thus, if one includes the empty final segment, there are precisely two order-types that arise as final segments of $𝜔$ , namely, $0$ and $𝜔$ itself. A finite ordinal $𝑛$ , in contrast, has precisely $𝑛 + 1$ many final segments, corresponding to each of the possible cuts between any of the elements or before all of them or after all of them, and these final segments, considered as orders themselves, all have different sizes and hence are not isomorphic.

He wanted to know whether an ordinal could have infinitely many different order-types for its tails.

Question. Is there an ordinal having infinitely many different isomorphism types for its final segments?

The answer is no, and I’d like to explain why. I’ll discuss two different arguments, the first being an easy direct argument aimed only at this answer, and the second being a more careful analysis aimed at understanding exactly how many and which order-types arise as the order type of a final segment of an ordinal $𝛼$ .

Theorem. Every ordinal has only finitely many order types of its final segments.

Proof: Suppose that $𝛼$ is an ordinal, and consider the order types of the final segments $[𝜂, 𝛼)$ , for $𝜂 \leq 𝛼$ . Note that as $𝜂$ increases, the final segment $[𝜂, 𝛼)$ becomes smaller as a suborder, and so it’s order type does not go up. And since these are well-orders, it can go down only finitely many times. So only finitely many order types arise, and the theorem is proved. QED

But let’s figure out exactly how many and which order types arise.

Theorem. The number of order types of final segments of an ordinal $𝛼$ is precisely $𝑛 + 1$ , where $𝑛$ is the number of terms in the Cantor normal form of $𝛼$ , and one can describe those order types in terms of the normal form of $𝛼$ .

Cantor proved that every ordinal $𝛼$ can be uniquely expressed as a finite sum $𝛼 = 𝜔 𝛽 𝑛 + \dots + 𝜔 𝛽 0,$ where $𝛽 𝑛 \geq \dots \geq 𝛽 0$ , and this is called the Cantor normal form of the ordinal. There are alternative forms, where one allows terms like $𝜔 𝛽 \cdot 𝑛$ for finite $𝑛$ , but in my favored formulation, one simply expands this into $𝑛$ terms with $𝜔 𝛽 + \dots + 𝜔 𝛽$ . In particular, the ordinal $𝜔 = 𝜔 1$ has exactly one term in its Cantor normal form, and a finite number $𝑛 = 𝜔 0 + \dots + 𝜔 0$ has exactly $𝑛$ terms in its Cantor normal form. So the statement of the theorem agrees with the calculations that we had made at the very beginning.

Proof: First, let’s observe that every nonempty final segment of an ordinal of the form $𝜔 𝛽$ is isomorphic to $𝜔 𝛽$ again. This amounts to the fact that ordinals of the form $𝜔 𝛽$ are additively indecomposable, or in other words, closed under ordinal addition, since the final segments of an ordinal $𝛼$ are precisely the ordinals $𝜁$ such that $𝛼 = 𝜉 + 𝜁$ for some $𝜉$ . If $𝛼$ is additively indecomposable, then it cannot be that $𝜁 < 𝛼$ , and so all final segments would be isomorphic to $𝛼$ . So let’s prove that $𝜔 𝛽$ is additively indecomposable. This is clear if $𝛽 = 0$ , since the only ordinal less than $𝜔 0 = 1$ is $0$ and $0 + 0 < 1$ . If $𝛽$ is a limit ordinal, then the ordinals $𝜔 𝜂$ for $𝜂 < 𝛽$ are unbounded in $𝜔 𝛽$ , and adding them stays below because $𝜔 𝜂 + 𝜔 𝜂 = 𝜔 𝜂 \cdot 2 \leq 𝜔 𝜂 \cdot 𝜔 = 𝜔 𝜂 + 1 < 𝜔 𝛽$ . If $𝛽 = 𝛿 + 1$ is a successor ordinal, then $𝜔 𝛽 = 𝜔 𝛿 + 1 = 𝜔 𝛿 \cdot 𝜔 = s u p 𝑛 < 𝜔 𝜔 𝛿 \cdot 𝑛$ , but again adding them stays below because $𝜔 𝛿 \cdot 𝑛 + 𝜔 𝛿 \cdot 𝑚 = 𝜔 𝛿 \cdot (𝑛 + 𝑚) < 𝜔 𝛿 \cdot 𝜔 = 𝜔 𝛽$ .

To prove the theorem, consider any ordinal $𝛼$ with Cantor normal form $𝛼 = 𝜔 𝛽 𝑛 + \dots + 𝜔 𝛽 0$ , where $𝛽 𝑛 \geq \dots \geq 𝛽 0$ . So as an order type, $𝛼$ consists of finitely many pieces, the first of type $𝜔 𝛽 𝑛$ , the next of type $𝜔 𝛽 𝑛 - 1$ and so on up to $𝜔 𝛽 0$ . Any final segment of $𝛼$ therefore consists of a final segment of one of these segments, together with all the segments after that segment (and omitting any segments prior to it, if any). But since these segments all have the form $𝜔 𝛽 𝑖$ , they are additively indecomposable and therefore are isomorphic to all their nonempty final segments. So any final segment of $𝛼$ is order-isomorphic to an ordinal whose Cantor normal form simply omits some (or none) of the terms from the front of the Cantor normal form of $𝛼$ . Since we may start with any of the $𝑛$ terms (or none), this gives precisely $𝑛 + 1$ many order types of the final segments of $𝛼$ , as claimed.

The argument shows, furthermore, that the possible order types of the final segments of $𝛼$ , where $𝛼 = 𝜔 𝛽 𝑛 + \dots + 𝜔 𝛽 0$ , are precisely the ordinals of the form $𝜔 𝛽 𝑘 + \dots + 𝜔 𝛽 0$ , omitting terms only from the front, where $𝑘 \leq 𝑛$ . QED

The axiom of determinacy for small sets

Posted on July 3, 2015 by Joel David Hamkins

I should like to argue that the axiom of determinacy is true for all games having a small payoff set. In particular, the size of the smallest non-determined set, in the sense of the axiom of determinacy, is the continuum; every set of size less than the continuum is determined, even when the continuum is enormous.

We consider two-player games of perfect information. Two players, taking turns, play moves from a fixed space $𝑋$ of possible moves, and thereby together build a particular play or instance of the game $⃗ 𝑎 = ⟨ 𝑎 0, 𝑎 1, \dots ⟩ \in 𝑋 𝜔$ . The winner of this instance of the game is determined according to whether the play $⃗ 𝑎$ is a member of some fixed payoff set $𝑈 \subset 𝑋 𝜔$ specifying the winning condition for this game. Namely, the first player wins in the case $⃗ 𝑎 \in 𝑈$ .

A strategy in such a game is a function $𝜎 : 𝑋 < 𝜔 \to 𝑋$ that instructs a particular player how to move next, given the sequence of partial play, and such a strategy is a winning strategy for that player, if all plays made against it are winning for that player. (The first player applies the strategy $𝜎$ only on even-length input, and the second player only to the odd-length inputs.) The game is determined, if one of the players has a winning strategy.

It is not difficult to see that if $𝑈$ is countable, then the game is determined. To see this, note first that if the space of moves $𝑋$ has at most one element, then the game is trivial and hence determined; and so we may assume that $𝑋$ has at least two elements. If the payoff set $𝑈$ is countable, then we may enumerate it as $𝑈 = {𝑠 0, 𝑠 1, \dots}$ . Let the opposing player now adopt the strategy of ensuring on the $𝑛 𝑡 ℎ$ move that the resulting play is different from $𝑠 𝑛$ . In this way, the opposing player will ensure that the play is not in $𝑈$ , and therefore win. So every game with a countable payoff set is determined.

Meanwhile, using the axiom of choice, we may construct a non-determined set even for the case $𝑋 = {0, 1}$ , as follows. Since a strategy is function from finite binary sequences to ${0, 1}$ , there are only continuum many strategies. By the axiom of choice, we may well-order the strategies in order type continuum. Let us define a payoff set $𝑈$ by a transfinite recursive procedure: at each stage, we will have made fewer than continuum many promises about membership and non-membership in $𝑈$ ; we consider the next strategy on the list; since there are continuum many plays that accord with that strategy for each particular player, we may make two additional promises about $𝑈$ by placing one of these plays into $𝑈$ and one out of $𝑈$ in such a way that this strategy is defeated as a winning strategy for either player. The result of the recursion is a non-determined set of size continuum.

So what is the size of the smallest non-determined set? For a lower bound, we argued above that every countable payoff set is determined, and so the smallest non-determined set must be uncountable, of size at least $ℵ 1$ . For an upper bound, we constructed a non-determined set of size continuum. Thus, if the continuum hypothesis holds, then the smallest non-determined set has size exactly continuum, which is $ℵ 1$ in this case. But what if the continuum hypothesis fails? I claim, nevertheless, that the smallest non-determined set still has size continuum.

Theorem. Every game whose winning condition is a set of size less than the continuum is determined.

Proof. Suppose that $𝑈 \subset 𝑋 𝜔$ is the payoff set of the game under consideration, so that $𝑈$ has size less than continuum. If $𝑋$ has at most one element, then the game is trivial and hence determined. So we may assume that $𝑋$ has at least two elements. Let us partition the elements of $𝑋 𝜔$ according to whether they have exactly the same plays for the second player. So there are at least continuum many classes in this partition. If $𝑈$ has size less than continuum, therefore, it must be disjoint from at least one (and in fact from most) of the classes of this partition (since otherwise we would have an injection from the continuum into $𝑈$ ). So there is a fixed sequence of moves for the second player, such that any instance of the game in which the second player makes those moves, the result is not in $𝑈$ and hence is a win for the second player. This is a winning strategy for the second player, and so the game is determined. QED

This proof generalizes the conclusion of the diagonalization argument against a countable payoff set, by showing that for any winning condition set of size less than continuum, there is a fixed play for the opponent (not depending on the play of the first player) that defeats it.

The proof of the theorem uses the axiom of choice in the step where we deduce that $𝑈$ must be disjoint from a piece of the partition, since there are continuum many such pieces and $𝑈$ had size less than the continuum. Without the axiom of choice, this conclusion does not follow. Nevertheless, what the proof does show without AC is that every set that does not surject onto $ℝ$ is determined, since if $𝑈$ contained an element from every piece of the partition it would surject onto $ℝ$ . Without AC, the assumption that $𝑈$ does not surject onto $ℝ$ is stronger than the assumption merely that it has size less the continuum, although these properties are equivalent in ZFC. Meanwhile, these issues are relevant in light of the model suggested by Asaf Karagila in the comments below, which shows that it is consistent with ZF without the axiom of choice that there are small non-determined sets. Namely, the result of Monro shows that it is consistent with ZF that $ℝ = 𝐴 ⊔ 𝐵$ , where both $𝐴$ and $𝐵$ have cardinality less than the continuum. In particular, in this model the continuum injects into neither $𝐴$ nor $𝐵$ , and consequently neither player can have a strategy to force the play into their side of this partition. Thus, both $𝐴$ and $𝐵$ are non-determined, even though they have size less than the continuum.

Determinacy for proper-class clopen games is equivalent to transfinite recursion along proper-class well-founded relations

Posted on July 2, 2015 by Joel David Hamkins

I’d like to continue a bit further my exploration of some principles of determinacy for proper-class games; it turns out that these principles have a surprising set-theoretic strength. A few weeks ago, I explained that the determinacy of proper-class open games and even clopen games implies Con(ZFC) and much more. Today, I’d like to prove that clopen determinacy is exactly equivalent over GBC to the principle of transfinite recursion along proper-class well-founded relations. Thus, GBC plus either of these principles is a strictly intermediate set theory between GBC and KM.

The principle of clopen determinacy for class games is the assertion that in any two-player infinite game of perfect information whose winning condition is a clopen class, there is a winning strategy for one of the players. Players alternately play moves in a playing space $𝑋$ , thereby creating a particular play $⃗ 𝑎 \in 𝑋 𝜔$ , and the winner is determined according to whether $⃗ 𝑎$ is in a certain fixed payoff class $𝑈 \subset 𝑋 𝜔$ or not. One has an open game when this winning condition class $𝑈$ is open in the product topology (using the discrete topology on $𝑋$ ). A game is open for a player if and only if every winning play for that player has an initial segment, all of whose extensions are also winning for that player. So the game is won for an open player at a finite stage of play. A clopen game, in contrast, has a payoff set that is open for both players. Clopen games can be equivalently cast in terms of the game tree, consisting of positions in the game where the winner is not yet determined, and where play terminates when the winner is known. Namely, a game is clopen exactly when this game tree is well-founded, so that in every play, the outcome is known already at a finite stage.

A strategy is a class function $𝜎 : 𝑋 < 𝜔 \to 𝑋$ that instructs the player what to play next, given a position of partial play, and the strategy is winning for a player if all plays that accord with it satisfy the winning condition for that player.

The principle of transfinite recursion along well-founded class relations is the assertion that we may undertake recursive definitions along any class well-founded partial order relation. That is, suppose that $⊲$ is a class well-founded partial order relation on a class $𝐴$ , and suppose that $𝜑 (𝐹, 𝑎, 𝑦)$ is a formula, using only first-order quantifiers but having a class variable $𝐹$ , which is functional in the sense that for any class $𝐹$ and any set $𝑎 \in 𝐴$ there is a unique $𝑦$ such that $𝜑 (𝐹, 𝑎, 𝑦)$ . The idea is that $𝜑 (𝐹, 𝑎, 𝑦)$ expresses the recursive rule to be iterated, and a solution of the recursion is a class function $𝐹$ such that $𝜑 (𝐹 ↾ 𝑎, 𝑎, 𝐹 (𝑎))$ holds for every $𝑎 \in 𝐴$ , where $𝐹 ↾ 𝑎$ means the restriction of $𝐹$ to the class ${𝑏 \in 𝐴 ∣ 𝑏 ⊲ 𝑎}$ . Thus, the value $𝐹 (𝑎)$ is determined by the class of previous values $𝐹 (𝑏)$ for $𝑏 ⊲ 𝑎$ . The principle of transfinite recursion along class well-founded relations is the assertion scheme that for every such well-founded partial order class $⟨ 𝐴, ⊲ ⟩$ and any recursive rule $𝜑$ as above, there is a solution.

In the case that the relation $⊲$ is set-like, which means that the predecessors ${𝑏 ∣ 𝑏 ⊲ 𝑎}$ of any point $𝑎$ form a set (rather than a proper class), then GBC easily proves that there is a unique solution class, which furthermore is definable from $⊲$ . Namely, one can show that every $𝑎 \in 𝐴$ has a partial solution that obeys the recursive rule at least up to $𝑎$ , and furthermore all such partial solutions agree below $𝑎$ , because there can be no $⊲$ -minimal violation of this. It follows that the class function $𝐹$ unifying these partial solutions is a total solution to the recursion. Similarly, GBC can prove that there are solutions to other transfinite recursion instances where the well-founded relation is not necessarily set-like, such as a recursion of length $O r d + O r d$ or even much longer.

Meanwhile, if GBC is consistent, then it cannot in general prove that transfinite recursions along non-set-like well-founded relations always succeed, since this principle would imply that there is a truth-predicate for first-order truth, as the Tarskian conditions are precisely such a recursion on a well-founded relation based on the complexity of formulas. (That relation is not set-like, since when considering the truth of $\exists 𝑥 𝜓 (𝑥, ⃗ 𝑎)$ , we want to consider the truth of $𝜓 (𝑏, ⃗ 𝑎)$ for any parameter $𝑏$ , and there are a proper class of such $𝑏$ .) Thus, GBC plus transfinite recursion (or plus clopen determinacy) is strictly stronger than GBC, although it is provable in Kelley-Morse set theory KM essentially the same as GBC proves the set-like special case.

Theorem. Assume GBC. Then the following are equivalent.

Clopen determinacy for class games. That is, for any two-player game of perfect information whose payoff class is both open and closed, there is a winning strategy for one of the players.
Transfinite recursion for proper class well-founded relations (not necessarily set-like).

Proof. ( $2 \to 1$ ) Assume the principle of transfinite recursion for proper class well-founded relations, and suppose we are faced with a clopen game. Consider the game tree $𝑇$ , consisting of positions arising during play, up to the moment that a winner is known. This tree is well-founded because the game is clopen. Let us label the terminal nodes of the tree with I or II according to who has won the game in that position, and more generally, let us label all the nodes of the tree with I or II according to the following transfinite recursion: if a node has I to play, then it will have label I if there is a move to a node labeled I, and otherwise II; and similarly when it is II to play. By the principle of transfinite recursion, there is a labeling of the entire tree that accords with this recursive rule. It is now easy to see that if the initial node is labeled with I, then player I has a winning strategy, which is simply to stay on the nodes labeled I. Note that player II cannot play in one move from a node labeled I to one labeled II. Similarly, if the initial node is labeled II, then player II has a winning strategy; and so the game is determined, as desired.

( $1 \to 2$ ) Conversely, let us assume the principle of clopen determinacy for class games. Suppose we are faced with a recursion along a class relation $⊲$ on a class $𝐴$ , using a recursion rule $𝜑 (𝐹, 𝑎, 𝑦)$ . We shall define a certain clopen game, and prove that any winning strategy for this game will produce a solution for the recursion.

It will be convenient to assume that $𝜑 (𝐹, 𝑎, 𝑦)$ is strongly functional, meaning that not only does it define a function as we have mentioned in $𝑉$ , but also that $𝜑 (𝐹, 𝑎, 𝑦)$ defines a function $(𝐹, 𝑎) \mapsto 𝑦$ when used over any model $⟨ 𝑉 𝜃, \in, 𝐹 ⟩$ for any class $𝐹 \subset 𝑉 𝜃$ . The strongly functional property can be achieved simply by replacing the formula with the assertion that $𝜑 (𝐹, 𝑎, 𝑦)$ , if $𝑦$ is unique such that this holds, and otherwise $𝑦 = \emptyset$ .

At first, let us consider a slightly easier game, which will be open rather than clopen; a bit later, we shall revise this game to a clopen game. The game is the recursion game, which will be very much like the truth-telling game of my previous post, Open determinacy for proper class games implies Con(ZFC) and much more. Namely, we have two players, the challenger and the truth-teller. The challenger will issues challenges about truth in a structure $⟨ 𝑉, \in, ⊲, 𝐹 ⟩$ , where $⊲$ is the well-founded class relation and $𝐹$ is a class function, not yet specified. Specifically, the challenger is allowed to ask about the truth of any formula $𝜑 (⃗ 𝑎)$ in this structure, and to inquire as to the value of $𝐹 (𝑎)$ for any particular $𝑎$ . The truth-teller, as before, will answer the challenges by saying either that $𝜑 (⃗ 𝑎)$ is true or false, and in the case $𝜑 (⃗ 𝑎) = \exists 𝑥 𝜓 (𝑥, ⃗ 𝑎)$ and the formula was declared true, by also giving a witness $𝑏$ and declaring $𝜓 (𝑏, ⃗ 𝑎)$ is true; and the truth-teller must specify a specific value for $𝐹 (𝑎)$ for any particular $𝑎$ . The truth-teller loses immediately, if she should ever violate Tarski’s recursive definition of truth; and she also loses unless she declares the recursive rules $𝜑 (𝐹 ↾ 𝑎, 𝑎, 𝐹 (𝑎))$ to be true. Since these violations occur at a finite stage of play if they do at all, the game is open for the challenger.

Lemma. The challenger has no winning strategy in the recursion game.

Proof. Suppose that $𝜎$ is a strategy for the challenger. So $𝜎$ is a class function that instructs the challenger how to play next, given a position of partial play. By the reflection theorem, there is an ordinal $𝜃$ such that $𝑉 𝜃$ is closed under $𝜎$ , and using the satisfaction class that comes from clopen determinacy, we may actually also arrange that $⟨ 𝑉 𝜃, \in, ⊲ \cap 𝑉 𝜃, 𝜎 \cap 𝑉 𝜃 ⟩ ≺ ⟨ 𝑉, \in, ⊲, 𝜎 ⟩$ . Consider the relation $⊲ \cap 𝑉 𝜃$ , which is a well-founded relation on $𝐴 \cap 𝑉 𝜃$ . The important point is that this relation is now a set, and in GBC we may certainly undertake transfinite recursions along well-founded set relations. Thus, there is a function $𝑓 : 𝐴 \cap 𝑉 𝜃 \to 𝑉 𝜃$ such that $⟨ 𝑉 𝜃, \in, 𝑓 ⟩$ satisfies $𝜑 (𝑓 ↾ 𝑎, 𝑎, 𝑓 (𝑎))$ for all $𝑎 \in 𝑉 𝜃$ , where $𝑓 ↾ 𝑎$ means restricting $𝑓$ to the predecessors of $𝑎$ in $𝑉 𝜃$ , and this may not be all the predecessors of $𝑎$ with respect to $⊲$ , which may not be set-like. Note that this is the place where we use our assumption that $𝜑$ was strongly functional, since we want to ensure that it can still be used to define a valid recursion over $⊲ \cap 𝑉 𝜃$ . (We are not claiming that $⟨ 𝑉 𝜃, \in, ⊲ \cap 𝑉 𝜃, 𝑓 ⟩$ models $Z F C (⊲, 𝑓)$ .)

Consider now the play of the recursion game in $𝑉$ , where the challenger uses the strategy $𝜎$ and the truth-teller plays in accordance with $⟨ 𝑉 𝜃, \in, ⊲ \cap 𝑉 𝜃, 𝑓 ⟩$ . Since $𝑉 𝜃$ was closed under $𝜎$ , the challenger will never issue challenges outside of $𝑉 𝜃$ . And since the function $𝑓$ fulfills the recursion $𝜑 (𝑓 ↾ 𝑎, 𝑎, 𝑓 (𝑎))$ in this structure, the truth-teller will not be trapped in any violation of the Tarski conditions or the recursion condition. Thus, the truth-teller will win this instance of the game, and so $𝜎$ was not a winning strategy for the challenger, as desired. QED

Lemma. The truth-teller has a winning strategy in the recursion game if and only if there is a solution of the recursion.

Proof. If there is a solution $𝐹$ of the recursion, then by clopen determinacy, we also get a satisfaction class for the structure $⟨ 𝑉, \in, ⊲, 𝐹 ⟩$ , and the truth-teller can answer all queries of the challenger by referring to what is actually true in this structure. This will be winning for the truth-teller, since the actual truth obeys the Tarskian conditions and the recursive rule.

Conversely, suppose that $𝜏$ is a winning strategy for the truth-teller in the recursion game. We claim that the truth assertions made by $𝜏$ do not depend on the order in which challenges are made by the challenger; they all cohere with one another. This is easy to see for formulas not involving $𝐹$ by induction on formulas, for if the truth of a formula $𝜓 (⃗ 𝑎)$ is independent of play, then also the truth of $\neg 𝜓 (⃗ 𝑎)$ is as well, and similarly if $\exists 𝑥 𝜓 (𝑥, ⃗ 𝑎)$ is declared true with witness $𝜓 (𝑏, ⃗ 𝑎)$ , then by induction $𝜓 (𝑏, ⃗ 𝑎)$ is independent of the play, in which case $\exists 𝑥 𝜓 (𝑥, ⃗ 𝑎)$ must always be declared true by $𝜏$ independently of the order of play by the challenger (although the particular witness $𝑏$ provided by $𝜏$ may depend on the play). Now, let us also argue that the values of $𝐹 (𝑎)$ declared by $𝜏$ are also independent of the order of play. If not, there is some $⊲$ -least $𝑎$ where this fails. (Note that such an $𝑎$ exists, since $𝜏$ is a class, and we can define from $𝜏$ the class of $𝑎$ for which the value of $𝐹 (𝑎)$ declared by $𝜏$ depends on the order of play; without $𝜏$ , one might have expected to need $Π 11$ -comprehension to find a minimal $𝑎$ where the recursion fails.) As in the truth-telling game, the truth assertions made by $𝜏$ about $⟨ 𝑉, \in, ⊲, 𝐹 ↾ 𝑎 ⟩$ , where $𝐹 ↾ 𝑎$ is the class function of values that are determined by $𝜏$ on $𝑏 ⊲ 𝑎$ , must not depend on the order of play. Since the recursion rule $𝜑 (𝐹 ↾ 𝑎, 𝑎, 𝑦)$ is functional, there is only one value $𝑦 = 𝐹 (𝑎)$ for which this formula can be truthfully held, and so if some play causes $𝜏$ to play a different value for $𝐹 (𝑎)$ , the challenger can in finitely many additional moves (bounded by the syntactic complexity of $𝜑$ ) trap the truth-teller in a violation of the Tarskian conditions or the recursion condition. Thus, the values of $𝐹 (𝑎)$ declared by $𝜏$ must in fact all cohere independently of the order of play, and so $𝜏$ is describing a class function $𝐹 : 𝐴 \to 𝑉$ such that $𝜑 (𝐹 ↾ 𝑎, 𝑎, 𝐹 (𝑎))$ is true for every $𝑎 \in 𝐴$ . So the recursion has a solution, as desired. QED

So far, we have established that the principle of open determinacy implies the principle of transfinite recursion along well-founded class relations. In order to improve this implication to use only clopen determinacy rather than open determinacy, we modify the game to become a clopen game rather than an open game.

Consider the clopen form of the recursion game, where we insist also that the challenger announce on the first move a natural number $𝑛$ , such that the challenger loses if the truth-teller survives for at least $𝑛$ moves. This is now a clopen game, since the winner will be known by that time, either because the truth-teller will violate the Tarski conditions or the recursion condition, or else the challenger’s limit on play will expire.

Since the modified version of the game is even harder for the challenger, there can still be no winning strategy for the challenger. So by the principle of clopen determinacy, there is a winning strategy $𝜏$ for the truth-teller. This strategy is allowed to make decisions based on the number $𝑛$ announced by the challenger on the first move, and it will no longer necessarily be the case that the theory declared true by $𝜏$ will be independent of the order of play. Nevertheless, it will be the case, we claim, that the theory declared true by $𝜏$ for all plays with sufficiently large $𝑛$ (and with sufficiently many remaining moves) will be independent of the order of play. One can see this by observing that if an assertion $𝜓 (⃗ 𝑎)$ is independent in this sense, then also $\neg 𝜓 (⃗ 𝑎)$ will be independent in this sense, for otherwise there would be plays with large $𝑛$ giving different answers for $\neg 𝜓 (⃗ 𝑎)$ and we could then challenge with $𝜓 (⃗ 𝑎)$ , which would have to give different answers or else $𝜏$ would not win. Similarly, since $𝜏$ is winning, one can see that allowing the challenger to specify a bound on the total length of play does not prevent the arguments above showing that $𝜏$ describes a coherent solution function $𝐹 : 𝐴 \to 𝑉$ satisfying the recursion $𝜑 (𝐹 ↾ 𝑎, 𝑎, 𝐹 (𝑎))$ , provided that one looks only at plays in which there are sufficiently many moves remaining. There cannot be a $⊲$ -least $𝑎$ where the value of $𝐹 (𝑎)$ is not determined in this sense, and so on as before.

Thus, we have proved that the principle of clopen determinacy for class games is equivalent to the principle of transfinite recursion along well-founded class relations. QED

The material in this post will become part of a joint project with Victoria Gitman and Thomas Johnstone. We are currently investigating several further related issues.

Open determinacy for proper class games implies Con(ZFC) and much more

Posted on June 17, 2015 by Joel David Hamkins

One of the intriguing lessons we have learned in the past half-century of set-theoretic developments is that there is a surprisingly robust connection between infinitary game theory and fundamental set-theoretic principles, including large cardinals. Assertions about the existence of strategies in infinite games often turn out to have an unexpected set-theoretic power. In this post, I should like to give another specific example of this, which Thomas Johnstone and I hit upon yesterday in an enjoyable day of mathematics.

Specifically, I’d like to prove that if we generalize the open-game concept from sets to classes, then assuming consistency, ZFC cannot prove that every definable open class game is determined, and indeed, over Gödel-Bernays set theory GBC the principle of open determinacy (and even just clopen determinacy) implies Con(ZFC) and much more.

To review a little, we are talking about games of perfect information, where two players alternately play elements from an allowed space $𝑋$ of possible moves, and together they build an infinite sequence $⃗ 𝑥 = ⟨ 𝑥 0, 𝑥 1, 𝑥 2, \dots ⟩$ in $𝑋 𝜔$ , which is the resulting play of this particular instance of the game. We have a fixed collection of plays $𝐴 \subset 𝑋 𝜔$ that is used to determine the winner, namely, the first player wins this particular instance of the game if the resulting play $⃗ 𝑥$ is in $𝐴$ , and otherwise the second player wins. A strategy for a player is a function $𝜎 : 𝑋 < 𝜔 \to 𝑋$ , which tells a player how to move next, given a finite position in the game. Such a strategy is winning for that player, if he or she always wins by following the instructions, no matter how the opponent plays. The game is determined, if one of the players has a winning strategy.

It is a remarkable but elementary fact that if the winning condition $𝐴$ is an open set, then the game is determined. One can prove this by using the theory of ordinal game values, and my article on transfinite game values in infinite chess contains an accessible introduction to the theory of game values. Basically, one defines that a position has game value zero (for player I, say), if the game has already been won at that stage, in the sense that every extension of that position is in the winning payoff set $𝐴$ . A position with player I to play has value $𝛼 + 1$ , if player I can move to a position with value $𝛼$ , and $𝛼$ is minimal. The value of a position with player II to play is the supremum of the values of all the positions that he or she might reach in one move, provided that those positions have a value. The point now is that if a position has a value, then player I can play so as strictly to decrease the value, and player II cannot play so as to increase it. So if a position has a value, then player I has a winning strategy, which is the value-reducing strategy. Conversely, if a position does not have a value, then player II can maintain that fact, and player I cannot play so as to give it a value; thus, in this case player II has a winning strategy, the value-maintaining strategy. Thus, we have proved the Gale-Stewart theorem: every open game is determined.

That proof relied on the space of moves $𝑋$ being a set, since we took a supremum over the values of the possible moves, and if $𝑋$ were a proper class, we couldn’t be sure to stay within the class of ordinals and the recursive procedure might break down. What I’d like to do is to consider more seriously the case where $𝑋$ is a proper class. Similarly, we allow the payoff collection $𝐴$ to be a proper class, and the strategies $𝜎 : 𝑋 < 𝜔 \to 𝑋$ are also proper classes. Can we still prove the Gale-Steward theorem for proper classes? The answer is no, unless we add set-theoretic strength. Indeed, even clopen determinacy has set-theoretic strength.

Theorem. (GBC) Clopen determinacy for proper classes implies Con(ZFC) and much more. Specifically, there is a clopen game, such the existence of a winning strategy is equivalent to the existence of a satisfaction class for first-order truth.

Proof. Let me first describe a certain open game, the truth-telling game, with those features, and I shall later modify it to a clopen game. The truth-telling game will have two players, which I call the challenger and the truth-teller. At any point in the game, the challenger plays by making an inquiry about a particular set-theoretic formula $𝜑 (⃗ 𝑎)$ with parameters. The truth-teller must reply to the inquiry by stating either true or false, and in the case that the formula $𝜑$ is an existential assertion $\exists 𝑥 𝜓 (𝑥, ⃗ 𝑎)$ declared to be true, then the truth teller must additionally identify a particular witness $𝑏$ and assert that $𝜓 (𝑏, ⃗ 𝑎)$ is true. So a play of the game consists of a sequence of such inquires and replies.

The truth-teller wins a play of the game, provided that she never violates the recursive Tarskian truth conditions. Thus, faced with an atomic formula, she must state true or false in accordance with the actual truth or falsity of that atomic formula, and similarly,
she must say true to $𝜑 \land 𝜓$ just in case she said true to both $𝜑$ and $𝜓$ separately (if those formulas had been issued by the challeger), and she must state opposite truth values for $𝜑$ and $\neg 𝜑$ , if both are issued as challenges.

This is an open game, since the challenger will win, if at all, at a finite stage of play, when the violation of the Tarskian truth conditions is first exhibited.

Lemma 1. The truth-teller has a winning strategy in the truth-telling game if and only if there is a satisfaction class for first-order truth.

Proof. Clearly, if there is a satisfaction class for first-order truth, then the truth-teller has a winning strategy, which is simply to answer all questions about truth by consulting the
satisfaction class. Since that class obeys the Tarskian conditions, she will win the game, no matter which challenges are issued.

Conversely, suppose that the truth-teller has a winning strategy $𝜏$ in the game. I claim that we may use $𝜏$ to build a satisfaction class for first-order truth. Specifically, let $𝑇$ be the collection of formulas $𝜑 (⃗ 𝑎)$ that are asserted to be true by $𝜏$ in any play according to $𝜏$ . I claim that $𝑇$ is a satisfaction class. We may begin by noting that since $𝑇$ must correctly state the truth of all atomic formulas, it follows that the particular answers that $𝜏$ gives on the atomic formulas does not depend on the order of the challenges issued by the challenger. Now, we argue by induction on formulas that the truth values issued by $𝜏$ does not depend on the order of the challenges. For example, if all plays in which $𝜑 (⃗ 𝑎)$ is issued as a challenge come out true, then all plays in which $\neg 𝜑 (⃗ 𝑎)$ is challenged will result in false, or else we would have a play in which $𝜏$ would violate the Tarskian truth conditions. Similarly, if $𝜑$ and $𝜓$ always come out the same way, then so does $𝜑 \land 𝜓$ . We don’t claim that $𝜏$ must always issue the same witness $𝑏$ for an existential $\exists 𝑥 𝜓 (𝑥, ⃗ 𝑎)$ , but if it ever says true to this statement, then it will provide some witness $𝑏$ , and for that statement $𝜓 (𝑏, ⃗ 𝑎)$ , the truth value stated by $𝜏$ is independent of the order of play by the challenger, by induction. Thus, by induction on formulas, the answers provided by the truth-teller strategy $𝜏$ gives us a satisfaction predicate for first-order truth. QED

Lemma 2. The challenger has no winning strategy in the truth-telling game.

Proof. Suppose that $𝐹$ is a strategy for the challenger. So $𝐹$ is a proper class function that directs the challenger to issue certain challenges, given the finite sequence of previous challenges and truth-telling answers. By the reflection theorem, there is a closed unbounded proper class of cardinals $𝜃$ , such that $𝐹 ″ 𝑉 𝜃 \subset 𝑉 𝜃$ . That is, $𝑉 𝜃$ is closed under $𝐹$ , in the sense that if all previous challenges and responses come from $𝑉 𝜃$ , then the next challenge will also come from $𝑉 𝜃$ . Since $⟨ 𝑉 𝜃, \in ⟩$ is a set, we have a satisfaction predicate on it. Consider the play, where the truth-teller replies to all inquires by consulting truth in $𝑉 𝜃$ , rather than truth in $𝑉$ . The point is that if the challenger follows $𝜏$ , then all the inquiries will involve only parameters $⃗ 𝑎$ in $𝑉 𝜃$ , provided that the truth-teller also always gives witnesses in $𝑉 𝜃$ , which in this particular play will be the case. Since the satisfaction predicate on $𝑉 𝜃$ does satisfy the Tarskian truth conditions, it follows that the truth-teller will win this instance of the game, and so $𝐹$ is not a winning strategy for the challenger. QED

Thus, if open determinacy holds for classes, then there is a satisfaction predicate for first-order truth.

This implies Con(ZFC) for reasons I explained on my post KM implies Con(ZFC) and much more, by appealing to the fact that we have the collection axiom relative to the class for the satisfaction predicate itself, and this is enough to verify that the nonstandard instances of collection also must be declared true in the satisfaction predicate.

But so far, we only have an open game, rather than a clopen game, since the truth-teller wins only by playing the game out for infinitely many steps. So let me describe how to modify the game to be clopen. Specifically, consider the version of the truth-telling game, where the challenger must also state on each move a specific ordinal $𝛼 𝑛$ , which descend during play $𝛼 0 > 𝛼 1 > \dots > 𝛼 𝑛$ . If the challenger gets to $0$ , then the truth-teller is declared the winner. For this modified game, the winner is known in finitely many moves, because either the truth-teller violates the Tarskian conditions or the challenger hits zero. So this is a clopen game. Since we made the game harder for the challenger, it follows that the challenger still can have no winning strategy. One can modify the proof of lemma 1 to say that if $𝜏$ is a winning strategy for the truth teller, then the truth assertions made by $𝜏$ in response to all plays with sufficiently large ordinals for the challenger all agree with one another independently of the order of the formulas issued by the challenger. Thus, there is a truth-telling strategy just in case there is a satisfaction class for first-order truth.

So clopen determinacy for class games implies the existence of a satisfaction class for first-order truth, and this implies Con(ZFC) and much more. QED

One may easily modify the game by allowing a fixed class parameter $𝐵$ , so that clopen determinacy implies that there is a satisfaction class relative to truth in $⟨ 𝑉, \in, 𝐵 ⟩$ .

Furthermore, we may also get iterated truth predicates. Specifically, consider the iterated truth-telling game, which in addition to the usual language of set theory, we have a hierarchy of predicates $T r 𝛼$ for any ordinal $𝛼$ . We now allow the challenger to ask about formulas in this expanded language, and the truth teller is required to obey not only the usual Tarskian recursive truth conditions, but also the requirements that $T r 𝛼 (𝜑 (⃗ 𝑎))$ is declared true just in case $𝜑 (⃗ 𝑎)$ uses only truth predicates $T r 𝛽$ for $𝛽 < 𝛼$ and also $𝜑 (⃗ 𝑎)$ is declared true (if this challenge was issued).

The main arguments as above generalize easily to show that the challenger cannot have a winning strategy in this iterated truth-telling game, and the truth-teller has a strategy just in case there is a satisfaction predicate for truth-about-truth iterated through the ordinals. Thus, the principle of open determinacy for proper class games implies Con(Con(ZFC)) and $C o n 𝛼 (Z F C)$ and so on.

Let me finish by mentioning that Kelley-Morse set theory is able to prove open determinacy for proper class games in much the same manner as we proved the Gale-Stewart theorem above, using well-ordered class meta-ordinals, rather than merely set ordinals, as well as in other ways. If there is interest, I can make a further post about that, so just ask in the comments!

The hypnagogic digraph, with applications to embeddings of the set-theoretic universe, JMM Special Session on Surreal Numbers, Seattle, January 2016

Posted on June 15, 2015 by Joel David Hamkins

JMM 2016 Seattle This will be an invited talk for the AMS-ASL special session on Surreal Numbers at the 2016 Joint Mathematics Meetings in Seattle, Washington, January 6-9, 2016.

Abstract. The hypnagogic digraph, a proper-class analogue of the countable random $ℚ$ -graded digraph, is a surreal-numbers-graded acyclic digraph exhibiting the set-pattern property (a form of existential-closure), making it set-homogeneous and universal for all class acyclic digraphs. A natural copy of this canonical structure arises during the course of the usual construction of the surreal number line, using as vertices the surreal-number numerals ${𝐴 ∣ 𝐵}$ . I shall explain the construction and elementary theory of the hypnagogic digraph and describe recent uses of it in connection with embeddings of the set-theoretic universe, such as in the proof that the countable models of set theory are linearly pre-ordered by embeddability.

Slides | schedule | related article | surreal numbers (Wikipedia)

Erin Carmody

Posted on June 7, 2015 by Joel David Hamkins

Erin Carmody successfully defended her dissertation under my supervision at the CUNY Graduate Center on April 24, 2015, and she earned her Ph.D. degree in May, 2015. Her dissertation follows the theme of killing them softly, proving many theorems of the form: given $𝜅$ with large cardinal property $𝐴$ , there is a forcing extension in which $𝜅$ no longer has property $𝐴$ , but still has large cardinal property $𝐵$ , which is very slightly weaker than $𝐴$ . Thus, she aims to enact very precise reductions in large cardinal strength of a given cardinal or class of large cardinals. In addition, as a part of the project, she developed transfinite meta-ordinal extensions of the degrees of hyper-inaccessibility and hyper-Mahloness, giving notions such as $(Ω 𝜔 2 + 5 + Ω 3 \cdot 𝜔 21 + Ω + 2)$ -inaccessible among others.

G+ profile | math genealogy | MathOverflow profile | NY Logic profile | ar $𝜒$ iv

Erin Carmody, “Forcing to change large cardinal strength,” Ph.D. dissertation for The Graduate Center of the City University of New York, May, 2015. ar $𝜒$ iv | PDF

Erin has accepted a professorship at Nebreska Wesleyan University for.the 2015-16 academic year.

Erin is also an accomplished artist, who has had art shows of her work in New York, and she has pieces for sale. Much of her work has an abstract or mathematical aspect, while some pieces exhibit a more emotional or personal nature. My wife and I have two of Erin’s paintings in our collection:

Transfinite Nim

Posted on May 27, 2015 by Joel David Hamkins

Shall we have a game of transfinite Nim? One of us sets up finitely many piles of wooden blocks, each pile having some ordinal height, possibly transfinite, and the other of us decides who shall make the first move. Taking turns, we each successively remove a top part of any one pile of our choosing, making it strictly shorter. Whoever takes the very last block wins. (It is fine to remove an entire pile on a turn or to remove blocks from a different pile on a later turn.)

In my challenge problem last week, for example, I set up six piles with heights:
$1 𝜔 + 3 𝜔 𝜔 + 5 𝜔 𝜔 + 3 + 𝜔 𝜔 \cdot 3 + 𝜔 \cdot 5 + 7 𝜖 0 𝜔 1$ Would you want to go first or second? What is the best move? In general, we can start with any finite number of piles of arbitrary ordinal heights — what is the general winning strategy?

Before proceeding with the transfinite case, however, let’s review the winning strategy in ordinary finite Nim, which I explained in my post last week concerning my visit to the 7th/8th grade Math Team at my son’s school. To say it quickly again, a finite Nim position is balanced, if when you consider the binary representations of the pile heights, there are an even number of ones in each binary place position. Another way to say this, and this is how I explained it to the school kids, is that if you think of each pile height as a sum of distinct powers of two, then any power of two that arises in any pile does so an even number of times overall for all the piles. The mathematical facts to establish are that (1) any move on a balanced position will unbalance it; and (2) any unbalanced position admits a balancing move. Since the winning move of taking the very last block is a balancing move, it follows that the winning strategy is to balance whatever position with which you are faced. At the start, if the position is unbalanced, then you should go first and balance it; if it is already balanced, then you should go second and adopt the balancing strategy. It may be interesting to note that this winning strategy is unique in the sense that any move that does not balance the position is a losing move, since the opposing player can adopt the balancing strategy from that point on. But of course there is often a choice of balancing moves.

Does this balancing strategy idea continue to apply to transfinite Nim? Yes! All we need to do is to develop a little of the theory of transfinite binary representation. Let me assume that you are all familiar with the usual ordinal arithmetic, for which $𝛼 + 𝛽$ is the ordinal whose order type is isomorphic to a copy of $𝛼$ followed by a copy of $𝛽$ , and $𝛼 \cdot 𝛽$ is the ordinal whose order type is isomorphic to $𝛽$ many copies of $𝛼$ . Consider now ordinal exponentiation, which can be defined recursively as follows:
$𝛼 0 = 1$ $𝛼 𝛽 + 1 = 𝛼 𝛽 \cdot 𝛼$ $𝛼 𝜆 = s u p 𝛽 < 𝜆 𝛼 𝛽 𝜆 l i m i t$ It turns out that $𝛼 𝛽$ is the order-type of the finite-support functions from $𝛽$ to $𝛼$ , under the suitable lexical order. Ordinal exponentiation should not be confused with cardinal exponentiation, since they are very different. For example, with ordinal exponentiation, one has $2 𝜔 = s u p 𝑛 < 𝜔 2 𝑛 = 𝜔,$ which of course is not the case with cardinal exponentiation. In this post, I use only ordinal exponentiation.

Theorem. Every ordinal $𝛽$ has a unique representation as a decreasing finite sum of ordinal powers of two. $𝛽 = 2 𝛽 𝑛 + \dots + 2 𝛽 0, 𝛽 𝑛 > \dots > 𝛽 0$

The proof is easy! We simply prove it by transfinite induction on $𝛽$ . If the theorem holds below an ordinal $𝛽$ , first let $2 𝛼$ be the largest power of two that is at most $𝛽$ , so that $𝛽 = 2 𝛼 + 𝛾$ for some ordinal $𝛾$ . It follows that $𝛾 < 2 𝛼$ , for otherwise we could have made $2 𝛼 + 1 \leq 𝛽$ . Thus, by induction, $𝛾$ has a representation with powers of two, and so we may simply add $2 𝛼$ at the front to represent $𝛽$ . To see that the representations are unique, first establish that any power of two is equal to or more than the supremum of the finite decreasing sums of any strictly smaller powers of two. From this, it follows that any representation of $𝛽$ as above must have used $2 𝛼$ just as we did for the first term, because otherwise it couldn’t be large enough, and then the representation of the remaining part $𝛾$ is unique by induction, and so we get uniqueness for the representation of $𝛽$ . QED

Thus, the theorem shows that every ordinal has a unique binary representation in the ordinals, with finitely many nonzero bits. Suppose that we are given a position in transfinite Nim with piles of ordinal heights $𝜂 0, \dots, 𝜂 𝑛$ . We define that such a position is balanced, if every power of two appearing in the representation of any of the piles appears an even number of times overall for all the piles.

The mathematical facts to establish are (1) any move on a balanced position will unbalance it; and (2) every unbalanced position has a balancing move. These facts can be proved in the transfinite case in essentially the same manner as the finite case. Namely, if a position is balanced, then any move affects only one pile, changing the ordinal powers of two that appear in it, and thereby destroy the balanced parity of whichever powers of two are affected. And if a position is unbalanced, then look at the largest unbalanced ordinal power of two appearing, and make a move on any pile having such a power of two in its representation, reducing it so as exactly to balance all the smaller powers of two appearing in the position.

Finally, those two facts again imply that the balancing strategy is a winning strategy, since the winning move of taking the last block or blocks is a balancing move, down to the all-zero position, which is balanced.

In the case of my challenge problem above, we may represent the ordinals in binary. We know how to do that in the case of 1, 3, 5 and 7, and actually those numbers are balanced. Here are some other useful binary representations:

$𝜔 + 3 = 2 𝜔 + 2 + 1$

$𝜔 𝜔 + 5 = (2 𝜔) 𝜔 + 5 = 2 𝜔 2 + 4 + 1$

$𝜔 𝜔 + 3 = (2 𝜔) 𝜔 + 3 = 2 𝜔 2 + 𝜔 \cdot 3$

$𝜔 𝜔 \cdot 3 = (2 𝜔) 𝜔 \cdot 3 = 2 𝜔 2 \cdot 2 + 2 𝜔 2 = 2 𝜔 2 + 1 + 2 𝜔 2$

$𝜔 \cdot 5 + 7 = 2 𝜔 \cdot 22 + 2 𝜔 + 7 = 2 𝜔 + 2 + 2 𝜔 + 4 + 2 + 1$

$𝜖 0 = 2 𝜖 0$

$𝜔 1 = 2 𝜔 1$

I emphasize again that this is ordinal exponentiation. The Nim position of the challenge problem above is easily seen to be unbalanced in several ways. For example, the $𝜔 1$ term among others appears only once. Thus, we definitely want to go first in this position. And since $𝜔 1$ is the largest unbalanced power of two and it appears only once, we know that we must play on the $𝜔 1$ pile. Once one represents all the ordinals in terms of their powers of two representation, one sees that the unique winning move is to reduce the $𝜔 1$ pile to have ordinal height
$𝜖 0 + 𝜔 𝜔 + 3 + 𝜔 𝜔 \cdot 2 + 𝜔 \cdot 4 .$ This will exactly balance all the smaller powers of two in the other piles and therefore leaves a balanced position overall. In general, the winning strategy in transfinite Nim, just as for finite Nim, is always to leave a balanced position.

Special honors to Pedro Sánchez Terraf for being the only one to post the winning move in the comments on the other post!

Win at Nim! The secret mathematical strategy for kids (with challange problems in transfinite Nim for the rest of us)

Posted on May 22, 2015 by Joel David Hamkins

Welcome to my latest instance of Math for Kids!

Today I had the pleasure to make an interactive mathematical presentation at my son’s school to the 7th / 8th grade Math Team, about 30 math-enthusiastic kids (twelve and thirteen years old) along with their math teachers and the chair of the school math department.

The topic was the game of Nim! This game has a secret mathematical strategy enabling anyone with that secret knowledge to win against those without it. It is a great game for kids, because with the strategy they can realistically expect to beat their parents, friends, siblings and parent’s friends almost every single time!

To play Nim, one player sets up a number of piles of blocks, and the opponent chooses whether to go first or second. The players take turns removing blocks — each player may remove any number of blocks (at least one) from any one pile, and it is fine to take a whole pile — whichever player takes the last block wins.

For the math team, we played a few demonstration games, in which I was able to beat all the brave challengers, and then the kids paired off to play each other and gain familiarity with the game. Then, it was time for the first strategy discussion.

What could the secret winning strategy be? I explained to the kids a trick that mathematicians often use when approaching a difficult problem, namely, to consider in detail some very simple special cases or boundary instances of the problem. It often happens that these special cases reveal a way of thinking about the problem that applies much more generally.

Perhaps one of the easiest special cases of Nim occurs when there is only one pile. If there is only one pile, then clearly one wants to go first, in order to make the winning move: take the entire pile!

Two balanced piles

A slightly less trivial and probably more informative case arises when there are exactly two piles. If the stacks have the same height, then the kids realized that the second player could make copying moves so as to preserve this balanced situation. The key insight now is that this copying strategy is a winning strategy, because if one can always copy, then in particular one will have a move whenever the opponent did, and so the opponent will never take the last block. With two piles, therefore, one wants always to make them balanced. If they are initially unbalanced, then choose to go first and follow the balancing strategy. If they are initially balanced, then choose to go second, and copy whatever moves your opponent makes to rebalance them.

A balanced position

With that insight, it is not difficult to see that it is winning to leave a position with any number of pairs of balanced piles. One can in effect play on each pair separately, because whenever the opponent makes a move on one of the piles, one can copy the move with the corresponding partner pile. In this way, we may count such a position overall as balanced. The more fundamental game-theoretic observation to make is that balanced piles in effect cancel each other out in any position, and one can ignore them when analyzing a position. When two balanced piles are present in a possibly more complicated position, one can pretend that they aren’t there, precisely because whenever your opponent plays on one of them, you can copy the move on the other, and so any winning strategy for the position in which those piles are absent can be converted into a winning strategy in which the balanced piles are present.

This idea now provides a complete winning strategy in the case that all piles have height one or two at most. One wants to leave a position with an even number of piles of each height. If only one height has an odd number of piles, then take a whole pile of that height. And if there are odd numbers of piles both of height one and two, then turn a height-two pile into a pile of height one, and this will make them both even. So any unbalanced position can be balanced, and any move on a balanced position will unbalance it.

1+2+3 counts as balanced

Let’s now consider that there may be piles of height three. For example, consider the basic position with piles of height one, two and three. The observation to make here is that any move on this position can be replied to with a move that leaves it balanced (check it yourself to be sure!). It follows that this position is winning to leave for the other player (and so one should go second on $1 + 2 + 3$ ). It would be nice if we could consider this position itself as already balanced in some sense. Indeed, we may incorporate this situation into the balancing idea if we think of the pile of height three as really consisting of two subpiles, one of height two and one of height one. In this way, the Nim position 1+2+3 counts as balanced, since the 3 counts as 2+1, which balances the other stacks. The 1+2+3 position has two stacks of height two and two of height one, when one regards the stack of height three as having a substack of height two and a substack of height one.

This way of thinking produces a complete winning strategy for Nim positions involving piles of height at most three. (And this is a strategy that can be mastered even by very young children — a few years ago I had talked about Nim with much younger children, Math for six-year-olds: Win at Nim!, first-graders at my daughter’s school, and at that time we concentrated on posititions with piles of height at most three. Older kids, however, can handle the full strategy.) Namely, the winning strategy in this case is to strive to balance the position, to make an even number overall of piles of height one and two, where we count piles of height three as one each of one and two. If you always give your opponent a balanced position, then you will win! Faced with an unbalanced position, it is a fact that you can always find a balancing move, and any move on an balanced position will unbalance it. If the game is just starting, and you are deciding whether to go first or second, you should determine whether it is balanced yet or not. If it unbalanced, then you should go first and make the balancing move; if it is already balanced, then you should go second and adopt the copying strategy, in which you re-balance the position with each move.

The general winning strategy, of course, goes beyond three. The key idea is to realize that what is really going on when we represent $3$ as $2 + 1$ is that we are using the binary representation of the number $3$ . To explain, I wrote the following numbers on the chalkboard $1, 2, 4, 8, 16, 32, 64, \dots$ and was very pleased when the kids immediately shouted out, “The powers of two!” I explained that any natural number can be expressed uniquely as a sum of distinct powers of two. Asked for a favorite number less than one hundred, one student suggested $88$ , and together we calculated $88 = 64 + 16 + 8,$ which means that the binary representation of $88$ is $1011000$ , which I read off as, “one $64$ , no $32$ s, one $16$ , one $8$ , no $4$ s, no $2$ s and no $1$ s. This is just the same as thinking of $9572$ as 9 thousands, 5 hundreds, 7 tens and 2 ones, using the powers of ten. It is interesting to learn that one may easily count very high on one hand using binary, up to 1023 on two hands!

The general strategy is to view every Nim pile as consisting of subpiles whose height is a power of two, and to make sure that one leaves a position that is balanced in the sense that every power of two has an even number of such instances in the position. So we think of $3$ as really $2 + 1$ for the purposes of balancing; $4$ counts as itself because it is a power of two, but $5$ counts as $4 + 1$ and $6$ counts as $4 + 2$ and $7$ as $4 + 2 + 1$ . Another way to describe the strategy is that we express all the pile heights in binary, and we want an even number of $1$ s in each binary place position.

The mathematical facts to verify are (1) any move on a balanced position in this powers-of-two sense will cause it to become unbalanced, and (2) any unbalanced position can be balanced in one move. It follows that leaving balanced positions is a winning strategy, because the winning move of taking the last block is a balancing move rather than an unbalancing move.

One can prove statement (1) by realizing that when you move a single stack, the binary representation changes, and so whichever binary digits changed will now become unbalanced. For statement (2), consider the largest unbalanced power of two $2 𝑘$ and move on any stack that contains a $2 𝑘$ size substack. Since $2 𝑘 - 1 = 111 \dots 11$ in binary, one can attain any binary pattern for the smaller height stacks by removing between $1$ and $2 𝑘$ many blocks. So one can balance the position.

As a practical matter, the proof of (2) also shows how one can find a (winning) balancing move, which can otherwise be difficult in some cases: look for the largest unbalanced power of two, and move on any pile containing such a subpile, making sure to leave a balanced position.

In most actual instances of Nim, the pile heights are rarely very tall, and so one is usually considering just $1$ , $2$ and $4$ as the powers of two that arise. A traditional starting configuration has piles of height 1, 3, 5, and 7, and this position is balanced, because one may view it as: $1, 2 + 1, 4 + 1, 4 + 2 + 1$ , and there are an even number of 1s, 2s and 4s.

It is interesting to consider also the Misère form of Nim, where one wants NOT to take the last block. This version of the game also has a secret mathematical strategy, which I shall reveal later on.

Challenge 1. What is the winning strategy in Misère Nim?

If you figure it out, please post a comment! I’ll post the solution later. One might naively expect that the winning strategy of Misère Nim is somehow totally opposite to the winning strategy of regular Nim, but in fact, the positions $1, 2, 3$ and $1, 3, 5, 7$ are winning for the second player both in Nim and also in Misère Nim. Indeed, I claim that all nontrivial Nim positions that are winning for regular Nim (with a suitable meaning of “nontrivial”) are also winning for Misère Nim. Can you prove it?

Another interesting generalization, for the set-theorists, is to consider transfinite Nim, where the piles can have transfinite ordinal height. So we have finitely many piles of ordinal height, perhaps infinite, and a move consists of making any one pile strictly shorter. Since there are no infinite descending sequence of ordinals, the game will terminate in finitely many moves, and the winner is whowever removes the last block.

Challenge 2. Who wins the transfinite Nim game with piles of heights: $1 𝜔 + 3 𝜔 𝜔 + 5 𝜔 𝜔 + 3 + 𝜔 𝜔 \cdot 3 + 𝜔 \cdot 5 + 7 𝜖 0 𝜔 1$ and what are the winning moves? What is the general winning strategy for transfinite Nim?

Post your solutions! You can also see my solution and further discussion.

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