## Set Theory Day at the CUNY Graduate Center, March 11, 2016

### Featured

Vika Gitman, Roman Kossak and Miha Habič have been very kind to organize what they have called Set Theory Day, to be held Friday March 11 at the CUNY Graduate Center in celebration of my 50th birthday. This will be an informal conference focussing on the research work of my various PhD graduate students, and all the lectures will be given by those who were or are currently a PhD student of mine. It will be great! I am very pleased to count among my former students many who have now become mathematical research colleagues and co-authors of mine, and I am looking forward to hearing the latest. If you want to hear what is going on with infinity, then please join us March 11 at the CUNY Graduate Center!

(The poster was designed by my student Erin Carmody, who graduated last year and now has a position at Nebraska Wesleyan.)

# Set-theoretic mereology

• J. D. Hamkins and M. Kikuchi, “Set-theoretic mereology.” (manuscript under review)
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Abstract. We consider a set-theoretic version of mereology based on the inclusion relation $\newcommand\of{\subseteq}\of$ and analyze how well it might serve as a foundation of mathematics. After establishing the non-definability of $\in$ from $\of$, we identify the natural axioms for $\of$-based mereology, which constitute a finitely axiomatizable, complete, decidable theory. Ultimately, for these reasons, we conclude that this form of set-theoretic mereology cannot by itself serve as a foundation of mathematics. Meanwhile, augmented forms of set-theoretic mereology, such as that obtained by adding the singleton operator, are foundationally robust.

In light of the comparative success of membership-based set theory in the foundations of mathematics, since the time of Cantor, Zermelo and Hilbert, a mathematical philosopher naturally wonders whether one might find a similar success for mereology, based upon a mathematical or set-theoretic parthood relation rather than the element-of relation $\in$. Can set-theoretic mereology serve as a foundation of mathematics? And what should be the central axioms of set-theoretic mereology?

We should like therefore to consider a mereological perspective in set theory, analyzing how well it might serve as a foundation while identifying the central axioms. Although set theory and mereology, of course, are often seen as being in conflict, what we take as the project here is to develop and investigate, within set theory, a set-theoretic interpretation of mereological ideas. Mereology, by placing its focus on the parthood relation, seems naturally interpreted in set theory by means of the inclusion relation $\of$, so that one set $x$ is a part of another $y$, just in case $x$ is a subset of $y$, written $x\of y$. This interpretation agrees with David Lewis’s Parts of Classes (1991) interpretation of set-theoretic mereology in the context of sets and classes, but we restrict our attention to the universe of sets. So in this article we shall consider the formulation of set-theoretic mereology as the study of the structure $\langle V,\of\rangle$, which we shall take as the canonical fundamental structure of set-theoretic mereology, where $V$ is the universe of all sets; this is in contrast to the structure $\langle V,{\in}\rangle$, usually taken as central in set theory. The questions are: How well does this mereological structure serve as a foundation of mathematics? Can we faithfully interpret the rest of mathematics as taking place in $\langle V,\of\rangle$ to the same extent that set theorists have argued (with whatever degree of success) that one may find faithful representations in $\langle V,{\in}\rangle$? Can we get by with merely the subset relation $\of$ in place of the membership relation $\in$?

Ultimately, we shall identify grounds supporting generally negative answers to these questions. On the basis of various mathematical results, our main philosophical thesis will be that the particular understanding of set-theoretic mereology via the inclusion relation $\of$ cannot adequately serve by itself as a foundation of mathematics. Specifically, the following theorem and corollary show that $\in$ is not definable from $\of$, and we take this to show that one may not interpret membership-based set theory itself within set-theoretic mereology in any straightforward, direct manner.

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

Corollary. One cannot define $\in$ from $\subseteq$ in any model of set theory, even allowing parameters in the definition.

A counterpoint to this is provided by the following theorem, however, which identifies a weak sense in which $\of$ may identify $\in$ up to definable automorphism of the universe.

Theorem. Assume ZFC in the universe $\langle V,\in\rangle$. Suppose that $\in^*$ is a definable class relation in $\langle V,{\in}\rangle$ for which $\langle V,\in^*\rangle$ is a model of set theory (a weak set theory suffices), such that the corresponding inclusion relation $$u\subseteq^* v\quad\iff\quad\forall a\,(a\in^* u\to a\in^* v)$$is the same as the usual inclusion relation $u\subseteq v$. Then the two membership relations are isomorphic $$\langle V,\in\rangle\cong\langle V,\in^*\rangle.$$

That counterpoint is not decisive, however, in light of the question whether we really need $\in^*$ to be a class with respect to $\in$, a question resolved by the following theorem, which shows that set-theoretic mereology does not actually determine the $\in$-isomorphism class or even the $\in$-theory of the $\in$-model in which it arises.

Theorem. For any two consistent theories extending ZFC, there are models $\langle W,{\in}\rangle$ and $\langle W,{\in^*}\rangle$ of those theories, respectively, with the same underlying set $W$ and the same induced inclusion relation $\of=\of^*$.

For example, we cannot determine in $\of$-based set-theoretic mereology whether the continuum hypothesis holds or fails, whether the axiom of choice holds or fails or whether there are large cardinals or not. Initially, the following central theorem may appear to be a positive result for mereology, since it identifies precisely what are the principles of set-theoretic mereology, considered as the theory of $\langle V,{\of}\rangle$. Namely, $\of$ is an atomic unbounded relatively complemented distributive lattice, and this is a finitely axiomatizable complete theory. So in a sense, this theory simply is the theory of $\of$-based set-theoretic mereology.

Theorem. Set-theoretic mereology, considered as the theory of $\langle V,\of\rangle$, is precisely the theory of an atomic unbounded relatively complemented distributive lattice, and furthermore, this theory is finitely axiomatizable, complete and decidable.

But upon reflection, since every finitely axiomatizable complete theory is decidable, the result actually appears to be devastating for set-theoretic mereology as a foundation of mathematics, because a decidable theory is much too simple to serve as a foundational theory for all mathematics. The full spectrum and complexity of mathematics naturally includes all the instances of many undecidable decision problems and so cannot be founded upon a decidable theory. Finally, it follows as a corollary that the structure consisting of the hereditarily finite sets under inclusion forms an elementary substructure of the full set-theoretic mereological universe $$\langle \text{HF},\of\rangle\prec\langle V,\of\rangle.$$ Consequently set-theoretic mereology cannot properly treat or even express the various concepts of infinity that arise in mathematics.

Some previous posts on this blog:

# Freiling’s axiom of symmetry, or throwing darts at the real line, Pizza Seminar, March 2016

This will be a talk I’ll give at the CUNY Graduate Center Graduate Student Colloquium, otherwise known as the Pizza Seminar in the math program, on Monday, March 28, 2016, 4-4:45 pm.  The talk will be aimed at a general audience of mathematics graduate students, and….Free Pizza!

Abstract. I shall give an elementary presentation of Freiling’s axiom of symmetry, which is the principle asserting that if $x\mapsto A_x$ is a function mapping every real $x\in[0,1]$ in the unit interval to a countable set of such reals $A_x\subset[0,1]$, then there are two reals $x$ and $y$ for which $x\notin A_y$ and $y\notin A_x$.  To argue for the truth of this principle, Freiling imagined throwing two darts at the real number line, landing at $x$ and $y$ respectively: almost surely, the location $y$ of the second dart is not in the set $A_x$ arising from that of the first dart, since that set is countable; and by symmetry, it shouldn’t matter which dart we imagine as being first. So it may seem that almost every pair must fulfill the principle. Nevertheless, the principle is independent of the axioms of ZFC and in fact it is provably equivalent to the failure of the continuum hypothesis.  I’ll introduce the continuum hypothesis in a general way and discuss these foundational matters, before providing a proof of the equivalence of $\neg$CH with the axiom of symmetry. The axiom of symmetry admits natural higher dimensional analogues, such as the case of maps $(x,y)\mapsto A_{x,y}$, where one seeks a triple $(x,y,z)$ for which no member is in the set arising from the other two, and these principles also have an equivalent formulation in terms of the size of the continuum.

Finally, and perhaps most importantly for those graduate students who are thinking about coming to the colloquium, without repeating myself let me say that it is my understanding that there will be free pizza available. Free pizza!

# Diamond on the ordinals

I was recently surprised to discover that if there is a definable well-ordering of the universe, then the diamond principle on the ordinals holds for definable classes, automatically. In fact, the diamond principle for definable classes is simply equivalent in ZFC to the existence of a definable well-ordering of the universe. It follows as a consequence that the diamond principle for definable classes, although seeming to be fundamentally scheme-theoretic, is actually expressible in the first-order language of set theory.

In set theory, the diamond principle asserts the existence of a sequence of objects $A_\alpha$, of growing size, such that any large object at the end is very often anticipated by these approximations.  In the case of diamond on the ordinals, what we will have is a definable sequence of $A_\alpha\subseteq\alpha$, such that for any definable class of ordinals $A$ and any definable class club set $C$, there are ordinals $\theta\in C$ with $A\cap\theta=A_\theta$.  This kind of principle typically allows one to undertake long constructions that will diagonalize against all the large objects, by considering and reacting to their approximations $A_\alpha$. Since every large object $A$ is often correctly approximated that way, this enables many such constructions to succeed.


Theorem. In $\ZFC$, if there is a definable well-ordering of the universe, then $\Diamond_{\Ord}$ holds for definable classes. That is, there is a $p$-definable sequence $\langle A_\alpha\mid\alpha<\Ord\rangle$, such that for any definable class $A\of\Ord$ and any definable closed unbounded class of ordinals $C\of\Ord$ (allowing parameters), there is some $\theta\in C$ with $A\cap\theta=A_\theta$.

Proof. The theorem is proved as a theorem scheme; namely, I shall provide a specific definition for the sequence $\vec A=\langle A_\alpha\mid\alpha<\Ord\rangle$, using the same parameter $p$ as the definition of the global well-order and with a definition of closely related syntactic complexity, and then prove as a scheme, a separate statement for each definable class $A\of\Ord$ and class club $C\of\Ord$, that there is some $\alpha\in C$ with $A\cap\alpha=A_\alpha$. The definitions of the classes $A$ and $C$ may involve parameters and have arbitrary complexity.

Let $\lhd$ be the definable well-ordering of the universe, definable by a specific formula using some parameter $p$. I define the $\Diamond_{\Ord}$-sequence $\vec A=\langle A_\alpha\mid\alpha<\Ord\rangle$ by transfinite recursion. Suppose that $\vec A\restrict\theta$ has been defined. I shall let $A_\theta=\emptyset$ unless $\theta$ is a $\beth$-fixed point above the rank of $p$ and there is a set $A\of\theta$ and a closed unbounded set $C\of\theta$, with both $A$ and $C$ definable in the structure $\langle V_\theta,\in\rangle$ (allowing parameters), such that $A\cap\alpha\neq A_\alpha$ for every $\alpha\in C$. In this case, I choose the least such pair $(A,C)$, minimizing first on the maximum of the logical complexities of the definitions of $A$ and of $C$, and then minimizing on the total length of the defining formulas of $A$ and $C$, and then minimizing on the Gödel codes of those formulas, and finally on the parameters used in the definitions, using the well-order $\lhd\restrict V_\theta$. For this minimal pair, let $A_\theta=A$. This completes the definition of the sequence $\vec A=\langle A_\alpha\mid\alpha\in\Ord\rangle$.

Let me remark on a subtle point, since the meta-mathematical issues loom large here. The definition of $\vec A$ is internal to the model, and at stage $\theta$ we ask about subsets of $\theta$ definable in $\langle V_\theta,\in\rangle$, using the truth predicate for this structure. If we were to run this definition inside an $\omega$-nonstandard model, it could happen that the minimal formula we get is nonstandard, and in this case, the set $A$ would not actually be definable by a standard formula. Also, even when $A$ is definable by a standard formula, it might be paired (with some constants), with a club set $C$ that is defined only by a nonstandard formula (and this is why we minimize on the maximum of the complexities of the definitions of $A$ and $C$ together). So one must give care in the main argument keeping straight the distinction between the meta-theoretic natural numbers and the internal natural numbers of the object theory $\ZFC$.

Let me now prove that the sequence $\vec A$ is indeed a $\Diamond_{\Ord}$-sequence for definable classes. The argument follows in spirit the classical proof of $\Diamond$ in $L$, subject to the mathematical issues I mentioned. If the sequence $\vec A$ is not a diamond sequence, then there is some definable class $A\of\Ord$, defined in $\langle V,\in\rangle$ by a specific formula $\varphi$ and parameter $z$, and definable club $C\of\Ord$, defined by some $\psi$ and parameter $y$, with $A\cap\alpha\neq A_\alpha$ for every $\alpha\in C$. We may assume without loss that these formulas are chosen so as to be minimal in the sense of the construction, so that the maximum of the complexities of $\varphi$ and $\psi$ are as small as possible, and the lengths of the formulas, and the Gödel codes and finally the parameters $z,y$ are $\lhd$-minimal, respectively, successively. Let $m$ be a sufficiently large natural number, larger than the complexity of the definitions of $\lhd$, $A$, $C$, and large enough so that the minimality condition we just discussed is expressible by a $\Sigma_m$ formula. Let $\theta$ be any $\Sigma_m$-correct ordinal above the ranks of the parameters used in the definitions. It follows that the restrictions $\lhd\restrict V_\theta$ and also $A\cap\theta$ and $C\cap\theta$ and $\vec A\restrict\theta$ are definable in $\langle V_\theta,\in\rangle$ by the same definitions and parameters as their counterparts in $V$, that $C\cap\theta$ is club in $\theta$, and that $A\cap\theta$ and $C\cap\theta$ form a minimal pair using those definitions with $A\cap\alpha\neq A_\alpha$ for any $\alpha\in C\cap\theta$. Thus, by the definition of $\vec A$, it follows that $A_\theta=A\cap\theta$. Since $C\cap\theta$ is unbounded in $\theta$ and $C$ is closed, it follows that $\theta\in C$, and so $A_\theta=A\cap\theta$ contradicts our assumption about $A$ and $C$. So there are no such counterexample classes, and thus $\vec A$ is a $\Diamond_{\Ord}$-sequence with respect to definable classes, as claimed.
QED

Theorem. The following are equivalent over $\ZFC$.

1. There is a definable well-ordering of the universe, using some set parameter $p$.
2. $V=\HOD_{\{p\}}$, for some set $p$.
3. $\Diamond_{\Ord}$ holds for definable classes. That is, there is a set parameter $p$ and a definable sequence $\vec A=\langle A_\alpha\mid\alpha<\Ord\rangle$, such that for any definable class $A\of\Ord$ and definable class club $C\of\Ord$, there is some $\alpha\in C$ with $A\cap\alpha=A_\alpha$.

Proof. Let me first give the argument, and then afterward discuss some issues about the formalization, which involves some subtle issues.

($1\to 2$) $\newcommand\rank{\text{rank}}$Suppose that $\lhd$ is a $p$-definable well-ordering of $V$, which means that every set has a $\lhd$-minimal element. Let us refine this order by defining $x\lhd’ y$, just in case $\rank(x)<\rank(y)$ or $\rank(x)=\rank(y)$ and $x\lhd y$. The new order is also a well-order, which now respects rank. In particular, the order $\lhd’$ is set-like, and so every object $x$ is the $\alpha^{th}$ element with respect to the $\lhd’$-order, for some ordinal $\alpha$. Thus, every object is definable from $p$ and an ordinal, and so $V=\HOD_{\{p\}}$, as desired.

($2\to 1$) If $V=\HOD_{\{p\}}$, then we have the canonical well-order of $\HOD$ using parameter $p$, similar to how one shows that the axiom of choice holds in $\HOD$. Namely, define $x\lhd y$ if and only if $\rank(x)<\rank(y)$, or the ranks are the same, but $x$ is definable from $p$ and ordinal parameters in some $V_\theta$ with a smaller $\theta$ than $y$ is, or the ranks are the same and the $\theta$ is the same, but $x$ is definable in that $V_\theta$ by a formula with a smaller Gödel code, or with the same formula but smaller ordinal parameters. It is easy to see that this is a $p$-definable well-ordering of the universe.

($1\to 3$) This is the content of the theorem above.

($3\to 1$) If $\vec A$ is a $p$-definable $\Diamond_{\Ord}$-sequence for definable classes, then it is easy to see that if $A$ is a set of ordinals, then $A$ must arise as $A_\alpha$ for unboundedly many $\alpha$. In $\ZFC$, using the axiom of choice, it is a standard fact that every set is coded by a set of ordinals. So let us define that $x\lhd y$, just in case $x$ is coded by a set of ordinals that appears earlier on $\vec A$ than any set of ordinals coding $y$. This is clearly a well-ordering, since the map sending $x$ to the ordinal $\alpha$ for which $A_\alpha$ codes $x$ is an $\Ord$-ranking of $\lhd$. So there is a $p$-definable well-ordering of the universe.
QED

An observant reader will notice some meta-mathematical issues concerning the previous theorem. The issue is that statements 1 and 2 are known to be expressible by statements in the first-order language of set theory, as single statements, but for statement 3 we have previously expressed it only as a scheme of first-order statements. So how can they be equivalent? The answer is that the full scheme-theoretic content of statement 3 follows already from instances in which the complexity of the definitions of $A$ and $C$ are bounded. Basically, once one gets the global well-order, then one can construct a $\Diamond_{\Ord}$-sequence that works for all definable classes. In this sense, we may regard the diamond principle $\Diamond_{\Ord}$ for definable classes as not really a scheme of statements, but rather equivalent to a single first-order assertion.

Lastly, let me consider the content of the theorems in Gödel-Bernays set theory or Kelley-Morse set theory. Of course, we know that there can be models of these theories that do not have $\Diamond_{\Ord}$ in the full second-order sense. For example, it is relatively consistent with ZFC that an inaccessible cardinal $\kappa$ does not have $\Diamond_\kappa$, and in this case, the structure $\langle V_\kappa,\in,V_{\kappa+1}\rangle$ will satisfy GBC and even KM, but it won’t have $\Diamond_{\Ord}$ with respect to all classes, even though it has a definable well-ordering of the universe (since there is such a well-ordering in $V_{\kappa+1}$). But meanwhile, there will be a $\Diamond_{\Ord}$-sequence that works with respect to classes that are definable from that well-ordering and parameters, simply by following the construction given in the theorem.

This leads to several extremely interesting questions, about which I am currently thinking, concerning instances where we can have $\Diamond_\kappa$ for definable classes in $V_\kappa$, even when the full $\Diamond_\kappa$ fails. Stay tuned!

# Set-theoretic mereology: the theory of the subset relation is decidable


To give a few examples, if $\newcommand\HF{\text{HF}}\HF$ is the set of hereditarily finite sets, then $\langle\HF,\of\rangle$, using the usual subset relation, is an unbounded atomic locally Boolean lattice. More generally, if $V$ is any model of set theory (even a very weak theory is sufficient), then $\langle V,\of\rangle$ is an unbounded atomic locally Boolean lattice.

I should like to prove here that the theory of unbounded atomic locally Boolean lattice orders is decidable, and furthermore admits elimination of quantifiers down to the language including the Boolean operations and the relations expressing the height or size of an object, $|x|\geq n$ and $|x|=n$.

Theorem. Every formula in the language of lattices is equivalent, over the theory of unbounded atomic locally Boolean lattices, to a quantifier-free formula in the language of the order $a\of b$, equality $a=b$, meet $a\intersect b$, join $a\union b$, relative complement $a-b$, constant $0$, the unary relation $|x|\geq n$, and the unary relation $|x|=n$, where $n$ is respectively any natural number.

Proof. We prove the result by induction on formulas. The collection of formulas equivalent to a quantifier-free formula in that language clearly includes all atomic formulas and is closed under Boolean combinations. So it suffices to eliminate the quantifier in a formula of the form $\exists x\, \varphi(x,\ldots)$, where $\varphi(x,\ldots)$ is quantifier-free in that language. Let us make a number of observations that will enable various simplifying assumptions about the form of $\varphi$.

Because equality of terms is expressible by the identity $a=b\iff a\of b\of a$, we do not actually need $=$ in the language (and here I refer to the use of equality in atomic formulas of the form $s=t$ where $s$ and $t$ are terms, and not to the incidental appearance of the symbol $=$ in the unary predicate $|x|=n$, which is an unrelated use of this symbol, a mere stylistic flourish). Similarly, in light of the equivalence $a\of b\iff |a-b|=0$, we do not need to make explicit reference to the order $a\of b$. So we may assume that all atomic assertions in $\varphi$ have the form $|t|\geq n$ or $|t|=n$ for some term $t$ in the language of meet, join, relative complement and $0$. We may omit the need for explicit negation in the formula by systematically applying the equivalences:
$$\neg(|t|\geq n)\iff \bigvee_{k<n}|t|=k\quad\text{ and}$$
$$\neg(|t|=n)\iff (|t|\geq n+1)\vee\bigvee_{k<n}|t|=k.$$
So we have reduced to the case where $\varphi$ is a positive Boolean combination of expressions of the form $|t|\geq n$ and $|t|=n$.

Let us consider the form of the terms $t$ that may arise in the formula. List all the variables $x=x_0,x_1,\ldots,x_N$ that arise in any of the terms appearing in $\varphi$, and consider the Venn diagram corresponding to these variables. The cells of this Venn diagram can each be described by a term of the form $\bigwedge_{i\leq N} \pm x_i$, which I shall refer to as a cell term, where $\pm x_i$ means that either $x_i$ appears or else we have subtracted $x_i$ from the other variables. Since we have only relative complements in a locally Boolean lattice, however, and not absolute complements, we need only consider the cells where at least one variable appears positively, since the exterior region in the Venn diagram is not actually represented by any term. In this way, every term in the language of locally Boolean lattices is a finite union of such cell terms, plus $\emptyset$ (which I suppose can be viewed as an empty union). Note that distinct cell terms are definitely representing disjoint objects in the lattice.

Next, by considering the possible sizes of $s-t$, $s\intersect t$ and $t-s$, observe that
$$|s\union t|\geq n\iff \bigvee_{i+j+k=n}(|s|\geq i+j)\wedge(|s\intersect t|\geq j)\wedge(|t|\geq j+k).$$
Through repeated application of this, we may reduce any assertion about $|t|$ for a term to a Boolean combination of assertions about cell terms. (Note that size assertions about $\emptyset$ are trivially settled by the theory and can be eliminated.)

Let us now focus on the quantified variable $x$ separately from the other variables, for it may appear either positively or negatively in such a cell term. More precisely, each cell term in the variables $x=x_0,x_1,\ldots,x_N$ is equivalent to $x\intersect c$ or $c-x$, for some cell term $c$ in the variables $x_1,\ldots,x_N$, that is, not including $x$, or to the term $x-(x_1\union\cdots\union x_N)$, which is the cell term for which $x$ is the only positive variable.

We have reduced the problem to the case where we want to eliminate the quantifier from $\exists x\, \varphi$, where $\varphi$ is a positive Boolean combination of size assertions about cell terms. We may express $\varphi$ in disjunctive normal form and then distribute the quantifier over the disjunct to reduce to the case where $\varphi$ is a conjunction of size assertions about cell terms. Each cell term has the form $x\intersect c$ or $c-x$ or $x-(x_1\union\cdots x_N)$, where $c$ is a cell term in the list of variables without $x$. Group the conjuncts of $\varphi$ that use the same cell term $c$ in this way together. The point now is that assertions about whether there is an object $x$ in the lattice such that certain cell terms obey various size requirements amount to the conjunction of various size requirements about cells in the variables not including $x$. For example, the assertion $$\exists x\,(|x\intersect c|\geq 3)\wedge(|x\intersect c|\geq 7)\wedge(|c-x|=2)$$ is equivalent (over the theory of unbounded atomic locally Boolean lattices) to the assertion $|c|\geq 9$, since we may simply let $x$ be all but $2$ atoms of $c$, and this will have size at least $7$, which is also at least $3$. If contradictory assertions are made, such as $\exists x\, (|x\intersect c|\geq 5\wedge |x\intersect c|=3)$, then the whole formula is equivalent to $\perp$, which can be expressed without quantifiers as $0\neq 0$.

Next, the key observation of the proof is that assertions about the existence of such $x$ for different cell terms in the variables not including $x$ will succeed or fail independently, since those cell terms are representing disjoint elements of the lattice, and so one may take the final witnessing $x$ to be the union of the witnesses for each piece. So to eliminate the quantifier, we simply group together the atomic assertions being made about the cell terms in the variables without $x$, and then express the existence assertion as a size requirement on those cell terms. For example, the assertion $$\exists x\, (|c\intersect x|\geq 5)\wedge(|c-x|=6)\wedge (|d\intersect x|\geq 7),$$ where $c$ and $d$ are distinct cell terms, is equivalent to $$(|c|\geq 11)\wedge(|d|\geq 7),$$ since $c$ and $d$ are disjoint and so we may let $x$ be the appropriate part of $c$ and a suitable piece of $d$. The only remaining complication concerns instances of the term $x-(x_1\union\cdots\union x_N)$. But for these, the thing to notice is that any single positive size assertion about this term is realizable in our theory, since we have assumed that the lattice is unbounded, and so there will always be as many atoms as desired disjoint from any finite list of objects. But we must again pay attention to whether the requirements expressed by distinct clauses are contradictory.

Altogether, I have provided a procedure for eliminating quantifiers from any assertion in the language of locally Boolean lattices, down to the language augmented by unary predicates expressing the size of an object. This procedure works in any unbounded atomic locally Boolean lattice, and so the theorem is proved. QED

Corollary. The theory of unbounded atomic locally Boolean lattices is complete.

Proof. Every sentence in this theory is equivalent by the procedure to a quantifier-free sentence in the stated language. But since such sentences have no variables, they must simply be a Boolean combination of trivial size assertions about $0$, such as $(|0|\geq 2)\vee \neg(|0|=5)$, whose truth value is settled by the theory. QED

Corollary. The structure of hereditarily finite sets $\langle\HF,\of\rangle$ is an elementary substructure of the entire set-theoretic universe $\langle V,\of\rangle$, with the inclusion relation.

Proof. These structures are both unbounded atomic locally Boolean lattices, and so they each support the quantifier-elimination procedure. But they agree on the truth of any quantifier-free assertion about the sizes of hereditarily finite sets, and so they they must agree on all truth assertions about objects in $\HF$. QED

Corollary. The structure $\langle V,\of\rangle$ has a decidable theory. The structure $\langle\HF,\of\rangle$ has a decidable elementary diagram, and hence a computably decidable presentation.

Proof. The theory is the theory of unbounded atomic locally Boolean lattices. Since the structure $\langle\HF,\of\rangle$ has a computable presentation via the Ackerman coding of hereditarily finite sets, for which the subset relation and the size relations are computable, it follows that we may also compute the truth of any formula by first reducing it to a quantifier-free assertions of those types. So this is a computably decidable presentation. QED

# Survey of Logic for Philosophers, CUNY GC PHIL 76500, Spring 2016

Survey of Logic for Philosophers
PHIL 76500
Program in Philosophy
Spring semester 2016
4 credits
Weds. 11:45-1:45
Room 5417

This seminar will be a graduate-level survey of topics in logic for philosophers, including propositional logic, modal logic, predicate logic, model theory, completeness, incompleteness, computability theory, set theory and the higher infinite. The material will thus be a mixture of philosophical logic and mathematical logic, and a goal of the course will be to develop the student’s ability to treat the main ideas of logic with clarity and precision.

Students will solve weekly problem sets and write a final paper on a topic chosen in consultation with me.

I shall post further information and resources here as the seminar approaches.

# 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.” (manuscript under review, Newton Institute preprint ni15066)
@ARTICLE{Hamkins:UpwardClosureAndAmalgamationInTheGenericMultiverse,
author = {Joel David Hamkins},
title = {Upward closure and amalgamation in the generic multiverse of a countable model of set theory},
journal = {},
year = {},
volume = {},
number = {},
pages = {},
month = {},
note = {manuscript under review, Newton Institute preprint ni15066},
url = {http://jdh.hamkins.org/upward-closure-and-amalgamation-in-the-generic-multiverse},
eprint = {1511.01074},
abstract = {},
keywords = {},
source = {},
}

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.

# The rearrangement number: how many rearrangements of a series suffice to verify absolute convergence? Vassar Math Colloquium, November 2015

This will be a talk for the Mathematics Colloquium at Vassar College, November 10, 2015, tea at 4:00 pm, talk at 4:15 pm, Rockefeller Hall 310

Abstract. The Riemann rearrangement theorem asserts that a series $\sum_n a_n$ is absolutely convergent if and only if every rearrangement $\sum_n a_{p(n)}$ of it is convergent, and furthermore, any conditionally convergent series can be rearranged so as to converge to any desired extended real value. How many rearrangements $p$ suffice to test for absolute convergence in this way? The rearrangement number, a new cardinal characteristic of the continuum introduced just recently, is the smallest size of a family of permutations, such that whenever the convergence and value of a convergent series is invariant by all these permutations, then it is absolutely convergent. The exact value of the rearrangement number turns out to be independent of the axioms of set theory. In this talk, I shall place the rearrangement number into a discussion of cardinal characteristics of the continuum, including an elementary introduction to the continuum hypothesis and an account of Freiling’s axiom of symmetry.

This talk is based in part on current joint work with Andreas Blass, Will Brian, myself, Michael Hardy and Paul Larson.

My notes are available here:

# A position in infinite chess with game value $\omega^4$

• C.~D.~A.~Evans, J. D. Hamkins, and N. L. Perlmutter, “A position in infinite chess with game value $\omega^4$.” (manuscript under review, Newton Institute preprint ni15065)
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Abstract.  We present a position in infinite chess exhibiting an ordinal game value of $\omega^4$, thereby improving on the previously largest-known values of $\omega^3$ and $\omega^3\cdot 4$.

This is a joint work with Cory Evans and Norman Perlmutter, continuing the research program of my previous article with Evans, Transfinite game values in infinite chess, namely, the research program of finding positions in infinite chess with large transfinite ordinal game values. In the previous article, Cory and I presented a position with game value $\omega^3$. In the current paper, with Norman Perlmutter now having joined us accompanied by some outstanding ideas, we present a new position having game value $\omega^4$, breaking the previous record.

A position in infinite chess with value $\omega^4$

In the new position, above, the kings sit facing each other in the throne room, an uneasy détente, while white makes steady progress in the rook towers. Meanwhile, at every step black, doomed, mounts increasingly desperate bouts of long forced play using the bishop cannon battery, with bishops flying with force out of the cannons, and then each making a long series of forced-reply moves in the terminal gateways. Ultimately, white wins with value omega^4, which exceeds the previously largest known values of omega^3.

In the throne room, if either black or white places a bishop on the corresponding diagonal entryway, then checkmate is very close. A key feature is that for white to place a white-square white bishop on the diagonal marked in red, it is immediate checkmate, whereas if black places a black-square black bishop on the blue diagonal, then checkmate comes three moves later.  The bishop cannon battery arrangement works because black threatens to release a bishop into the free region, and if white does not reply to those threats, then black will be three steps ahead, but otherwise, only two.

The throne room

The rook towers are similar to the corresponding part of the previous $\omega^3$ position, and this is where white undertakes most of his main line progress towards checkmate.  Black will move the key bishop out as far as he likes on the first move, past $n$ rook towers, and the resulting position will have value $\omega^3\cdot n$.  These towers are each activated in turn, leading to a long series of play for white, interrupted at every opportunity by black causing a dramatic spectacle of forced-reply moves down in the bishop cannon battery.

The rook towers

At every opportunity, black mounts a long distraction down in the bishop cannon battery.  Shown here is one bishop cannon. The cannonballs fire out of the cannon with force, in the sense that when each green bishop fires out, then white must reply by moving the guard pawns into place.

Bishop cannon

Upon firing, each bishop will position itself so as to attack the entrance diagonal of a long bishop gateway terminal wing.  This wing is arranged so that black can make a series of forced-reply threats successively, by moving to the attack squares (marked with the blue squares). Black is threatening to exit through the gateway doorway (in brown), but white can answer the threat by moving the white bishop guards (red) into position. Thus, each bishop coming out of a cannon (with force) can position itself at a gateway terminal of length $g$, making $g$ forced-reply moves in succession.  Since black can initiate firing with an arbitrarily large cannon, this means that at any moment, black can cause a forced-reply delay with game value $\omega^2$. Since the rook tower also has value $\omega^2$ by itself, the overall position has value $\omega^4=\omega^2\cdot\omega^2$.

Bishop gateway terminal wing

With future developments in mind, we found that one can make a more compact arrangement of the bishop cannon battery, freeing up a quarter board for perhaps another arrangement that might lead to a higher ordinal values.

Alternative compact version of bishop cannon battery

Read more about it in the article, which is available at the arxiv (pdf).

# The rearrangement number, CUNY set theory seminar, November 2015

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

The Riemann rearrangement theorem states that a convergent real series $\sum_n a_n$ is absolutely convergent if and only if the value of the sum is invariant under all rearrangements $\sum_n a_{p(n)}$ by any permutation $p$ on the natural numbers; furthermore, if the series is merely conditionally convergent, then one may find rearrangements for which the new sum $\sum_n a_{p(n)}$ has any desired (extended) real value or which becomes non-convergent.  In recent joint work with Andreas Blass, Will Brian, myself, Michael Hardy and Paul Larson, based on an exchange in reply to a Hardy’s MathOverflow question on the topic, we investigate the minimal size of a family of permutations that can be used in this manner to test an arbitrary convergent series for absolute convergence.


# Open determinacy for games on the ordinals is stronger than ZFC, CUNY Logic Workshop, October 2015

This will be a talk for the CUNY Logic Workshop on October 2, 2015.

Abstract. The principle of open determinacy for class games — two-player games of perfect information with plays of length $\omega$, 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 Gödel-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 $\text{Tr}_\alpha$ 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 elementary transfinite recursion ETR over well-founded class relations. Meanwhile, the principle of open determinacy for class games is provable in the stronger theory GBC+$\Pi^1_1$-comprehension, a proper fragment of Kelley-Morse set theory KM.

This is joint work with Victoria Gitman, with the helpful participation of Thomas Johnstone.

Related article and posts:

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

$\newcommand\HOD{\text{HOD}}$The axiom $V=\HOD$, 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 $\HOD$, 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 $V=\HOD$ holds, if for every set $x$, there is an ordinal $\theta$ with $x\in V_\theta$, for which which $x$ is definable by some formula $\psi(x)$ in the structure $\langle V_\theta,{\in}\rangle$ using ordinal parameters. Since $V_\theta$ is a set, we may freely make reference to first-order truth in $V_\theta$ without requiring any truth predicate in $V$. Certainly any such $x$ as this is also ordinal-definable in $V$, since we may use $\theta$ and the Gödel-code of $\psi$ also as parameters, and note that $x$ is the unique object such that it is in $V_\theta$ and satisfies $\psi$ in $V_\theta$. (Note that inside an $\omega$-nonstandard model of set theory, we may really need to use $\psi$ as a parameter, since it may be nonstandard, and $x$ may not be definable in $V_\theta$ 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 $\varphi(x,\vec\alpha)$ with ordinal parameters $\vec\alpha$, then it follows by the reflection theorem that $x$ is defined by $\varphi(x,\vec\alpha)$ inside some $V_\theta$. So this formulation of $V=HOD$ is expressible and exactly captures the desired second-order property that every set is ordinal-definable.

Consider next the axiom $V=\HOD(b)$, asserting that every set is definable from ordinal parameters and parameter $b$. Officially, as before, $V=\HOD(b)$ asserts that for every $x$, there is an ordinal $\theta$, formula $\psi$ and ordinals $\vec \alpha<\theta$, such that $x$ is the unique object in $V_\theta$ for which $\langle V_\theta,{\in}\rangle\models\psi(x,\vec\alpha,b)$, 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 $\exists b\,\left( V=\HOD(b)\right)$, asserting that the universe is $\HOD$ 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 $\exists b\, (V=\HOD(b))$ 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 $\text{ZFC}+V=\HOD(b)$, and $V[G]$ is a forcing extension of $V$ by generic filter $G\subset\mathbb{P}\in V$. By the ground-model definability theorem, it follows that $V$ is definable in $V[G]$ from parameter $P(\mathbb{P})^V$. 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 $\mathbb{P}$-names used to form $V[G]$, it follows that $V[G]=\HOD(b,P(\mathbb{P})^V,G)$, and so $V[G]$ is $\HOD$ of a set, as desired.

Conversely, suppose that $W$ is a ground of $V$, so that $V=W[G]$ for some $W$-generic filter $G\subset\mathbb{P}\in W$, and $V=\HOD(b)$ for some set $b$. Let $\dot b$ be a name for which $\dot b_G=b$. Every object $x\in W$ is definable in $W[G]$ from $b$ and ordinal parameters $\vec\alpha$, so there is some formula $\psi$ for which $x$ is unique such that $\psi(x,b,\vec\alpha)$. Thus, there is some condition $p\in\mathbb{P}$ such that $x$ is unique such that $p\Vdash\psi(\check x,\dot b,\check{\vec\alpha})$. If $\langle p_\beta\mid\beta<|\mathbb{P}|\rangle$ is a fixed enumeration of $\mathbb{P}$ in $W$, then $p=p_\beta$ for some ordinal $\beta$, and we may therefore define $x$ in $W$ using ordinal parameters, along with $\dot b$ and the fixed enumeration of $\mathbb{P}$. So $W$ thinks the universe is $\HOD$ 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 $\exists b\, (V=\HOD(b))$ is constant throughout the generic multiverse. QED

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

Proof. We are working in ZFC. Suppose that $V=\HOD(b)$. 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 $c$, $P(\omega)^V$ 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 $c$, $b$, $P(\omega)^V$, $G$ and $H$ are all definable without parameters. Since $V\subset V[c][G][H]$ has a closure point at $\omega$, it satisfies the $\omega_1$-approximation and cover properties, and therefore the class $V$ is definable in $V[c][G][H]$ using $P(\omega)^V$ 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 $\HOD(b)^V=V$ is ordinal-definable in $V[c][G][H]$. And since $c$, $G$ and $H$ are also definable without parameters, we have $V[c][G][H]\models V=\HOD$, as desired. QED

Corollary. The following are equivalent.

1. The universe is $\HOD$ of a set: $\exists b\, (V=\HOD(b))$.
2. Somewhere in the generic multiverse, the universe is $\HOD$ of a set.
3. Somewhere in the generic multiverse, the axiom $V=\HOD$ holds.
4. The axiom $V=\HOD$ 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 $V=\HOD$, 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 $V=\HOD$ to fail, simply by adding a Cohen real. If $V=\HOD$ is true, then by the first theorem, every forcing extension has $V=\HOD(b)$ for some $b$, in which case $V=\HOD$ remains forceable, by the second theorem. QED

# Quoted in New York magazine: The Chalk for Math Professors

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.

Related MathOverflow post.

# Different models of set theory with the same subset relation

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 $\langle V,\in\rangle$, there is a definable relation $\in^*$, different from $\in$, such that $\langle V,\in^*\rangle$ is a model of set theory, in fact isomorphic to the original universe $\langle V,\in\rangle$, for which the corresponding inclusion relation $$u\subseteq^* v\iff \forall a\, (a\in^* u\to a\in^* v)$$ is identical to the usual inclusion relation $u\subseteq v$.

Proof. Let $\theta:V\to V$ be any definable non-identity permutation of the universe, and let $\tau:u\mapsto \theta[u]=\{\ \theta(a)\mid a\in u\ \}$ be the function determined by pointwise image under $\theta$. Since $\theta$ is bijective, it follows that $\tau$ is also a bijection of $V$ to $V$, since every set is the $\theta$-image of a unique set. Furthermore, $\tau$ is an automorphism of $\langle V,\subseteq\rangle$, since $$u\subseteq v\iff\theta[u]\subseteq\theta[v]\iff\tau(u) \subseteq\tau(v).$$ 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 $\tau(\{a\})=\{\theta(a)\}$, it follows that any instance of nontriviality $\theta(a)\neq a$ in $\theta$ leads immediately to an instance of nontriviality in $\tau$.

Using the map $\tau$, define $a\in^* b\iff\tau(a)\in\tau(b)$. By definition, therefore, $\tau$ is an isomorphism of $\langle V,\in^*\rangle\cong\langle V,\in\rangle$. Let us show that $\in^*\neq \in$. Since $\theta$ is nontrivial, there is an $\in$-minimal set $a$ with $\theta(a)\neq a$. By minimality, $\theta[a]=a$ and so $\tau(a)=a$. But as mentioned, $\tau(\{a\})=\{\theta(a)\}\neq\{a\}$. So we have $a\in\{a\}$, but $\tau(a)=a\notin\{\theta(a)\}=\tau(\{a\})$ 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 $\forall a\, (\tau(a)\in\tau(u)\to \tau(a)\in\tau(v))$; but since $\tau$ is surjective, this holds if and only if $\tau(u)\subseteq \tau(v)$, 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 $\tau$ is an isomorphism of the structure $\langle V,{\in^*},{\subseteq}\rangle$ with $\langle V,{\in},{\subseteq}\rangle$, 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 $\tau$ 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 $\langle V,\in\rangle$ 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 $\langle V,\in\rangle$. Suppose that $\in^*$ is a class relation for which $\langle V,\in^*\rangle$ is a model of set theory (a weak set theory suffices), such that the corresponding inclusion relation $$u\subseteq^* v\iff\forall a\,(a\in^* u\to a\in^* v)$$is the same as the usual inclusion relation $u\subseteq v$. Then the two membership relations are isomorphic $$\langle V,\in\rangle\cong\langle V,\in^*\rangle.$$

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 $\eta(a)$ such that $x\in^* a\iff x=\eta(a)$. By extensionality and since every object has its singleton, it follows that $\eta:V\to V$ is both one-to-one and onto. Let $\theta=\eta^{-1}$ be the inverse permutation.

Observe that $a\in u\iff \{a\}\subseteq u\iff \{a\}\subseteq^* u\iff\eta(a)\in^* u$. Thus, $$b\in^* u\iff \theta(b)\in u.$$

Using $\in$-recursion, define $b^*=\{\ \theta(a^*)\mid a\in b\ \}$. 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 $\theta^{-1}$ to the elements of $b^*$ and then using the induction assumption to find the unique $a$ from $a^*$ for each $\theta(a^*)\in b^*$, 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 $\theta(b^*)$ for any $b$. Since $\theta$ is surjective, this means there is $\theta(y_1)\in y_0$ with $y_1\neq b^*$ for any $b$. Continuing, there is $y_{n+1}$ with $\theta(y_{n+1})\in y_n$ and $y_{n+1}\neq b^*$ for any $b$. Let $z=\{\ \theta(y_n)\mid n\in\omega\ \}$. Since $x\in^* u\iff \theta(x)\in u$, it follows that the $\in^*$-elements of $z$ are precisely the $y_n$’s. But $\theta(y_{n+1})\in y_n$, 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 $$a\in b\iff\theta(a^*)\in b^*\iff a^*\in^* b^*,$$ it follows that the map $b\mapsto b^*$ is an isomorphism of $\langle V,\in\rangle$ with $\langle V,\in^*\rangle$, 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 $\langle V,\in\rangle$ universe.

Question. Can there be two models of set theory $\langle M,\in\rangle$ and $\langle M,\in^*\rangle$, 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

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.