# The universal finite set

• J. D. Hamkins and W. Woodin, “The universal finite set,” ArXiv e-prints, pp. 1-16, 2017. (manuscript under review)
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Abstract. We define a certain finite set in set theory $\{x\mid\varphi(x)\}$ and prove that it exhibits a universal extension property: it can be any desired particular finite set in the right set-theoretic universe and it can become successively any desired larger finite set in top-extensions of that universe. Specifically, ZFC proves the set is finite; the definition $\varphi$ has complexity $\Sigma_2$, so that any affirmative instance of it $\varphi(x)$ is verified in any sufficiently large rank-initial segment of the universe $V_\theta$; the set is empty in any transitive model and others; and if $\varphi$ defines the set $y$ in some countable model $M$ of ZFC and $y\subseteq z$ for some finite set $z$ in $M$, then there is a top-extension of $M$ to a model $N$ in which $\varphi$ defines the new set $z$. Thus, the set shows that no model of set theory can realize a maximal $\Sigma_2$ theory with its natural number parameters, although this is possible without parameters. Using the universal finite set, we prove that the validities of top-extensional set-theoretic potentialism, the modal principles valid in the Kripke model of all countable models of set theory, each accessing its top-extensions, are precisely the assertions of S4. Furthermore, if ZFC is consistent, then there are models of ZFC realizing the top-extensional maximality principle.

Woodin had established the universal algorithm phenomenon, showing that there is a Turing machine program with a certain universal top-extension property in models of arithmetic (see also work of Blanck and Enayat 2017 and upcoming paper of mine with Gitman and Kossak; also my post The universal algorithm: a new simple proof of Woodin’s theorem). Namely, the program provably enumerates a finite set of natural numbers, but it is relatively consistent with PA that it enumerates any particular desired finite set of numbers, and furthermore, if $M$ is any model of PA in which the program enumerates the set $s$ and $t$ is any (possibly nonstandard) finite set in $M$ with $s\subseteq t$, then there is a top-extension of $M$ to a model $N$ in which the program enumerates exactly the new set $t$. So it is a universal finite computably enumerable set, which can in principle be any desired finite set of natural numbers in the right arithmetic universe and become any desired larger finite set in a suitable larger arithmetic universe.

I had inquired whether there is a set-theoretic analogue of this phenomenon, using $\Sigma_2$ definitions in set theory in place of computable enumerability (see The universal definition — it can define any mathematical object you like, in the right set-theoretic universe). The idea was that just as a computably enumerable set is one whose elements are gradually revealed as the computation proceeds, a $\Sigma_2$-definable set in set theory is precisely one whose elements become verified at some level $V_\theta$ of the cumulative set-theoretic hierarchy as it grows. In this sense, $\Sigma_2$ definability in set theory is analogous to computable enumerability in arithmetic.

Main Question. Is there a universal $\Sigma_2$ definition in set theory, one which can define any desired particular set in some model of \ZFC\ and always any desired further set in a suitable top-extension?

I had noticed in my earlier post that one can do this using a $\Pi_3$ definition, or with a $\Sigma_2$ definition, if one restricts to models of a certain theory, such as $V\neq\text{HOD}$ or the eventual GCH, or if one allows $\{x\mid\varphi(x)\}$ sometimes to be a proper class.

Here, we provide a fully general affirmative answer with the following theorem.

Main Theorem. There is a formula $\varphi(x)$ of complexity $\Sigma_2$ in the language of set theory, provided in the proof, with the following properties:

1. ZFC proves that $\{x\mid \varphi(x)\}$ is a finite set.
2. In any transitive model of \ZFC\ and others, this set is empty.
3. If $M$ is a countable model of ZFC in which $\varphi$ defines the set $y$ and $z\in M$ is any finite set in $M$ with $y\subseteq z$, then there is a top-extension of $M$ to a model $N$ in which $\varphi$ defines exactly $z$.

By taking the union of the set defined by $\varphi$, an arbitrary set can be achieved; so the finite-set result as stated in the main theorem implies the arbitrary set case as in the main question. One can also easily deduce a version of the theorem to give a universal countable set or a universal set of some other size (for example, just take the union of the countable elements of the universal set). One can equivalently formulate the main theorem in terms of finite sequences, rather than sets, so that the sequence is extended as desired in the top-extension. The sets $y$ and $z$ in statement (3) may be nonstandard finite, if $M$ if $\omega$-nonstandard.

We use this theorem to establish the fundamental validities of top-extensional set-theoretic potentialism. Specifically, in the potentialist system consisting of the countable models of ZFC, with each accessing its top extensions, the modal validities with respect to substitution instances in the language of set theory, with parameters, are exactly the assertions of S4. When only sentences are considered, the validities are between S4 and S5, with both endpoints realized.

In particular, we prove that if ZFC is consistent, then there is a model $M$ of ZFC with the top-extensional maximality principle: any sentence $\sigma$ in the language of set theory which is true in some top extension $M^+$ and all further top extensions of $M^+$, is already true in $M$.

This principle is true is any model of set theory with a maximal $\Sigma_2$ theory, but it is never true when $\sigma$ is allowed to have natural-number parameters, and in particular, it is never true in any $\omega$-standard model of set theory.

Click through to the arXiv for more, the full article in pdf.

• J. D. Hamkins and W. Woodin, “The universal finite set,” ArXiv e-prints, pp. 1-16, 2017. (manuscript under review)
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# Models of set theory with the same reals and the same cardinals, but which disagree on the continuum hypothesis

I’d like to describe a certain interesting and surprising situation that can happen with models of set theory.

Theorem. If $\newcommand\ZFC{\text{ZFC}}\ZFC$ set theory is consistent, then there are two models of $\ZFC$ set theory $M$ and $N$ for which

• $M$ and $N$ have the same real numbers $$\newcommand\R{\mathbb{R}}\R^M=\R^N.$$
• $M$ and $N$ have the ordinals and the same cardinals $$\forall\alpha\qquad \aleph_\alpha^M=\aleph_\alpha^N$$
• But $M$ thinks that the continuum hypothesis $\newcommand\CH{\text{CH}}\CH$ is true, while $N$ thinks that $\CH$ is false.

This is a little strange, since the two models have the set $\R$ in common and they agree on the cardinal numbers, but $M$ thinks that $\R$ has size $\aleph_1$ and $N$ will think that $\R$ has size $\aleph_2$.  In particular, $M$ can well-order the reals in order type $\omega_1$ and $N$ can do so in order-type $\omega_2$, even though the two models have the same reals and they agree that these order types have different cardinalities.

Another abstract way to describe what is going on is that even if two models of set theory, even transitive models, agree on which ordinals are cardinals, they needn’t agree on which sets are equinumerous, for sets they have in common, even for the reals.

Let me emphasize that it is the requirement that the models have the same cardinals that makes the problem both subtle and surprising. If you drop that requirement, then the problem is an elementary exercise in forcing: start with any model $V$, and first force $\CH$ to fail in $V[H]$ by adding a lot of Cohen reals, then force to $V[G]$ by collapsing the continuum to $\aleph_1$. This second step adds no new reals and forces $\CH$, and so $V[G]$ and $V[H]$ will have the same reals, while $V[H]$ thinks $\CH$ is true and $V[G]$ thinks $\CH$ is false. The problem becomes nontrivial and interesting mainly when you insist that cardinals are not collapsed.

In fact, the situation described in the theorem can be forced over any given model of $\ZFC$.

Theorem. Every model of set theory $V\models\ZFC$ has two set-forcing extensions $V[G]$ and $V[H]$ for which

• $V[G]$ and $V[H]$ have the same real numbers $$\newcommand\R{\mathbb{R}}\R^{V[G]}=\R^{V[H]}.$$
• $V[G]$ and $V[H]$ have the same cardinals $$\forall\alpha\qquad \aleph_\alpha^{V[G]}=\aleph_\alpha^{V[H]}$$
• But $V[G]$ thinks that the continuum hypothesis $\CH$ is true, while $V[H]$ thinks that $\CH$ is false.

Proof. Start in any model $V\models\ZFC$, and by forcing if necessary, let’s assume $\CH$ holds in $V$. Let $H\subset\text{Add}(\omega,\omega_2)$ be $V$-generic for the forcing to add $\omega_2$ many Cohen reals. So $V[H]$ satisfies $\neg\CH$ and has the same ordinals and cardinals as $V$.

Next, force over $V[H]$ using the forcing from $V$ to collapse $\omega_2$ to $\omega_1$, forming the extension $V[H][g]$, where $g$ is the generic bijection between those ordinals. Since we used the forcing in $V$, which is countably closed there, it makes sense to consider $V[g]$.  In this extension, the forcing $\text{Add}(\omega,\omega_1^V)$ and $\text{Add}(\omega,\omega_2^V)$ are isomorphic. Since $H$ is $V[g]$-generic for the latter, let $G=g\mathrel{“}H$ be the image of this filter in $\text{Add}(\omega,\omega_1)$, which is therefore $V[g]$-generic for the former. So $V[g][G]=V[g][H]$. Since the forcing $\text{Add}(\omega,\omega_1)$ is c.c.c., it follows that $V[G]$ also has the same cardinals as $V$ and hence also the same as in $V[H]$.

If we now view these extensions as $V[G][g]=V[H][g]$ and note that the coutable closure of $g$ in $V$ implies that $g$ adds no new reals over either $V[G]$ or $V[H]$, it follows that $\R^{V[G]}=\R^{V[H]}$. So the two models have the same reals and the same cardinals. But $V[G]$ has $\CH$ and $V[H]$ has $\neg\CH$, in light of the forcing, and so the proof is complete. QED

Let me prove the following surprising generalization.

Theorem. If $V$ is any model of $\ZFC$ and $V[G]$ is the forcing extension obtained by adding $\kappa$ many Cohen reals, for some uncountable $\kappa$, then for any other uncountable cardinal $\lambda$, there is another forcing extension $V[H]$ where $H$ is $V$-generic for the forcing to add $\lambda$ many Cohen reals, yet $\R^{V[G]}=\R^{V[H]}$.

Proof. Start in $V[G]$, and let $g$ be $V[G]$-generic to collapse $\lambda$ to $\kappa$, using the collapse forcing of the ground model $V$. This forcing is countably closed in $V$ and therefore does not add reals over $V[G]$. In $V[g]$, the two forcing notions $\text{Add}(\omega,\kappa)$ and $\text{Add}(\omega,\lambda)$ are isomorphic. Thus, since $G$ is $V[g]$-generic for the former poset, it follows that the image $H=g\mathrel{“}G$ is $V[g]$-generic for the latter poset. So $V[H]$ is generic over $V$ for adding $\lambda$ many Cohen reals. By construction, we have $V[G][g]=V[H][g]$, and since $g$ doesn’t add reals, it follows that $\R^{V[G]}=\R^{V[H]}$, as desired. QED

I have a vague recollection of having first heard of this problem many years ago, perhaps as a graduate student, although I don’t quite recall where it was or indeed what the construction was — the argument above is my reconstruction (which I have updated and extended from my initial post). If someone could provide a reference in the comments for due credit, I’d be appreciative.  The problem appeared a few years ago on MathOverflow.

# The inclusion relations of the countable models of set theory are all isomorphic

• J. D. Hamkins and M. Kikuchi, “The inclusion relations of the countable models of set theory are all isomorphic,” , 2017. (manuscript under review)
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Abstract. The structures $\langle M,\newcommand\of{\subseteq}\of^M\rangle$ arising as the inclusion relation of a countable model of sufficient set theory $\langle M,\in^M\rangle$, whether well-founded or not, are all isomorphic. These structures $\langle M,\of^M\rangle$ are exactly the countable saturated models of the theory of set-theoretic mereology: an unbounded atomic relatively complemented distributive lattice. A very weak set theory suffices, even finite set theory, provided that one excludes the $\omega$-standard models with no infinite sets and the $\omega$-standard models of set theory with an amorphous set. Analogous results hold also for class theories such as Gödel-Bernays set theory and Kelley-Morse set theory.

Set-theoretic mereology is the study of the inclusion relation $\of$ as it arises within set theory. In any set-theoretic context, with the set membership relation $\in$, one may define the corresponding inclusion relation $\of$ and investigate its properties. Thus, every model of set theory $\langle M,\in^M\rangle$ gives rise to a corresponding model of set-theoretic mereology $\langle M,\of^M\rangle$, the reduct to the inclusion relation.

In our previous article,

J. D. Hamkins and M. Kikuchi, Set-theoretic mereology, Logic and Logical Philosophy, special issue “Mereology and beyond, part II”, vol. 25, iss. 3, pp. 1-24, 2016.

we had identified exactly the complete theory of these mereological structures $\langle M,\of^M\rangle$. Namely, if $\langle M,\in^M\rangle$ is a model of set theory, even for extremely weak theories, including set theory without the infinity axiom, then the corresponding mereological reduct $\langle M,\of^M\rangle$ is an unbounded atomic relatively complemented distributive lattice. We call this the theory of set-theoretic mereology. By a quantifier-elimination argument that we give in our earlier paper, partaking of Tarski’s Boolean-algebra invariants and Ersov’s work on lattices, this theory is complete, finitely axiomatizable and decidable.  We had proved among other things that $\in$ is never definable from $\of$ in any model of set theory and furthermore, some models of set-theoretic mereology can arise as the inclusion relation of diverse models of set theory, with different theories. Furthermore, we proved that $\langle\text{HF},\subseteq\rangle\prec\langle V,\subseteq\rangle$.

After that work, we found it natural to inquire:

Question. Which models of set-theoretic mereology arise as the inclusion relation $\of$ of a model of set theory?

More precisely, given a model $\langle M,\newcommand\sqof{\sqsubseteq}\sqof\rangle$ of set-theoretic mereology, under what circumstances can we place a binary relation $\in^M$ on $M$ in such a way that $\langle M,\in^M\rangle$ is a model of set theory and the inclusion relation $\of$ defined in $\langle M,\in^M\rangle$ is precisely the given relation $\sqof$? One can view this question as seeking a kind of Stone-style representation of the mereological structure $\langle M,\sqof\rangle$, because such a model $M$ would provide a representation of $\langle M,\sqof\rangle$ as a relative field of sets via the model of set theory $\langle M,\in^M\rangle$.

A second natural question was to wonder how much of the theory of the original model of set theory can be recovered from the mereological reduct.

Question. If $\langle M,\of^M\rangle$ is the model of set-theoretic mereology arising as the inclusion relation $\of$ of a model of set theory $\langle M,\in^M\rangle$, what part of the theory of $\langle M,\in^M\rangle$ is determined by the structure $\langle M,\of^M\rangle$?

In the case of the countable models of ZFC, these questions are completely answered by our main theorems.

Main Theorems.

1. All countable models of set theory $\langle M,\in^M\rangle\models\text{ZFC}$ have isomorphic reducts $\langle M,\of^M\rangle$ to the inclusion relation.
2. The same holds for models of considerably weaker theories such as KP and even finite set theory, provided one excludes the $\omega$-standard models without infinite sets and the $\omega$-standard models having an amorphous set.
3. These inclusion reducts $\langle M,\of^M\rangle$ are precisely the countable saturated models of set-theoretic mereology.
4. Similar results hold for class theory: all countable models of Gödel-Bernays set theory have isomorphic reducts to the inclusion relation, and this reduct is precisely the countably infinite saturated atomic Boolean algebra.

Specifically, we show that the mereological reducts $\langle M,\of^M\rangle$ of the models of sufficient set theory are always $\omega$-saturated, and from this it follows on general model-theoretic grounds that they are all isomorphic, establishing statements (1) and (2). So a countable model $\langle M,\sqof\rangle$ of set-theoretic mereology arises as the inclusion relation of a model of sufficient set theory if and only if it is $\omega$-saturated, establishing (3) and answering the first question. Consequently, in addition, the mereological reducts $\langle M,\of^M\rangle$ of the countable models of sufficient set theory know essentially nothing of the theory of the structure $\langle M,\in^M\rangle$ from which they arose, since $\langle M,\of^M\rangle$ arises equally as the inclusion relation of other models $\langle M,\in^*\rangle$ with any desired sufficient alternative set theory, a fact which answers the second question. Our analysis works with very weak set theories, even finite set theory, provided one excludes the $\omega$-standard models with no infinite sets and the $\omega$-standard models with an amorphous set, since the inclusion reducts of these models are not $\omega$-saturated. We also prove that most of these results do not generalize to uncountable models, nor even to the $\omega_1$-like models.

Our results have some affinity with the classical results in models of arithmetic concerned with the additive reducts of models of PA. Restricting a model of set theory to the inclusion relation $\of$ is, after all, something like restricting a model of arithmetic to its additive part. Lipshitz and Nadel (1978) proved that a countable model of Presburger arithmetic (with $+$ only) can be expanded to a model of PA if and only if it is computably saturated. We had hoped at first to prove a corresponding result for the mereological reducts of the models of set theory. In arithmetic, the additive reducts are not all isomorphic, since the standard system of the PA model is fully captured by the additive reduct. Our main result for the countable models of set theory, however, turned out to be stronger than we had expected, since the inclusion reducts are not merely computably saturated, but fully $\omega$-saturated, and this is why they are all isomorphic. Meanwhile, Lipshitz and Nadel point out that their result does not generalize to uncountable models of arithmetic, and similarly ours also does not generalize to uncountable models of set theory.

The work leaves the following question open:

Question. Are the mereological reducts $\langle M,\of^M\rangle$ of all the countable models $\langle M,\in^M\rangle$ of ZF with an amorphous set all isomorphic?

We expect the answer to come from a deeper understanding of the Tarski-Ersov invariants for the mereological structures combined with knowledge of models of ZF with amorphous sets.

This is joint work with Makoto Kikuchi.

# All countable models of set theory have the same inclusion relation up to isomorphism, CUNY Logic Workshop, April 2017

This will be a talk for the CUNY Logic Workshop, April 28, 2:00-3:30 in room 6417 at the CUNY Graduate Center.

Abstract.  Take any countable model of set theory $\langle M,\in^M\rangle\models\text{ZFC}$, whether well-founded or not, and consider the corresponding inclusion relation $\langle M,\newcommand\of{\subseteq}\of^M\rangle$.  All such models, we prove, are isomorphic. Indeed, if $\langle M,\in^M\rangle$ is a countable model of set theory — a very weak theory suffices, including finite set theory, if one excludes the $\omega$-standard models with no infinite sets and the $\omega$-standard models with an amorphous set — then the corresponding inclusion reduct $\langle M,\of^M\rangle$ is an $\omega$-saturated model of the theory we have called set-theoretic mereology. Since this is a complete theory, it follows by the back-and-forth construction that all such countable saturated models are isomorphic. Thus, the inclusion relation $\langle M,\of^M\rangle$ knows essentially nothing about the theory of the set-theoretic structure $\langle M,\in^M\rangle$ from which it arose. Analogous results hold also for class theories such as Gödel-Bernays set theory and Kelley-Morse set theory.

This is joint work with Makoto Kikuchi, and our paper is available at

J. D. Hamkins and M. Kikuchi, The inclusion relations of the countable models of set theory are all isomorphic, manuscript under review.

Our previous work, upon which these results build, is available at:

J. D. Hamkins and M. Kikuchi, Set-theoretic mereology, Logic and Logical Philosophy, special issue “Mereology and beyond, part II”, vol. 25, iss. 3, pp. 1-24, 2016.

# Same structure, different truths, Stanford University CSLI, May 2016

This will be a talk for the Workshop on Logic, Rationality, and Intelligent Interaction at the CSLI, Stanford University, May 27-28, 2016.

Abstract. To what extent does a structure determine its theory of truth? I shall discuss several surprising mathematical results illustrating senses in which it does not, for the satisfaction relation of first-order logic is less absolute than one might have expected. Two models of set theory, for example, can have exactly the same natural numbers and the same arithmetic structure $\langle\mathbb{N},+,\cdot,0,1,<\rangle$, yet disagree on what is true in this structure; they have the same arithmetic, but different theories of arithmetic truth; two models of set theory can have the same natural numbers and a computable linear order in common, yet disagree on whether it is a well-order; two models of set theory can have the same natural numbers and the same reals, yet disagree on projective truth; two models of set theory can have a rank initial segment of the universe $\langle V_\delta,{\in}\rangle$ in common, yet disagree about whether it is a model of ZFC. These theorems and others can be proved with elementary classical model-theoretic methods, which I shall explain. On the basis of these observations, Ruizhi Yang (Fudan University, Shanghai) and I argue that the definiteness of the theory of truth for a structure, even in the case of arithmetic, cannot be seen as arising solely from the definiteness of the structure itself in which that truth resides, but rather is a higher-order ontological commitment.

Slides | Main article: Satisfaction is not absolute | CLSI 2016 | Abstract at CLSI

# Upward closure and amalgamation in the generic multiverse of a countable model of set theory

• J. D. Hamkins, “Upward closure and amalgamation in the generic multiverse of a countable model of set theory,” RIMS Kyôkyûroku, pp. 17-31, 2016.
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Abstract. I prove several theorems concerning upward closure and amalgamation in the generic multiverse of a countable transitive model of set theory. Every such model $W$ has forcing extensions $W[c]$ and $W[d]$ by adding a Cohen real, which cannot be amalgamated in any further extension, but some nontrivial forcing notions have all their extensions amalgamable. An increasing chain $W[G_0]\subseteq W[G_1]\subseteq\cdots$ has an upper bound $W[H]$ if and only if the forcing had uniformly bounded essential size in $W$. Every chain $W\subseteq W[c_0]\subseteq W[c_1]\subseteq \cdots$ of extensions adding Cohen reals is bounded above by $W[d]$ for some $W$-generic Cohen real $d$.

This article is based upon I talk I gave at the conference on Recent Developments in Axiomatic Set Theory at the Research Institute for Mathematical Sciences (RIMS) at Kyoto University, Japan in September, 2015, and I am extremely grateful to my Japanese hosts, especially Toshimichi Usuba, for supporting my research visit there and also at the CTFM conference at Tokyo Institute of Technology just preceding it. This article includes material adapted from section section 2 of Set-theoretic geology, joint with G. Fuchs, myself and J. Reitz, and also includes a theorem that was proved in a series of conversations I had with Giorgio Venturi at the Young Set Theory Workshop 2011 in Bonn and continuing at the London 2011 summer school on set theory at Birkbeck University London.

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

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 $M\models\text{ZFC}$, 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
$$M[c_0]\subset M[c_1]\subset\cdots\subset M[c_n]\subset\cdots,$$
we may find an $M$-generic Cohen real $d$ such that $M[c_n]\subset M[d]$ for every natural number $n$.

Proof. $\newcommand\Add{\text{Add}}$Suppose that we have such a tower of forcing extensions $M[c_0]\subset M[c_1]\subset\cdots$, 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:
$$M[b_0]\subset M[b_0\times b_1]\subset\cdots\subset M[b_0\times b_1\times\cdots\times b_n]\subset\cdots,$$
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 $\langle b_n\mid n<\omega\rangle$ is $M$-generic for $\Add(\omega,\omega)$, 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 $\omega$-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 $2^\omega$, 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 $2^{<\omega}$, which is one of the standard isomorphic copies of Cohen forcing. We will think of $d$ as a binary function on the plane $d:\omega\times\omega\to 2$, where the $n^{th}$ slice $d_n$ is the corresponding function $\omega\to 2$ 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 $\omega\times\omega\to 2$ 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 $\omega$ many Cohen reals. We consider the dense set $D_n$. We may factor $\Add(\omega,\omega)$ as $\Add(\omega,n)\times\Add(\omega,[n,\omega))$. Since $d_0\times\cdots\times d_{n-1}$ 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 $d_0\times\cdots\times d_{n-1}$. 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 $\Add(\omega,\omega)$, yet every $b_n$ will differ only finitely from $d_n$ and hence be an element of $M[d]$. So we have $M[b_0]\cdots[b_n]\subset M[d]$, 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.

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

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 $M[G_0]\subset M[G_1]\subset M[G_2]\subset\cdots$, 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 $\langle G_n\mid n<\omega\rangle$ 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.

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

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 $\Delta_0$-elementary. Indeed, the embeddability result is surprising precisely because we can easily prove that in many of these instances, there can be no $\Delta_0$-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 $\mathbb{Q}$-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 $\omega_1$-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 $0^\sharp$ 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 $\langle P(\omega),\in\rangle$ embeds into $\langle P(\omega)^L,\in\rangle$.

# Embeddings of the universe into the constructible universe, current state of knowledge, CUNY Set Theory Seminar, March 2015

This will be a talk for the CUNY Set Theory Seminar, March 6, 2015.

I shall describe the current state of knowledge concerning the question of whether there can be an embedding of the set-theoretic universe into the constructible universe.

Question.(Hamkins) Can there be an embedding $j:V\to L$ of the set-theoretic universe $V$ into the constructible universe $L$, when $V\neq L$?

The notion of embedding here is merely that $$x\in y\iff j(x)\in j(y),$$ and such a map need not be elementary nor even $\Delta_0$-elementary. It is not difficult to see that there can generally be no $\Delta_0$-elementary embedding $j:V\to L$, when $V\neq L$.

Nevertheless, the question arises very naturally in the context of my previous work on the embeddability phenomenon, Every countable model of set theory embeds into its own constructible universe, where the title theorem is the following.

Theorem.(Hamkins) Every countable model of set theory $\langle M,\in^M\rangle$, including every countable transitive model of set theory, has an embedding $j:\langle M,\in^M\rangle\to\langle L^M,\in^M\rangle$ into its own constructible universe.

The methods of proof also established that the countable models of set theory are linearly pre-ordered by embeddability: given any two models, one of them embeds into the other; or equivalently, one of them is isomorphic to a submodel of the other. Indeed, one model $\langle M,\in^M\rangle$ embeds into another $\langle N,\in^N\rangle$ just in case the ordinals of the first $\text{Ord}^M$ order-embed into the ordinals of the second $\text{Ord}^N$. (And this implies the theorem above.)

In the proof of that theorem, the embeddings $j:M\to L^M$ are defined completely externally to $M$, and so it was natural to wonder to what extent such an embedding might be accessible inside $M$. And I realized that I could not generally refute the possibility that such a $j$ might even be a class in $M$.

Currently, the question remains open, but we have some partial progress, and have settled it in a number of cases, including the following, on which I’ll speak:

• If there is an embedding $j:V\to L$, then for a proper class club of cardinals $\lambda$, we have $(2^\lambda)^V=(\lambda^+)^L$.
• If $0^\sharp$ exists, then there is no embedding $j:V\to L$.
• If $0^\sharp$ exists, then there is no embedding $j:V\to L$ and indeed no embedding $j:P(\omega)\to L$.
• If there is an embedding $j:V\to L$, then the GCH holds above $\aleph_0$.
• In the forcing extension $V[G]$ obtained by adding $\omega_1$ many Cohen reals (or more), there is no embedding $j:V[G]\to L$, and indeed, no $j:P(\omega)^{V[G]}\to V$. More generally, after adding $\kappa^+$ many Cohen subsets to $\kappa$, for any regular cardinal $\kappa$, then in $V[G]$ there is no $j:P(\kappa)\to V$.
• If $V$ is a nontrivial set-forcing extension of an inner model $M$, then there is no embedding $j:V\to M$. Indeed, there is no embedding $j:P(\kappa^+)\to M$, if the forcing has size $\kappa$. In particular, if $V$ is a nontrivial forcing extension, then there is no embedding $j:V\to L$.
• Every countable set $A$ has an embedding $j:A\to L$.

This is joint work of myself, W. Hugh Woodin, Menachem Magidor, with contributions also by David Aspero, Ralf Schindler and Yair Hayut.

See my related MathOverflow question: Can there be an embedding $j:V\to L$ from the set-theoretic universe $V$ to the constructible universe $L$, when $V\neq L$?

Talk Abstract

# Incomparable $\omega_1$-like models of set theory

• G. Fuchs, V. Gitman, and J. D. Hamkins, “Incomparable $\omega_1$-like models of set theory,” Mathematical Logic Quarterly, pp. 1-11, 2017.
@article {FuchsGitmanHamkins2017:IncomparableOmega1-likeModelsOfSetTheory,
author = {Fuchs, Gunter and Gitman, Victoria and Hamkins, Joel David},
title = {Incomparable $\omega_1$-like models of set theory},
journal = {Mathematical Logic Quarterly},
issn = {1521-3870},
url = {},
doi = {10.1002/malq.201500002},
pages = {1--11},
year = {2017},
month = {March},
eprint = {1501.01022},
archivePrefix = {arXiv},
primaryClass = {math.LO},
url = {http://jdh.hamkins.org/incomparable-omega-one-like-models-of-set-theory},
}

This is joint work with Gunter Fuchs and Victoria Gitman.

Abstract. We show that the analogues of the Hamkins embedding theorems, proved for the countable models of set theory, do not hold when extended to the uncountable realm of $\omega_1$-like models of set theory. Specifically, under the $\diamondsuit$ hypothesis and suitable consistency assumptions, we show that there is a family of $2^{\omega_1}$ many $\omega_1$-like models of $\text{ZFC}$, all with the same ordinals, that are pairwise incomparable under embeddability; there can be a transitive $\omega_1$-like model of ZFC that does not embed into its own constructible universe; and there can be an $\omega_1$-like model of PA whose structure of hereditarily finite sets is not universal for the $\omega_1$-like models of set theory.

In this article, we consider the question of whether the embedding theorems of my article, Every countable model of set theory embeds into its own constructible universe, which concern the countable models of set theory, might extend to the realm of uncountable models. Specifically, in that paper I had proved that (1) any two countable models of set theory are comparable by embeddability; indeed, (2) one countable model of set theory embeds into another just in case the ordinals of the first order-embed into the ordinals of the second; consequently, (3) every countable model of set theory embeds into its own constructible universe; and furthermore, (4) every countable model of set theory embeds into the hereditarily finite sets $\langle\text{HF},{\in}\rangle^M$ of any nonstandard model of arithmetic $M\models\text{PA}$. The question we consider here is, do the analogous results hold for uncountable models? Our answer is that they do not. Indeed, we shall prove that the corresponding statements do not hold even in the special case of $\omega_1$-like models of set theory, which otherwise among uncountable models often exhibit a special affinity with the countable models. Specifically, we shall construct large families of pairwise incomparable $\omega_1$-like models of set theory, even though they all have the same ordinals; we shall construct $\omega_1$-like models of set theory that do not embed into their own $L$; and we shall construct $\omega_1$-like models of \PA\ that are not universal for all $\omega_1$-like models of set theory.

The embedding theorems are expressed collectively in the theorem below. An embedding of one model $\langle M,{\in^M}\rangle$ of set theory into another $\langle N,{\in^N}\rangle$ is simply a function $j:M\to N$ for which $x\in^My\longleftrightarrow j(x)\in^Nj(y)$, for all $x,y\in M$, and in this case we say that $\langle M,{\in^M}\rangle$ embeds into $\langle N,{\in^N}\rangle$; note by extensionality that every embedding is injective. Thus, an embedding is simply an isomorphism of $\langle M,{\in^M}\rangle$ with its range, which is a submodel of $\langle N,{\in^N}\rangle$. Although this is the usual model-theoretic embedding concept for relational structures, the reader should note that it is a considerably weaker embedding concept than commonly encountered in set theory, because this kind of embedding need not be elementary nor even $\Delta_0$-elementary, although clearly every embedding as just defined is elementary at least for quantifier-free assertions. So we caution the reader not to assume a greater degree of elementarity beyond quantifier-free elementarity for the embeddings appearing in this paper.

Theorem.

1. For any two countable models of set theory $\langle M,\in^M\rangle$ and $\langle N,\in^N\rangle$, one of them embeds into the other.

2. Indeed, such an $\langle M,{\in^M}\rangle$ embeds into $\langle N,{\in^N}\rangle$ if and only if the ordinals of $M$ order-embed into the ordinals of $N$.

3. Consequently, every countable model $\langle M,\in^M\rangle$ of set theory embeds into its own constructible universe $\langle L^M,\in^M\rangle$.

4. Furthermore, every countable model of set theory embeds into the hereditary finite sets $\langle \text{HF},{\in}\rangle^M$ of any nonstandard model of arithmetic $M\models\text{PA}$. Indeed, $\text{HF}^M$ is universal for all countable acyclic binary relations.

One can begin to get an appreciation for the difference in embedding concepts by observing that ZFC proves that there is a nontrivial embedding $j:V\to V$, namely, the embedding recursively defined as follows $$j(y)=\bigl\{\ j(x)\ \mid\ x\in y\ \bigr\}\cup\bigl\{\{\emptyset,y\}\bigr\}.$$

We leave it as a fun exercise to verify that $x\in y\longleftrightarrow j(x)\in j(y)$ for the embedding $j$ defined by this recursion. (See my paper Every countable model of set theory embeds into its own constructible universe; but to give a hint here for the impatient, note that every $j(y)$ is nonempty and also $\emptyset\notin j(y)$; it follows that inside $j(y)$ we may identify the pair $\{\emptyset,y\}\in j(y)$; it follows that $j$ is injective and furthermore, the only way to have $j(x)\in j(y)$ is from $x\in y$.} Contrast this situation with the well-known Kunen inconsistency, which asserts that there can be no nontrivial $\Sigma_1$-elementary embedding $j:V\to V$. Similarly, the same recursive definition applied in $L$ leads to nontrivial embeddings $j:L\to L$, regardless of whether $0^\sharp$ exists. But again, the point is that embeddings are not necessarily even $\Delta_0$-elementary, and the familiar equivalence of the existence of $0^\sharp$ with a nontrivial “embedding” $j:L\to L$ actually requires a $\Delta_0$-elementary embedding.)

We find it interesting to note in contrast to the theorem above that there is no such embedding phenomenon in the the context of the countable models of Peano arithmetic (where an embedding of models of arithmetic is a function preserving all atomic formulas in the language of arithmetic). Perhaps the main reason for this is that embeddings between models of PA are automatically $\Delta_0$-elementary, as a consequence of the MRDP theorem, whereas this is not true for models of set theory, as the example above of the recursively defined embedding $j:V\to V$ shows, since this is an embedding, but it is not $\Delta_0$-elementary, in light of $j(\emptyset)\neq\emptyset$. For countable models of arithmetic $M,N\models\text{PA}$, one can show that there is an embedding $j:M\to N$ if and only if $N$ satisfies the $\Sigma_1$-theory of $M$ and the standard system of $M$ is contained in the standard system of $N$. It follows that there are many instances of incomparability. Meanwhile, it is a consequence of statement (4) that the embedding phenomenon recurs with the countable models of finite set theory $\text{ZFC}^{\neg\infty}$, that is, with $\langle\text{HF},{\in}\rangle^M$ for $M\models\text{PA}$, since all nonstandard such models are universal for all countable acyclic binary relations, and so in the context of countable models of $\text{ZFC}^{\neg\infty}$ there are precisely two bi-embeddability classes, namely, the standard model, which is initial, and the nonstandard countable models, which are universal.

Our main theorems are as follows.

Theorem.

1. If $\diamondsuit$ holds and ZFC is consistent, then there is a family $\mathcal C$ of $2^{\omega_1}$ many pairwise incomparable $\omega_1$-like models of ZFC, meaning that there is no embedding between any two distinct models in $\mathcal C$.

2. The models in statement (1) can be constructed so that their ordinals order-embed into each other and indeed, so that the ordinals of each model is a universal $\omega_1$-like linear order. If ZFC has an $\omega$-model, then the models of statement (1) can be constructed so as to have precisely the same ordinals.

3. If $\diamondsuit$ holds and ZFC is consistent, then there is an $\omega_1$-like model $M\models\text{ZFC}$ and an $\omega_1$-like model $N\models\text{PA}$ such that $M$ does not embed into $\langle\text{HF},{\in}\rangle^N$.

4. If there is a Mahlo cardinal, then in a forcing extension of $L$, there is a transitive $\omega_1$-like model $M\models\text{ZFC}$ that does not embed into its own constructible universe $L^M$.

Note that the size of the family $\mathcal C$ in statement (1) is as large as it could possibly be, given that any two elements in a pairwise incomparable family of structures must be non-isomorphic and there are at most $2^{\omega_1}$ many isomorphism types of $\omega_1$-like models of set theory or indeed of structures of size $\omega_1$ in any first-order finite language. Statement (2) shows that the models of the family $\mathcal C$ serve as $\omega_1$-like counterexamples to the assertion that one model of set theory embeds into another whenever the ordinals of the first order-embed into the ordinals of the second.

# Upward closure in the toy multiverse of all countable models of set theory

The toy multiverse of all countable models of set theory is upward closed under countably many successive forcing extensions of bounded size…

I’d like to explain a topic from my recent paper

G. Fuchs, J. D. Hamkins, J. ReitzSet-theoretic geology, to appear in the Annals of Pure and Applied Logic.

We just recently made the final revisions, and the paper is available if you follow the title link through to the arxiv. Most of the geology article proceeds from a downward-oriented focus on forcing, looking from a universe $V$ down to its grounds, the inner models $W$ over which $V$ might have arisen by forcing $V=W[G]$. Thus, the set-theoretic geology project arrives at deeper and deeper grounds and the mantle and inner mantle concepts.

One section of the paper, however, has an upward-oriented focus, namely, $\S2$ A brief upward glance, and it is that material about which I’d like to write here, because I find it to be both interesting and comparatively accessible, but also because the topic proceeds from a different perspective than the rest of the geology paper, and so I am a little fearful that it may get lost there.

First is the observation that I first heard from W. Hugh Woodin in the early 1990s.


Observation. If $W$ is a countable model of ZFC set theory, then there are forcing extensions $W[c]$ and $W[d]$, both obtained by adding a Cohen real, which are non-amalgamable in the sense that there can be no model of ZFC with the same ordinals as $W$ containing both $W[c]$ and $W[d]$. Thus, the family of forcing extensions of $W$ is not upward directed.

Proof. Since $W$ is countable, let $z$ be a real coding the entirety of $W$. Enumerate the dense subsets $\langle D_n\mid n<\omega\rangle$ of the Cohen forcing $\text{Add}(\omega,1)$ in $W$. We construct $c$ and $d$ in stages. We begin by letting $c_0$ be any element of $D_0$. Let $d_0$ consist of exactly as many $0$s as $|c_0|$, followed by a $1$, followed by $z(0)$, and then extended to an element of $D_0$. Continuing, $c_{n+1}$ extends $c_n$ by adding $0$s until the length of $d_n$, and then a $1$, and then extending into $D_{n+1}$; and $d_{n+1}$ extends $d_n$ by adding $0$s to the length of $c_{n+1}$, then a $1$, then $z(n)$, then extending into $D_{n+1}$. Let $c=\bigcup c_n$ and $d=\bigcup d_n$. Since we met all the dense sets in $W$, we know that $c$ and $d$ are $W$-generic Cohen reals, and so we may form the forcing extensions $W[c]$ and $W[d]$. But if $W\subset U\models\text{ZFC}$ and both $c$ and $d$ are in $U$, then in $U$ we may reconstruct the map $n\mapsto\langle c_n,d_n\rangle$, by giving attention to the blocks of $0$s in $c$ and $d$. From this map, we may reconstruct $z$ in $U$, which reveals all the ordinals of $W$ to be countable, a contradiction if $U$ and $W$ have the same ordinals. QED

Most of the results here concern forcing extensions of an arbitrary countable model of set theory, which of course includes the case of ill-founded models. Although there is no problem with forcing extensions of ill-founded models, when properly carried out, the reader may prefer to focus on the case of countable transitive models for the results in this section, and such a perspective will lose very little of the point of our observations.

The method of the observation above is easily generalized to produce three $W$-generic Cohen reals $c_0$, $c_1$ and $c_2$, such that any two of them can be amalgamated, but the three of them cannot. More generally:

Observation. If $W$ is a countable model of ZFC set theory, then for any finite $n$ there are $W$-generic Cohen reals $c_0,c_1,\ldots,c_{n-1}$, such that any proper subset of them are mutually $W$-generic, so that one may form the generic extension $W[\vec c]$, provided that $\vec c$ omits at least one $c_i$, but there is no forcing extension $W[G]$ simultaneously extending all $W[c_i]$ for $i<n$. In particular, the sequence $\langle c_0,c_1,\ldots,c_{n-1}\rangle$ cannot be added by forcing over $W$.

Let us turn now to infinite linearly ordered sequences of forcing extensions. We show first in the next theorem and subsequent observation that one mustn’t ask for too much; but nevertheless, after that we shall prove the surprising positive result, that any increasing sequence of forcing extensions over a countable model $W$, with forcing of uniformly bounded size, is bounded above by a single forcing extension $W[G]$.

Theorem. If $W$ is a countable model of ZFC, then there is an increasing sequence of set-forcing extensions of $W$ having no upper bound in the generic multiverse of $W$. $$W[G_0]\of W[G_1]\of\cdots\of W[G_n]\of\cdots$$

Proof. Since $W$ is countable, there is an increasing sequence $\langle\gamma_n\mid n<\omega\rangle$ of ordinals that is cofinal in the ordinals of $W$. Let $G_n$ be $W$-generic for the collapse forcing $\text{Coll}(\omega,\gamma_n)$, as defined in $W$. (By absorbing the smaller forcing, we may arrange that $W[G_n]$ contains $G_m$ for $m<n$.) Since every ordinal of $W$ is eventually collapsed, there can be no set-forcing extension of $W$, and indeed, no model with the same ordinals as $W$, that contains every $W[G_n]$. QED

But that was cheating, of course, since the sequence of forcing notions is not even definable in $W$, as the class $\{\gamma_n\mid n<\omega\}$ is not a class of $W$. A more intriguing question would be whether this phenomenon can occur with forcing notions that constitute a set in $W$, or (equivalently, actually) whether it can occur using always the same poset in $W$. For example, if $W[c_0]\of W[c_0][c_1]\of W[c_0][c_1][c_2]\of\cdots$ is an increasing sequence of generic extensions of $W$ by adding Cohen reals, then does it follow that there is a set-forcing extension $W[G]$ of $W$ with $W[c_0]\cdots[c_n]\of W[G]$ for every $n$? For this, we begin by showing that one mustn’t ask for too much:

Observation. If $W$ is a countable model of ZFC, then there is a sequence of forcing extensions $W\of W[c_0]\of W[c_0][c_1]\of W[c_0][c_1][c_2]\of\cdots$, adding a Cohen real at each step, such that there is no forcing extension of $W$ containing the sequence $\langle c_n\mid n<\omega\rangle$.

Proof. Let $\langle d_n\mid n<\omega\rangle$ be any $W$-generic sequence for the forcing to add $\omega$ many Cohen reals over $W$. Let $z$ be any real coding the ordinals of $W$. Let us view these reals as infinite binary sequences. Define the real $c_n$ to agree with $d_n$ on all digits except the initial digit, and set $c_n(0)=z(n)$. That is, we make a single-bit change to each $d_n$, so as to code one additional bit of $z$. Since we have made only finitely many changes to each $d_n$, it follows that $c_n$ is an $W$-generic Cohen real, and also $W[c_0]\cdots[c_n]=W[d_0]\cdots [d_n]$. Thus, we have $$W\of W[c_0]\of W[c_0][c_1]\of W[c_0][c_1][c_2]\of\cdots,$$ adding a generic Cohen real at each step. But there can be no forcing extension of $W$ containing $\langle c_n\mid n<\omega\rangle$, since any such extension would have the real $z$, revealing all the ordinals of $W$ to be countable. QED

We can modify the construction to allow $z$ to be $W$-generic, but collapsing some cardinals of $W$. For example, for any cardinal $\delta$ of $W$, we could let $z$ be $W$-generic for the collapse of $\delta$. Then, if we construct the sequence $\langle c_n\mid n<\omega\rangle$ as above, but inside $W[z]$, we get a sequence of Cohen real extensions $$W\of W[c_0]\of W[c_0][c_1]\of W[c_0][c_1][c_2]\of\cdots$$ such that $W[\langle c_n\mid n<\omega\rangle]=W[z]$, which collapses $\delta$.

But of course, the question of whether the models $W[c_0][c_1]\cdots[c_n]$ have an upper bound is not the same question as whether one can add the sequence $\langle c_n\mid n<\omega\rangle$, since an upper bound may not have this sequence. And in fact, this is exactly what occurs, and we have a surprising positive result:

Theorem. Suppose that $W$ is a countable model of \ZFC, and $$W[G_0]\of W[G_1]\of\cdots\of W[G_n]\of\cdots$$ is an increasing sequence of forcing extensions of $W$, with $G_n\of\Q_n\in W$ being $W$-generic. If the cardinalities of the $\Q_n$’s in $W$ are bounded in $W$, then there is a set-forcing extension $W[G]$ with $W[G_n]\of W[G]$ for all $n<\omega$.

Proof. Let us first make the argument in the special case that we have $$W\of W[g_0]\of W[g_0][g_1]\of\cdots\of W[g_0][g_1]\cdots[g_n]\of\cdots,$$ where each $g_n$ is generic over the prior model for forcing $\Q_n\in W$. That is, each extension $W[g_0][g_1]\cdots[g_n]$ is obtained by product forcing $\Q_0\cross\cdots\cross\Q_n$ over $W$, and the $g_n$ are mutually $W$-generic. Let $\delta$ be a regular cardinal with each $\Q_n$ having size at most $\delta$, built with underlying set a subset of $\delta$. In $W$, let $\theta=2^\delta$, let $\langle \R_\alpha\mid\alpha<\theta\rangle$ enumerate all posets of size at most $\delta$, with unbounded repetition, and let $\P=\prod_{\alpha<\theta}\R_\alpha$ be the finite-support product of these posets. Since each factor is $\delta^+$-c.c., it follows that the product is $\delta^+$-c.c. Since $W$ is countable, we may build a filter $H\of\P$ that is $W$-generic. In fact, we may find such a filter $H\of\P$ that meets every dense set in $\bigcup_{n<\omega}W[g_0][g_1]\cdots[g_n]$, since this union is also countable. In particular, $H$ and $g_0\cross\cdots\cross g_n$ are mutually $W$-generic for every $n<\omega$. The filter $H$ is determined by the filters $H_\alpha\of\R_\alpha$ that it adds at each coordinate.

Next comes the key step. Externally to $W$, we may find an increasing sequence $\langle \theta_n\mid n<\omega\rangle$ of ordinals cofinal in $\theta$, such that $\R_{\theta_n}=\Q_n$. This is possible because the posets are repeated unboundedly, and $\theta$ is countable in $V$. Let us modify the filter $H$ by surgery to produce a new filter $H^*$, by changing $H$ at the coordinates $\theta_n$ to use $g_n$ rather than $H_{\theta_n}$. That is, let $H^*_{\theta_n}=g_n$ and otherwise $H^*_\alpha=H_\alpha$, for $\alpha\notin\{\theta_n\mid n<\omega\}$. It is clear that $H^*$ is still a filter on $\P$. We claim that $H^*$ is $W$-generic. To see this, suppose that $A\of\P$ is any maximal antichain in $W$. By the $\delta^+$-chain condition and the fact that $\text{cof}(\theta)^W>\delta$, it follows that the conditions in $A$ have support bounded by some $\gamma<\theta$. Since the $\theta_n$ are increasing and cofinal in $\theta$, only finitely many of them lay below $\gamma$, and we may suppose that there is some largest $\theta_m$ below $\gamma$. Let $H^{**}$ be the filter derived from $H$ by performing the surgical modifications only on the coordinates $\theta_0,\ldots,\theta_m$. Thus, $H^*$ and $H^{**}$ agree on all coordinates below $\gamma$. By construction, we had ensured that $H$ and $g_0\cross\cdots\cross g_m$ are mutually generic over $W$ for the forcing $\P\cross\Q_0\cross\cdots\cross\Q_m$. This poset has an automorphism swapping the latter copies of $\Q_i$ with their copy at $\theta_i$ in $\P$, and this automorphism takes the $W$-generic filter $H\cross g_0\cross\cdots\cross g_m$ exactly to $H^{**}\cross H_{\theta_0}\cross\cdots \cross H_{\theta_m}$. In particular, $H^{**}$ is $W$-generic for $\P$, and so $H^{**}$ meets the maximal antichain $A$. Since $H^*$ and $H^{**}$ agree at coordinates below $\gamma$, it follows that $H^*$ also meets $A$. In summary, we have proved that $H^*$ is $W$-generic for $\P$, and so $W[H^*]$ is a set-forcing extension of $W$. By design, each $g_n$ appears at coordinate $\theta_n$ in $H^*$, and so $W[g_0]\cdots[g_n]\of W[H^*]$ for every $n<\omega$, as desired.

Finally, we reduce the general case to this special case. Suppose that $W[G_0]\of W[G_1]\of\cdots\of W[G_n]\of\cdots$ is an increasing sequence of forcing extensions of $W$, with $G_n\of\Q_n\in W$ being $W$-generic and each $\Q_n$ of size at most $\kappa$ in $W$. By the standard facts surrounding finite iterated forcing, we may view each model as a forcing extension of the previous model $$W[G_{n+1}]=W[G_n][H_n],$$ where $H_n$ is $W[G_n]$-generic for the corresponding quotient forcing $\Q_n/G_n$ in $W[G_n]$. Let $g\of\text{Coll}(\omega,\kappa)$ be $\bigcup_n W[G_n]$-generic for the collapse of $\kappa$, so that it is mutually generic with every $G_n$. Thus, we have the increasing sequence of extensions $W[g][G_0]\of W[g][G_1]\of\cdots$, where we have added $g$ to each model. Since each $\Q_n$ is countable in $W[g]$, it is forcing equivalent there to the forcing to add a Cohen real. Furthermore, the quotient forcing $\Q_n/G_n$ is also forcing equivalent in $W[g][G_n]$ to adding a Cohen real. Thus, $W[g][G_{n+1}]=W[g][G_n][H_n]=W[g][G_n][h_n]$, for some $W[g][G_n]$-generic Cohen real $h_n$. Unwrapping this recursion, we have $W[g][G_{n+1}]=W[g][G_0][h_1]\cdots[h_n]$, and consequently $$W[g]\of W[g][G_0]\of W[g][G_0][h_1]\of W[g][G_0][h_1][h_2]\of\cdots,$$ which places us into the first case of the proof, since this is now product forcing rather than iterated forcing. QED

Definition. A collection $\{W[G_n]\mid n<\omega\}$ of forcing extensions of $W$ is finitely amalgamable over $W$ if for every $n<\omega$ there is a forcing extension $W[H]$ with $W[G_m]\of W[H]$ for all $m\leq n$. It is amalgamable over $W$ if there is $W[H]$ such that $W[G_n]\of W[H]$ for all $n<\omega$.

The next corollary shows that we cannot improve the non-amalgamability result of the initial observation to the case of infinitely many Cohen reals, with all finite subsets amalgamable.

Corollary. If $W$ is a countable model of ZFC and $\{W[G_n]\mid n<\omega\}$ is a finitely amalgamable collection of forcing extensions of $W$, using forcing of bounded size in $W$, then this collection is fully amalgamable. That is, there is a forcing extension $W[H]$ with $W[G_n]\of W[H]$ for all $n<\omega$.

Proof. Since the collection is finitely amalgamable, for each $n<\omega$ there is some $W$-generic $K$ such that $W[G_m]\of W[K]$ for all $m\leq n$. Thus, we may form the minimal model $W[G_0][G_1]\cdots[G_n]$ between $W$ and $W[K]$, and thus $W[G_0][G_1]\cdots [G_n]$ is a forcing extension of $W$. We are thus in the situation of the theorem, with an increasing chain of forcing extensions. $$W\of W[G_0]\of W[G_0][G_1]\of\cdots\of W[G_0][G_1]\cdots[G_n]\of\cdots$$ Therefore, by the theorem, there is a model $W[H]$ containing all these extensions, and in particular, $W[G_n]\of W[H]$, as desired. QED

Please go to the paper for more details and discussion.

# The pluralist perspective on the axiom of constructibility, MidWest PhilMath Workshop, Notre Dame, October 2014

This will be a featured talk at the Midwest PhilMath Workshop 15, held at Notre Dame University October 18-19, 2014.  W. Hugh Woodin and I will each give one-hour talks in a session on Perspectives on the foundations of set theory, followed by a one-hour discussion of our talks.

Abstract. I shall argue that the commonly held $V\neq L$ via maximize position, which rejects the axiom of constructibility V = L on the basis that it is restrictive, implicitly takes a stand in the pluralist debate in the philosophy of set theory by presuming an absolute background concept of ordinal. The argument appears to lose its force, in contrast, on an upwardly extensible concept of set, in light of the various facts showing that models of set theory generally have extensions to models of V = L inside larger set-theoretic universes.

Set-theorists often argue against the axiom of constructibility V=L on the grounds that it is restrictive, that we have no reason to suppose that every set should be constructible and that it places an artificial limitation on set-theoretic possibility to suppose that every set is constructible. Penelope Maddy, in her work on naturalism in mathematics, sought to explain this perspective by means of the MAXIMIZE principle, and further to give substance to the concept of what it means for a theory to be restrictive, as a purely formal property of the theory. In this talk, I shall criticize Maddy’s proposal, pointing out that neither the fairly-interpreted-in relation nor the (strongly) maximizes-over relation is transitive, and furthermore, the theory ZFC + there is a proper class of inaccessible cardinals’ is formally restrictive on Maddy’s account, contrary to what had been desired. Ultimately, I shall argue that the V≠L via maximize position loses its force on a multiverse conception of set theory with an upwardly extensible concept of set, in light of the classical facts that models of set theory can generally be extended to models of V=L. I shall conclude the talk by explaining various senses in which V=L remains compatible with strength in set theory.

This talk will be based on my paper, A multiverse perspective on the axiom of constructibility.

Slides

# Satisfaction is not absolute

• J. D. Hamkins and R. Yang, “Satisfaction is not absolute,” to appear in the Review of Symbolic Logic, pp. 1-34, 2014.
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author = {Joel David Hamkins and Ruizhi Yang},
title = {Satisfaction is not absolute},
journal = {to appear in the Review of Symbolic Logic},
year = {2014},
volume = {},
number = {},
pages = {1--34},
month = {},
note = {},
abstract = {},
keywords = {},
source = {},
eprint = {1312.0670},
archivePrefix = {arXiv},
primaryClass = {math.LO},
url = {http://jdh.hamkins.org/satisfaction-is-not-absolute},
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}


Abstract. We prove that the satisfaction relation $\mathcal{N}\satisfies\varphi[\vec a]$ of first-order logic is not absolute between models of set theory having the structure $\mathcal{N}$ and the formulas $\varphi$ all in common. Two models of set theory can have the same natural numbers, for example, and the same standard model of arithmetic $\langle\N,{+},{\cdot},0,1,{\lt}\rangle$, yet disagree on their theories of arithmetic truth; two models of set theory can have the same natural numbers and the same arithmetic truths, yet disagree on their truths-about-truth, at any desired level of the iterated truth-predicate hierarchy; two models of set theory can have the same natural numbers and the same reals, yet disagree on projective truth; two models of set theory can have the same $\langle H_{\omega_2},{\in}\rangle$ or the same rank-initial segment $\langle V_\delta,{\in}\rangle$, yet disagree on which assertions are true in these structures.

On the basis of these mathematical results, we argue that a philosophical commitment to the determinateness of the theory of truth for a structure cannot be seen as a consequence solely of the determinateness of the structure in which that truth resides. The determinate nature of arithmetic truth, for example, is not a consequence of the determinate nature of the arithmetic structure $\N=\{ 0,1,2,\ldots\}$ itself, but rather, we argue, is an additional higher-order commitment requiring its own analysis and justification.

Many mathematicians and philosophers regard the natural numbers $0,1,2,\ldots\,$, along with their usual arithmetic structure, as having a privileged mathematical existence, a Platonic realm in which assertions have definite, absolute truth values, independently of our ability to prove or discover them. Although there are some arithmetic assertions that we can neither prove nor refute—such as the consistency of the background theory in which we undertake our proofs—the view is that nevertheless there is a fact of the matter about whether any such arithmetic statement is true or false in the intended interpretation. The definite nature of arithmetic truth is often seen as a consequence of the definiteness of the structure of arithmetic $\langle\N,{+},{\cdot},0,1,{\lt}\rangle$ itself, for if the natural numbers exist in a clear and distinct totality in a way that is unambiguous and absolute, then (on this view) the first-order theory of truth residing in that structure—arithmetic truth—is similarly clear and distinct.

Feferman provides an instance of this perspective when he writes (Feferman 2013, Comments for EFI Workshop, p. 6-7) :

In my view, the conception [of the bare structure of the natural numbers] is completely clear, and thence all arithmetical statements are definite.

It is Feferman’s thence’ to which we call attention.  Martin makes a similar point (Martin, 2012, Completeness or incompleteness of basic mathematical concepts):

What I am suggesting is that the real reason for confidence in first-order completeness is our confidence in the full determinateness of the concept of the natural numbers.

Many mathematicians and philosophers seem to share this perspective. The truth of an arithmetic statement, to be sure, does seem to depend entirely on the structure $\langle\N,{+},{\cdot},0,1,{\lt}\rangle$, with all quantifiers restricted to $\N$ and using only those arithmetic operations and relations, and so if that structure has a definite nature, then it would seem that the truth of the statement should be similarly definite.

Nevertheless, in this article we should like to tease apart these two ontological commitments, arguing that the definiteness of truth for a given mathematical structure, such as the natural numbers, the reals or higher-order structures such as $H_{\omega_2}$ or $V_\delta$, does not follow from the definite nature of the underlying structure in which that truth resides. Rather, we argue that the commitment to a theory of truth for a structure is a higher-order ontological commitment, going strictly beyond the commitment to a definite nature for the underlying structure itself.

We make our argument in part by proving that different models of set theory can have a structure identically in common, even the natural numbers, yet disagree on the theory of truth for that structure.

Theorem.

• Two models of set theory can have the same structure of arithmetic $$\langle\N,{+},{\cdot},0,1,{\lt}\rangle^{M_1}=\langle\N,{+},{\cdot},0,1,{\lt}\rangle^{M_2},$$yet disagree on the theory of arithmetic truth.
• Two models of set theory can have the same natural numbers and a computable linear order in common, yet disagree about whether it is a well-order.
• Two models of set theory that have the same natural numbers and the same reals, yet disagree on projective truth.
• Two models of set theory can have a transitive rank initial segment in common $$\langle V_\delta,{\in}\rangle^{M_1}=\langle V_\delta,{\in}\rangle^{M_2},$$yet disagree about whether it is a model of ZFC.

The proofs use only elementary classical methods, and might be considered to be a part of the folklore of the subject of models of arithmetic. The paper includes many further examples of the phenomenon, and concludes with a philosophical discussion of the issue of definiteness, concerning the question of whether one may deduce definiteness-of-truth from definiteness-of-objects and definiteness-of-structure.

# On the axiom of constructibility and Maddy’s conception of restrictive theories, Logic Workshop, February 2013

This is a talk for the CUNY Logic Workshop on February 15, 2013.

This talk will be based on my paper, A multiverse perspective on the axiom of constructibility.

Set-theorists often argue against the axiom of constructibility $V=L$ on the grounds that it is restrictive, that we have no reason to suppose that every set should be constructible and that it places an artificial limitation on set-theoretic possibility to suppose that every set is constructible.  Penelope Maddy, in her work on naturalism in mathematics, sought to explain this perspective by means of the MAXIMIZE principle, and further to give substance to the concept of what it means for a theory to be restrictive, as a purely formal property of the theory.

In this talk, I shall criticize Maddy’s specific proposal.  For example, it turns out that the fairly-interpreted-in relation on theories is not transitive, and similarly the maximizes-over and strongly-maximizes-over relations are not transitive.  Further, the theory ZFC + `there is a proper class of inaccessible cardinals’ is formally restrictive on Maddy’s proposal, although this is not what she had desired.

Ultimately, I argue that the $V\neq L$ via maximize position loses its force on a multiverse conception of set theory, in light of the classical facts that models of set theory can generally be extended to (taller) models of $V=L$.  In particular, every countable model of set theory is a transitive set inside a model of $V=L$.  I shall conclude the talk by explaining various senses in which $V=L$ remains compatible with strength in set theory.