# The $\Sigma_1$-definable universal finite sequence

• J. D. Hamkins and K. J. Williams, “The $\Sigma_1$-definable universal finite sequence,” ArXiv e-prints, 2019. (undeer review)
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Abstract. We introduce the $\Sigma_1$-definable universal finite sequence and prove that it exhibits the universal extension property amongst the countable models of set theory under end-extension. That is, (i) the sequence is $\Sigma_1$-definable and provably finite; (ii) the sequence is empty in transitive models; and (iii) if $M$ is a countable model of set theory in which the sequence is $s$ and $t$ is any finite extension of $s$ in this model, then there is an end extension of $M$ to a model in which the sequence is $t$. Our proof method grows out of a new infinitary-logic-free proof of the Barwise extension theorem, by which any countable model of set theory is end-extended to a model of $V=L$ or indeed any theory true in a suitable submodel of the original model. The main theorem settles the modal logic of end-extensional potentialism, showing that the potentialist validities of the models of set theory under end-extensions are exactly the assertions of S4. Finally, we introduce the end-extensional maximality principle, which asserts that every possibly necessary sentence is already true, and show that every countable model extends to a model satisfying it.

• The universal algorithm,
• J. D. Hamkins and H. W. Woodin, “The universal finite set,” ArXiv e-prints, pp. 1-16, 2017. (manuscript under review)
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• The modal logic of arithmetic potentialism,
• J. D. Hamkins, “The modal logic of arithmetic potentialism and the universal algorithm,” ArXiv e-prints, pp. 1-35, 2018. (under review)
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• A new proof of the Barwise extension theorem
• Kameryn’s blog post about the paper

# Representing Ordinal Numbers with Arithmetically Interesting Sets of Real Numbers

• D. D. Blair, J. D. Hamkins, and K. O’Bryant, “Representing Ordinal Numbers with Arithmetically Interesting Sets of Real Numbers,” Mathematics arXiv, 2019. (under review)
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Abstract. For a real number $x$ and set of natural numbers $A$, define $x∗A = \{xa \mod 1 \mid a \in A\}\subseteq [0,1)$. We consider relationships between $x$, $A$, and the order-type of $x∗A$. For example, for every irrational $x$ and order-type $\alpha$, there is an $A$ with $x ∗ A \simeq\alpha$, but if $\alpha$ is an ordinal, then $A$ must be a thin set. If, however, $A$ is restricted to be a subset of the powers of $2$, then not every order type is possible, although arbitrarily large countable well orders arise.

# Kelley-Morse set theory does not prove the class Fodor principle

• V. Gitman, J. D. Hamkins, and A. Karagila, “Kelley-Morse set theory does not prove the class Fodor theorem.” (manuscript under review)
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Abstract.
We show that Kelley-Morse (KM) set theory does not prove the class Fodor principle, the assertion that every regressive class function $F:S\to\newcommand\Ord{\text{Ord}}\Ord$ defined on a stationary class $S$ is constant on a stationary subclass. Indeed, it is relatively consistent with KM for any infinite $\lambda$ with $\omega\leq\lambda\leq\Ord$ that there is a class function $F:\Ord\to\lambda$ that is not constant on any stationary class. Strikingly, it is consistent with KM that there is a class $A\subseteq\omega\times\Ord$, such that each section $A_n=\{\alpha\mid (n,\alpha)\in A\}$ contains a class club, but $\bigcap_n A_n$ is empty. Consequently, it is relatively consistent with KM that the class club filter is not $\sigma$-closed.

The class Fodor principle is the assertion that every regressive class function $F:S\to\Ord$ defined on a stationary class $S$ is constant on a stationary subclass of $S$. This statement can be expressed in the usual second-order language of set theory, and the principle can therefore be sensibly considered in the context of any of the various second-order set-theoretic systems, such as Gödel-Bernays (GBC) set theory or Kelley-Morse (KM) set theory. Just as with the classical Fodor’s lemma in first-order set theory, the class Fodor principle is equivalent, over a weak base theory, to the assertion that the class club filter is normal. We shall investigate the strength of the class Fodor principle and try to find its place within the natural hierarchy of second-order set theories. We shall also define and study weaker versions of the class Fodor principle.

If one tries to prove the class Fodor principle by adapting one of the classical proofs of the first-order Fodor’s lemma, then one inevitably finds oneself needing to appeal to a certain second-order class-choice principle, which goes beyond the axiom of choice and the global choice principle, but which is not available in Kelley-Morse set theory. For example, in one standard proof, we would want for a given $\Ord$-indexed sequence of non-stationary classes to be able to choose for each member of it a class club that it misses. This would be an instance of class-choice, since we seek to choose classes here, rather than sets. The class choice principle $\text{CC}(\Pi^0_1)$, it turns out, is sufficient for us to make these choices, for this principle states that if every ordinal $\alpha$ admits a class $A$ witnessing a $\Pi^0_1$-assertion $\varphi(\alpha,A)$, allowing class parameters, then there is a single class $B\subseteq \Ord\times V$, whose slices $B_\alpha$ witness $\varphi(\alpha,B_\alpha)$; and the property of being a class club avoiding a given class is $\Pi^0_1$ expressible.

Thus, the class Fodor principle, and consequently also the normality of the class club filter, is provable in the relatively weak second-order set theory $\text{GBC}+\text{CC}(\Pi^0_1)$. This theory is known to be weaker in consistency strength than the theory $\text{GBC}+\Pi^1_1$-comprehension, which is itself strictly weaker in consistency strength than KM.

But meanwhile, although the class choice principle is weak in consistency strength, it is not actually provable in KM; indeed, even the weak fragment $\text{CC}(\Pi^0_1)$ is not provable in KM. Those results were proved several years ago by the first two authors, but they can now be seen as consequences of the main result of this article (see corollary 15. In light of that result, however, one should perhaps not have expected to be able to prove the class Fodor principle in KM.

Indeed, it follows similarly from arguments of the third author in his dissertation that if $\kappa$ is an inaccessible cardinal, then there is a forcing extension $V[G]$ with a symmetric submodel $M$ such that $V_\kappa^M=V_\kappa$, which implies that $\mathcal M=(V_\kappa,\in, V^M_{\kappa+1})$ is a model of Kelley-Morse, and in $\mathcal M$, the class Fodor principle fails in a very strong sense.

In this article, adapting the ideas of Karagila to the second-order set-theoretic context and using similar methods as in Gitman and Hamkins’s previous work on KM, we shall prove that every model of KM has an extension in which the class Fodor principle fails in that strong sense: there can be a class function $F:\Ord\to\omega$, which is not constant on any stationary class. In particular, in these models, the class club filter is not $\sigma$-closed: there is a class $B\subseteq\omega\times\Ord$, each of whose vertical slices $B_n$ contains a class club, but $\bigcap B_n$ is empty.

Main Theorem. Kelley-Morse set theory KM, if consistent, does not prove the class Fodor principle. Indeed, if there is a model of KM, then there is a model of KM with a class function $F:\Ord\to \omega$, which is not constant on any stationary class; in this model, therefore, the class club filter is not $\sigma$-closed.

We shall also investigate various weak versions of the class Fodor principle.

Definition.

1. For a cardinal $\kappa$, the class $\kappa$-Fodor principle asserts that every class function $F:S\to\kappa$ defined on a stationary class $S\subseteq\Ord$ is constant on a stationary subclass of $S$.
2. The class ${<}\Ord$-Fodor principle is the assertion that the $\kappa$-class Fodor principle holds for every cardinal $\kappa$.
3. The bounded class Fodor principle asserts that every regressive class function $F:S\to\Ord$ on a stationary class $S\subseteq\Ord$ is bounded on a stationary subclass of $S$.
4. The very weak class Fodor principle asserts that every regressive class function $F:S\to\Ord$ on a stationary class $S\subseteq\Ord$ is constant on an unbounded subclass of $S$.

We shall separate these principles as follows.

Theorem. Suppose KM is consistent.

1. There is a model of KM in which the class Fodor principle fails, but the class ${<}\Ord$-Fodor principle holds.
2. There is a model of KM in which the class $\omega$-Fodor principle fails, but the bounded class Fodor principle holds.
3. There is a model of KM in which the class $\omega$-Fodor principle holds, but the bounded class Fodor principle fails.
4. $\text{GB}^-$ proves the very weak class Fodor principle.

Finally, we show that the class Fodor principle can neither be created nor destroyed by set forcing.

Theorem. The class Fodor principle is invariant by set forcing over models of $\text{GBC}^-$. That is, it holds in an extension if and only if it holds in the ground model.

Let us conclude this brief introduction by mentioning the following easy negative instance of the class Fodor principle for certain GBC models. This argument seems to be a part of set-theoretic folklore. Namely, consider an $\omega$-standard model of GBC set theory $M$ having no $V_\kappa^M$ that is a model of ZFC. A minimal transitive model of ZFC, for example, has this property. Inside $M$, let $F(\kappa)$ be the least $n$ such that $V_\kappa^M$ fails to satisfy $\Sigma_n$-collection. This is a definable class function $F:\Ord^M\to\omega$ in $M$, but it cannot be constant on any stationary class in $M$, because by the reflection theorem there is a class club of cardinals $\kappa$ such that $V_\kappa^M$ satisfies $\Sigma_n$-collection.

• V. Gitman, J. D. Hamkins, and A. Karagila, “Kelley-Morse set theory does not prove the class Fodor theorem.” (manuscript under review)
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# The axiom of well-ordered replacement is equivalent to full replacement over Zermelo + foundation

In recent work, Alfredo Roque Freire and I have realized that the axiom of well-ordered replacement is equivalent to the full replacement axiom, over the Zermelo set theory with foundation.

The well-ordered replacement axiom is the scheme asserting that if $I$ is well-ordered and every $i\in I$ has unique $y_i$ satisfying a property $\phi(i,y_i)$, then $\{y_i\mid i\in I\}$ is a set. In other words, the image of a well-ordered set under a first-order definable class function is a set.

Alfredo had introduced the theory Zermelo + foundation + well-ordered replacement, because he had noticed that it was this fragment of ZF that sufficed for an argument we were mounting in a joint project on bi-interpretation. At first, I had found the well-ordered replacement theory a bit awkward, because one can only apply the replacement axiom with well-orderable sets, and without the axiom of choice, it seemed that there were not enough of these to make ordinary set-theoretic arguments possible.

But now we know that in fact, the theory is equivalent to ZF.

Theorem. The axiom of well-ordered replacement is equivalent to full replacement over Zermelo set theory with foundation.

$$\text{ZF}\qquad = \qquad\text{Z} + \text{foundation} + \text{well-ordered replacement}$$

Proof. Assume Zermelo set theory with foundation and well-ordered replacement.

Well-ordered replacement is sufficient to prove that transfinite recursion along any well-order works as expected. One proves that every initial segment of the order admits a unique partial solution of the recursion up to that length, using well-ordered replacement to put them together at limits and overall.

Applying this, it follows that every set has a transitive closure, by iteratively defining $\cup^n x$ and taking the union. And once one has transitive closures, it follows that the foundation axiom can be taken either as the axiom of regularity or as the $\in$-induction scheme, since for any property $\phi$, if there is a set $x$ with $\neg\phi(x)$, then let $A$ be the set of elements $a$ in the transitive closure of $\{x\}$ with $\neg\phi(a)$; an $\in$-minimal element of $A$ is a set $a$ with $\neg\phi(a)$, but $\phi(b)$ for all $b\in a$.

Another application of transfinite recursion shows that the $V_\alpha$ hierarchy exists. Further, we claim that every set $x$ appears in the $V_\alpha$ hierarchy. This is not immediate and requires careful proof. We shall argue by $\in$-induction using foundation. Assume that every element $y\in x$ appears in some $V_\alpha$. Let $\alpha_y$ be least with $y\in V_{\alpha_y}$. The problem is that if $x$ is not well-orderable, we cannot seem to collect these various $\alpha_y$ into a set. Perhaps they are unbounded in the ordinals? No, they are not, by the following argument. Define an equivalence relation $y\sim y’$ iff $\alpha_y=\alpha_{y’}$. It follows that the quotient $x/\sim$ is well-orderable, and thus we can apply well-ordered replacement in order to know that $\{\alpha_y\mid y\in x\}$ exists as a set. The union of this set is an ordinal $\alpha$ with $x\subseteq V_\alpha$ and so $x\in V_{\alpha+1}$. So by $\in$-induction, every set appears in some $V_\alpha$.

The argument establishes the principle: for any set $x$ and any definable class function $F:x\to\text{Ord}$, the image $F\mathrel{\text{”}}x$ is a set. One proves this by defining an equivalence relation $y\sim y’\leftrightarrow F(y)=F(y’)$ and observing that $x/\sim$ is well-orderable.

We can now establish the collection axiom, using a similar idea. Suppose that $x$ is a set and every $y\in x$ has a witness $z$ with $\phi(y,z)$. Every such $z$ appears in some $V_\alpha$, and so we can map each $y\in x$ to the smallest $\alpha_y$ such that there is some $z\in V_{\alpha_y}$ with $\phi(y,z)$. By the observation of the previous paragraph, the set of $\alpha_y$ exists and so there is an ordinal $\alpha$ larger than all of them, and thus $V_\alpha$ serves as a collecting set for $x$ and $\phi$, verifying this instance of collection.

From collection and separation, we can deduce the replacement axiom $\Box$

I’ve realized that this allows me to improve an argument I had made some time ago, concerning Transfinite recursion as a fundamental principle. In that argument, I had proved that ZC + foundation + transfinite recursion is equivalent to ZFC, essentially by showing that the principle of transfinite recursion implies replacement for well-ordered sets. The new realization here is that we do not need the axiom of choice in that argument, since transfinite recursion implies well-ordered replacement, which gives us full replacement by the argument above.

Corollary. The principle of transfinite recursion is equivalent to the replacement axiom over Zermelo set theory with foundation.

$$\text{ZF}\qquad = \qquad\text{Z} + \text{foundation} + \text{transfinite recursion}$$

There is no need for the axiom of choice.

# Set-theoretic blockchains

• M. E. Habič, J. D. Hamkins, L. D. Klausner, J. Verner, and K. J. Williams, “Set-theoretic blockchains,” Archive for Mathematical Logic, 2019.
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Abstract. Given a countable model of set theory, we study the structure of its generic multiverse, the collection of its forcing extensions and ground models, ordered by inclusion. Mostowski showed that any finite poset embeds into the generic multiverse while preserving the nonexistence of upper bounds. We obtain several improvements of his result, using what we call the blockchain construction to build generic objects with varying degrees of mutual genericity. The method accommodates certain infinite posets, and we can realize these embeddings via a wide variety of forcing notions, while providing control over lower bounds as well. We also give a generalization to class forcing in the context of second-order set theory, and exhibit some further structure in the generic multiverse, such as the existence of exact pairs.

# Topological models of arithmetic

• A. Enayat, J. D. Hamkins, and B. Wcisło, “Topological models of arithmetic,” ArXiv e-prints, 2018. (under review)
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The first author had inquired whether a nonstandard model of arithmetic could be continuously presented on the rational numbers.

Main Question. (Enayat, 2009) Are there continuous functions $\oplus$ and $\otimes$ on the rational numbers $\Q$, such that $\langle\Q,\oplus,\otimes\rangle$ is a nonstandard model of arithmetic?

By a model of arithmetic, what we mean here is a model of the first-order Peano axioms PA, although we also consider various weakenings of this theory. The theory PA asserts of a structure $\langle M,+,\cdot\rangle$ that it is the non-negative part of a discretely ordered ring, plus the induction principle for assertions in the language of arithmetic. The natural numbers $\newcommand\N{\mathbb{N}}\langle \N,+,\cdot\rangle$, for example, form what is known as the standard model of PA, but there are also many nonstandard models, including continuum many non-isomorphic countable models.

We answer the question affirmatively, and indeed, the main theorem shows that every countable model of PA is continuously presented on $\Q$. We define generally that a topological model of arithmetic is a topological space $X$ equipped with continuous functions $\oplus$ and $\otimes$, for which $\langle X,\oplus,\otimes\rangle$ satisfies the desired arithmetic theory. In such a case, we shall say that the underlying space $X$ continuously supports a model of arithmetic and that the model is continuously presented upon the space $X$.

Question. Which topological spaces support a topological model of arithmetic?

In the paper, we prove that the reals $\R$, the reals in any finite dimension $\R^n$, the long line and Cantor space do not support a topological model of arithmetic, and neither does any Suslin line. Meanwhile, there are many other spaces that do support topological models, including many uncountable subspaces of the plane $\R^2$. It remains an open question whether any uncountable Polish space, including the Baire space, can support a topological model of arithmetic.

Let me state the main theorem and briefly sketch the proof.

Main Theorem. Every countable model of PA has a continuous presentation on the rationals $\Q$.

Proof. We shall prove the theorem first for the standard model of arithmetic $\langle\N,+,\cdot\rangle$. Every school child knows that when computing integer sums and products by the usual algorithms, the final digits of the result $x+y$ or $x\cdot y$ are completely determined by the corresponding final digits of the inputs $x$ and $y$. Presented with only final segments of the input, the child can nevertheless proceed to compute the corresponding final segments of the output.

\begin{equation*}\small\begin{array}{rcr}
\end{array}\end{equation*}

This phenomenon amounts exactly to the continuity of addition and multiplication with respect to what we call the final-digits topology on $\N$, which is the topology having basic open sets $U_s$, the set of numbers whose binary representations ends with the digits $s$, for any finite binary string $s$. (One can do a similar thing with any base.) In the $U_s$ notation, we include the number that would arise by deleting initial $0$s from $s$; for example, $6\in U_{00110}$. Addition and multiplication are continuous in this topology, because if $x+y$ or $x\cdot y$ has final digits $s$, then by the school-child’s observation, this is ensured by corresponding final digits in $x$ and $y$, and so $(x,y)$ has an open neighborhood in the final-digits product space, whose image under the sum or product, respectively, is contained in $U_s$.

Let us make several elementary observations about the topology. The sets $U_s$ do indeed form the basis of a topology, because $U_s\cap U_t$ is empty, if $s$ and $t$ disagree on some digit (comparing from the right), or else it is either $U_s$ or $U_t$, depending on which sequence is longer. The topology is Hausdorff, because different numbers are distinguished by sufficiently long segments of final digits. There are no isolated points, because every basic open set $U_s$ has infinitely many elements. Every basic open set $U_s$ is clopen, since the complement of $U_s$ is the union of $U_t$, where $t$ conflicts on some digit with $s$. The topology is actually the same as the metric topology generated by the $2$-adic valuation, which assigns the distance between two numbers as $2^{-k}$, when $k$ is largest such that $2^k$ divides their difference; the set $U_s$ is an open ball in this metric, centered at the number represented by $s$. (One can also see that it is metric by the Urysohn metrization theorem, since it is a Hausdorff space with a countable clopen basis, and therefore regular.) By a theorem of Sierpinski, every countable metric space without isolated points is homeomorphic to the rational line $\Q$, and so we conclude that the final-digits topology on $\N$ is homeomorphic to $\Q$. We’ve therefore proved that the standard model of arithmetic $\N$ has a continuous presentation on $\Q$, as desired.

But let us belabor the argument somewhat, since we find it interesting to notice that the final-digits topology (or equivalently, the $2$-adic metric topology on $\N$) is precisely the order topology of a certain definable order on $\N$, what we call the final-digits order, an endless dense linear order, which is therefore order-isomorphic and thus also homeomorphic to the rational line $\Q$, as desired.

Specifically, the final-digits order on the natural numbers, pictured in figure 1, is the order induced from the lexical order on the finite binary representations, but considering the digits from right-to-left, giving higher priority in the lexical comparison to the low-value final digits of the number. To be precise, the final-digits order $n\triangleleft m$ holds, if at the first point of disagreement (from the right) in their binary representation, $n$ has $0$ and $m$ has $1$; or if there is no disagreement, because one of them is longer, then the longer number is lower, if the next digit is $0$, and higher, if it is $1$ (this is not the same as treating missing initial digits as zero). Thus, the even numbers appear as the left half of the order, since their final digit is $0$, and the odd numbers as the right half, since their final digit is $1$, and $0$ is directly in the middle; indeed, the highly even numbers, whose representations end with a lot of zeros, appear further and further to the left, while the highly odd numbers, which end with many ones, appear further and further to the right. If one does not allow initial $0$s in the binary representation of numbers, then note that zero is represented in binary by the empty sequence. It is evident that the final-digits order is an endless dense linear order on $\N$, just as the corresponding lexical order on finite binary strings is an endless dense linear order.

The basic open set $U_s$ of numbers having final digits $s$ is an open set in this order, since any number ending with $s$ is above a number with binary form $100\cdots0s$ and below a number with binary form $11\cdots 1s$ in the final-digits order; so $U_s$ is a union of intervals in the final-digits order. Conversely, every interval in the final-digits order is open in the final-digits topology, because if $n\triangleleft x\triangleleft m$, then this is determined by some final segment of the digits of $x$ (appending initial $0$s if necessary), and so there is some $U_s$ containing $x$ and contained in the interval between $n$ and $m$. Thus, the final-digits topology is the precisely same as the order topology of the final-digits order, which is a definable endless dense linear order on $\N$. Since this order is isomorphic and hence homeomorphic to the rational line $\Q$, we conclude again that $\langle \N,+,\cdot\rangle$ admits a continuous presentation on $\Q$.

We now complete the proof by considering an arbitrary countable model $M$ of PA. Let $\triangleleft^M$ be the final-digits order as defined inside $M$. Since the reasoning of the above paragraphs can be undertaken in PA, it follows that $M$ can see that its addition and multiplication are continuous with respect to the order topology of its final-digits order. Since $M$ is countable, the final-digits order of $M$ makes it a countable endless dense linear order, which by Cantor’s theorem is therefore order-isomorphic and hence homeomorphic to $\Q$. Thus, $M$ has a continuous presentation on the rational line $\Q$, as desired. $\Box$

The executive summary of the proof is: the arithmetic of the standard model $\N$ is continuous with respect to the final-digits topology, which is the same as the $2$-adic metric topology on $\N$, and this is homeomorphic to the rational line, because it is the order topology of the final-digits order, a definable endless dense linear order; applied in a nonstandard model $M$, this observation means the arithmetic of $M$ is continuous with respect to its rational line $\Q^M$, which for countable models is isomorphic to the actual rational line $\Q$, and so such an $M$ is continuously presentable upon $\Q$.

Let me mention the following order, which it seems many people expect to use instead of the final-digits order as we defined it above. With this order, one in effect takes missing initial digits of a number as $0$, which is of course quite reasonable.

The problem with this order, however, is that the order topology is not actually the final-digits topology. For example, the set of all numbers having final digits $110$ in this order has a least element, the number $6$, and so it is not open in the order topology. Worse, I claim that arithmetic is not continuous with respect to this order. For example, $1+1=2$, and $2$ has an open neighborhood consisting entirely of even numbers, but every open neighborhood of $1$ has both odd and even numbers, whose sums therefore will not all be in the selected neighborhood of $2$. Even the successor function $x\mapsto x+1$ is not continuous with respect to this order.

Finally, let me mention that a version of the main theorem also applies to the integers $\newcommand\Z{\mathbb{Z}}\Z$, using the following order.

Go to the article to read more.

• A. Enayat, J. D. Hamkins, and B. Wcisło, “Topological models of arithmetic,” ArXiv e-prints, 2018. (under review)
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# Open class determinacy is preserved by forcing

• J. D. Hamkins and H. W. Woodin, “Open class determinacy is preserved by forcing,” ArXiv e-prints, pp. 1-14, 2018. (under review)
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Abstract. The principle of open class determinacy is preserved by pre-tame class forcing, and after such forcing, every new class well-order is isomorphic to a ground-model class well-order. Similarly, the principle of elementary transfinite recursion ETR${}_\Gamma$ for a fixed class well-order $\Gamma$ is preserved by pre-tame class forcing. The full principle ETR itself is preserved by countably strategically closed pre-tame class forcing, and after such forcing, every new class well-order is isomorphic to a ground-model class well-order. Meanwhile, it remains open whether ETR is preserved by all forcing, including the forcing merely to add a Cohen real.

The principle of elementary transfinite recursion ETR — according to which every first-order class recursion along any well-founded class relation has a solution — has emerged as a central organizing concept in the hierarchy of second-order set theories from Gödel-Bernays set theory GBC up to Kelley-Morse set theory KM and beyond. Many of the principles in the hierarchy can be seen as asserting that certain class recursions have solutions.

In addition, many of these principles, including ETR and its variants, are equivalently characterized as determinacy principles for certain kinds of class games. Thus, the hierarchy is also fruitfully unified and organized by determinacy ideas.

This hierarchy of theories is the main focus of study in the reverse mathematics of second-order set theory, an emerging subject aiming to discover the precise axiomatic principles required for various arguments and results in second-order set theory. The principle ETR itself, for example, is equivalent over GBC to the principle of clopen determinacy for class games and also to the existence of iterated elementary truth predicates (see Open determinacy for class games); since every clopen game is also an open game, the principle ETR is naturally strengthened by the principle of open determinacy for class games, and this is a strictly stronger principle (see Hachtman and Sato); the weaker principle ETR${}_\text{Ord}$, meanwhile, asserting solutions to class recursions of length Ord, is equivalent to the class forcing theorem, which asserts that every class forcing notion admits a forcing relation, to the existence of set-complete Boolean completions of any class partial order, to the existence of Ord-iterated elementary truth predicates, to the determinacy of clopen games of rank at most Ord+1, and to other natural set-theoretic principles (see The exact strength of the class forcing theorem).

Since one naturally seeks in any subject to understand how one’s fundamental principles and tools interact, we should like in this article to consider how these second-order set-theoretic principles are affected by forcing. These questions originated in previous work of Victoria Gitman and myself, and question 1 also appears in the dissertation of Kameryn Williams, which was focused on the structure of models of second-order set theories.

It is well-known, of course, that ZFC, GBC, and KM are preserved by set forcing and by tame class forcing, and this is true for other theories in the hierarchy, such as GBC$+\Pi^1_n$-comprehension and higher levels of the second-order comprehension axiom. The corresponding forcing preservation theorem for ETR and for open class determinacy, however, has not been known.

Question 1. Is ETR preserved by forcing?

Question 2. Is open class determinacy preserved by forcing?

We intend to ask in each case about class forcing as well as set forcing. Question 1 is closely connected with the question of whether forcing can create new class well-order order types, longer than any class well-order in the ground model. Specifically, Victoria Gitman and I had observed earlier that ETR${}_\Gamma$ for a specific class well-order $\Gamma$ is preserved by pre-tame class forcing, and this would imply that the full principle ETR would also be preserved, if no fundamentally new class well-orders are created by the forcing. In light of the fact that forcing over models of ZFC adds no new ordinals, that would seem reasonable, but proof is elusive, and the question remains open. Can forcing add new class well-orders, perhaps very tall ones that are not isomorphic to any ground model class well-order? Perhaps there are some very strange models of GBC that gain new class well-order order types in a forcing extension.

Question 3. Assume GBC. After forcing, must every new class well-order be isomorphic to a ground-model class well-order? Does ETR imply this?

The main theorem of this article provides a full affirmative answer to question 2, and partial affirmative answers to questions 2 and 3.

Main Theorem.

1. Open class determinacy is preserved by pre-tame class forcing. After such forcing, every new class well-order is isomorphic to a ground-model class well-order.
2. The principle ETR${}_\Gamma$, for any fixed class well order $\Gamma$, is preserved by pre-tame class forcing.
3. The full principle ETR is preserved by countably strategically closed pre-tame class forcing. After such forcing, every new class well-order is isomorphic to a ground-model class well-order.

We should like specifically to highlight the fact that questions 1 and 3 remain open even in the case of the forcing to add a Cohen real. Is ETR preserved by the forcing to add a Cohen real? After adding a Cohen real, is every new class well-order isomorphic to a ground-model class well-order? One naturally expects affirmative answers, especially in a model of ETR.

For more, click through the arxiv for a pdf of the full article.

• J. D. Hamkins and H. W. Woodin, “Open class determinacy is preserved by forcing,” ArXiv e-prints, pp. 1-14, 2018. (under review)
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# The subseries number

• J. Brendle, W. Brian, and J. D. Hamkins, “The subseries number,” Fund. Math., vol. 247, iss. 1, pp. 49-85, 2019.
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Abstract. Every conditionally convergent series of real numbers has a divergent  subseries. How many subsets of the natural numbers are needed so that every conditionally convergent series diverges on the subseries corresponding to one of these sets? The answer to this question is defined to be the subseries number, a new cardinal characteristic of the continuum. This cardinal is bounded below by $\aleph_1$ and  above by the cardinality of the continuum, but it is not provably equal to either. We define three natural variants of the subseries number, and compare them with each other, with their corresponding rearrangement numbers, and with several well-studied cardinal characteristics of the continuum. Many consistency results are obtained from these comparisons, and we obtain another by computing the value of the subseries number in the Laver model.

This paper grew naturally out of our previous paper, The rearrangement number, which considered the minimal number of permutations of $\mathbb{N}$ which suffice to reveal the conditional convergence of all conditionally convergent series. I had defined the subseries number in my answer to a MathOverflow question, On Hamkins’s answer to a question of Michael Hardy’s, asked by M. Rahman in response to the earlier MO questions on the rearrangement number.

In the paper, we situation the subseries number  (German sharp s) with respect to other cardinal characteristics, including the rearrangement numbers.

# The modal logic of arithmetic potentialism and the universal algorithm

• J. D. Hamkins, “The modal logic of arithmetic potentialism and the universal algorithm,” ArXiv e-prints, pp. 1-35, 2018. (under review)
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Abstract. Natural potentialist systems arise from the models of arithmetic when they are considered under their various natural extension concepts, such as end-extensions, arbitrary extension, $\Sigma_n$-elementary extensions, conservative extensions and more. For these potentialist systems, I prove, a propositional modal assertion is valid in a model of arithmetic, with respect to assertions in the language of arithmetic with parameters, exactly when it is an assertion of S4. Meanwhile, with respect to sentences, the validities of a model are always between S4 and S5, and these bounds are sharp in that both endpoints are realized. The models validating exactly S5 are the models of the arithmetic maximality principle, which asserts that every possibly necessary statement is already true, and these models are equivalently characterized as those satisfying a maximal $\Sigma_1$ theory. The main proof makes fundamental use of the universal algorithm, of which this article provides a self-contained account.

In this article, I consider the models of arithmetic under various natural extension concepts, including end-extensions, arbitrary extensions, $\Sigma_n$-elementary extensions, conservative extensions and more. Each extension concept gives rise to an arithmetic potentialist system, a Kripke model of possible arithmetic worlds, and the main goal is to discover the modal validities of these systems.

For most of the extension concepts, a modal assertion is valid with respect to assertions in the language of arithmetic, allowing parameters, exactly when it is an assertion of the modal theory S4. For sentences, however, the modal validities form a theory between S4 and S5, with both endpoints being realized. A model of arithmetic validates S5 with respect to sentences just in case it is a model of the arithmetic maximality principle, and these models are equivalently characterized as those realizing a maximal $\Sigma_1$ theory.

The main argument relies fundamentally on the universal algorithm, the theorem due to Woodin that there is a Turing machine program that can enumerate any finite sequence in the right model of arithmetic, and furthermore this model can be end-extended so as to realize any further extension of that sequence available in the model. In the paper, I give a self-contained account of this theorem using my simplified proof.

The paper concludes with philosophical remarks on the nature of potentialism, including a discussion of how the linear inevitability form of potentialism is actually much closer to actualism than the more radical forms of potentialism, which exhibit branching possibility. I also propose to view the philosphy of ultrafinitism in modal terms as a form of potentialism, pushing the issue of branching possibility in ultrafinitism to the surface.

# The universal finite set

• J. D. Hamkins and H. 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 H. W. Woodin, “The universal finite set,” ArXiv e-prints, pp. 1-16, 2017. (manuscript under review)
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# The set-theoretic universe is not necessarily a class-forcing extension of HOD

• J. D. Hamkins and J. Reitz, “The set-theoretic universe $V$ is not necessarily a class-forcing extension of HOD,” ArXiv e-prints, 2017. (manuscript under review)
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Abstract. In light of the celebrated theorem of Vopěnka, proving in ZFC that every set is generic over $\newcommand\HOD{\text{HOD}}\HOD$, it is natural to inquire whether the set-theoretic universe $V$ must be a class-forcing extension of $\HOD$ by some possibly proper-class forcing notion in $\HOD$. We show, negatively, that if ZFC is consistent, then there is a model of ZFC that is not a class-forcing extension of its $\HOD$ for any class forcing notion definable in $\HOD$ and with definable forcing relations there (allowing parameters). Meanwhile, S. Friedman (2012) showed, positively, that if one augments $\HOD$ with a certain ZFC-amenable class $A$, definable in $V$, then the set-theoretic universe $V$ is a class-forcing extension of the expanded structure $\langle\HOD,\in,A\rangle$. Our result shows that this augmentation process can be necessary. The same example shows that $V$ is not necessarily a class-forcing extension of the mantle, and the method provides a counterexample to the intermediate model property, namely, a class-forcing extension $V\subseteq V[G]$ by a certain definable tame forcing and a transitive intermediate inner model $V\subseteq W\subseteq V[G]$ with $W\models\text{ZFC}$, such that $W$ is not a class-forcing extension of $V$ by any class forcing notion with definable forcing relations in $V$. This improves upon a previous example of Friedman (1999) by omitting the need for $0^\sharp$.

In 1972, Vopěnka proved the following celebrated result.

Theorem. (Vopěnka) If $V=L[A]$ where $A$ is a set of ordinals, then $V$ is a forcing extension of the inner model $\HOD$.

The result is now standard, appearing in Jech (Set Theory 2003, p. 249) and elsewhere, and the usual proof establishes a stronger result, stated in ZFC simply as the assertion: every set is generic over $\HOD$. In other words, for every set $a$ there is a forcing notion $\mathbb{B}\in\HOD$ and a $\HOD$-generic filter $G\subseteq\mathbb{B}$ for which $a\in\HOD[G]\subseteq V$. The full set-theoretic universe $V$ is therefore the union of all these various set-forcing generic extensions $\HOD[G]$.

It is natural to wonder whether these various forcing extensions $\HOD[G]$ can be unified or amalgamated to realize $V$ as a single class-forcing extension of $\HOD$ by a possibly proper class forcing notion in $\HOD$. We expect that it must be a very high proportion of set theorists and set-theory graduate students, who upon first learning of Vopěnka’s theorem, immediately ask this question.

Main Question. Must the set-theoretic universe $V$ be a class-forcing extension of $\HOD$?

We intend the question to be asking more specifically whether the universe $V$ arises as a bona-fide class-forcing extension of $\HOD$, in the sense that there is a class forcing notion $\mathbb{P}$, possibly a proper class, which is definable in $\HOD$ and which has definable forcing relation $p\Vdash\varphi(\tau)$ there for any desired first-order formula $\varphi$, such that $V$ arises as a forcing extension $V=\HOD[G]$ for some $\HOD$-generic filter $G\subseteq\mathbb{P}$, not necessarily definable.

In this article, we shall answer the question negatively, by providing a model of ZFC that cannot be realized as such a class-forcing extension of its $\HOD$.

Main Theorem. If ZFC is consistent, then there is a model of ZFC which is not a forcing extension of its $\HOD$ by any class forcing notion definable in that $\HOD$ and having a definable forcing relation there.

Throughout this article, when we say that a class is definable, we mean that it is definable in the first-order language of set theory allowing set parameters.

The main theorem should be placed in contrast to the following result of Sy Friedman.

Theorem. (Friedman 2012) There is a definable class $A$, which is strongly amenable to $\HOD$, such that the set-theoretic universe $V$ is a generic extension of $\langle \HOD,\in,A\rangle$.

This is a postive answer to the main question, if one is willing to augment $\HOD$ with a class $A$ that may not be definable in $\HOD$. Our main theorem shows that in general, this kind of augmentation process is necessary.

It is natural to ask a variant of the main question in the context of set-theoretic geology.

Question. Must the set-theoretic universe $V$ be a class-forcing extension of its mantle?

The mantle is the intersection of all set-forcing grounds, and so the universe is close in a sense to the mantle, perhaps one might hope that it is close enough to be realized as a class-forcing extension of it. Nevertheless, the answer is negative.

Theorem. If ZFC is consistent, then there is a model of ZFC that does not arise as a class-forcing extension of its mantle $M$ by any class forcing notion with definable forcing relations in $M$.

We also use our results to provide some counterexamples to the intermediate-model property for forcing. In the case of set forcing, it is well known that every transitive model $W$ of ZFC set theory that is intermediate $V\subseteq W\subseteq V[G]$ a ground model $V$ and a forcing extension $V[G]$, arises itself as a forcing extension $W=V[G_0]$.

In the case of class forcing, however, this can fail.

Theorem. If ZFC is consistent, then there are models of ZFC set theory $V\subseteq W\subseteq V[G]$, where $V[G]$ is a class-forcing extension of $V$ and $W$ is a transitive inner model of $V[G]$, but $W$ is not a forcing extension of $V$ by any class forcing notion with definable forcing relations in $V$.

Theorem. If ZFC + Ord is Mahlo is consistent, then one can form such a counterexample to the class-forcing intermediate model property $V\subseteq W\subseteq V[G]$, where $G\subset\mathbb{B}$ is $V$-generic for an Ord-c.c. tame definable complete class Boolean algebra $\mathbb{B}$, but nevertheless $W$ does not arise by class forcing over $V$ by any definable forcing notion with a definable forcing relation.

More complete details, please go to the paper (click through to the arxiv for a pdf).

• J. D. Hamkins and J. Reitz, “The set-theoretic universe $V$ is not necessarily a class-forcing extension of HOD,” ArXiv e-prints, 2017. (manuscript under review)
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# Inner-model reflection principles

• N. Barton, A. E. Caicedo, G. Fuchs, J. D. Hamkins, and J. Reitz, “Inner-model reflection principles,” ArXiv e-prints, 2017. (manuscript under review)
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Abstract. We introduce and consider the inner-model reflection principle, which asserts that whenever a statement $\varphi(a)$ in the first-order language of set theory is true in the set-theoretic universe $V$, then it is also true in a proper inner model $W\subsetneq V$. A stronger principle, the ground-model reflection principle, asserts that any such $\varphi(a)$ true in $V$ is also true in some nontrivial ground model of the universe with respect to set forcing. These principles each express a form of width reflection in contrast to the usual height reflection of the Lévy-Montague reflection theorem. They are each equiconsistent with ZFC and indeed $\Pi_2$-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH.

Every set theorist is familiar with the classical Lévy-Montague reflection principle, which explains how truth in the full set-theoretic universe $V$ reflects down to truth in various rank-initial segments $V_\theta$ of the cumulative hierarchy. Thus, the Lévy-Montague reflection principle is a form of height-reflection, in that truth in $V$ is reflected vertically downwards to truth in some $V_\theta$.

In this brief article, in contrast, we should like to introduce and consider a form of width-reflection, namely, reflection to nontrivial inner models. Specifically, we shall consider the following reflection principles.

Definition.

1. The inner-model reflection principle asserts that if a statement $\varphi(a)$ in the first-order language of set theory is true in the set-theoretic universe $V$, then there is a proper inner model $W$, a transitive class model of ZF containing all ordinals, with $a\in W\subsetneq V$ in which $\varphi(a)$ is true.
2. The ground-model reflection principle asserts that if $\varphi(a)$ is true in $V$, then there is a nontrivial ground model $W\subsetneq V$ with $a\in W$ and $W\models\varphi(a)$.
3. Variations of the principles arise by insisting on inner models of a particular type, such as ground models for a particular type of forcing, or by restricting the class of parameters or formulas that enter into the scheme.
4. The lightface forms of the principles, in particular, make their assertion only for sentences, so that if $\sigma$ is a sentence true in $V$, then $\sigma$ is true in some proper inner model or ground $W$, respectively.

We explain how to force the principles, how to separate them, how they are consequences of various large cardinal assumptions, consequences of the maximality principle and of the inner model hypothesis. Kindly proceed to the article (pdf available at the arxiv) for more.

• N. Barton, A. E. Caicedo, G. Fuchs, J. D. Hamkins, and J. Reitz, “Inner-model reflection principles,” ArXiv e-prints, 2017. (manuscript under review)
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This article grew out of an exchange held by the authors on math.stackexchange
in response to an inquiry posted by the first author concerning the nature of width-reflection in comparison to height-reflection:  What is the consistency strength of width reflection?

# Boolean ultrapowers, the Bukovský-Dehornoy phenomenon, and iterated ultrapowers

• G. Fuchs and J. D. Hamkins, “The Bukovský-Dehornoy phenomenon for Boolean ultrapowers,” ArXiv e-prints, 2017. (under review)
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Abstract. We show that while the length $\omega$ iterated ultrapower by a normal ultrafilter is a Boolean ultrapower by the Boolean algebra of Příkrý forcing, it is consistent that no iteration of length greater than $\omega$ (of the same ultrafilter and its images) is a Boolean ultrapower. For longer iterations, where different ultrafilters are used, this is possible, though, and we give Magidor forcing and a generalization of Příkrý forcing as examples. We refer to the discovery that the intersection of the finite iterates of the universe by a normal measure is the same as the generic extension of the direct limit model by the critical sequence as the Bukovský-Dehornoy phenomenon, and we develop a criterion (the existence of a simple skeleton) for when a version of this phenomenon holds in the context of Boolean ultrapowers.

# The exact strength of the class forcing theorem

• V. Gitman, J. D. Hamkins, P. Holy, P. Schlicht, and K. Williams, “The exact strength of the class forcing theorem,” ArXiv e-prints, 2017. (manuscript under review)
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We shall characterize the exact strength of the class forcing theorem, which asserts that every class forcing notion $\P$ has a corresponding forcing relation $\forces_\P$, a relation satisfying the forcing relation recursion. When there is such a forcing relation, then statements true in any corresponding forcing extension are forced and forced statements are true in those extensions.

Unlike the case of set forcing, where one may prove in ZFC that every set forcing notion has corresponding forcing relations, for class forcing it is consistent with Gödel-Bernays set theory GBC that there is a proper class forcing notion lacking a corresponding forcing relation, even merely for the atomic formulas. For certain forcing notions, the existence of an atomic forcing relation implies Con(ZFC) and much more, and so the consistency strength of the class forcing theorem strictly exceeds GBC, if this theory is consistent. Nevertheless, the class forcing theorem is provable in stronger theories, such as Kelley-Morse set theory. What is the exact strength of the class forcing theorem?

Our project here is to identify the strength of the class forcing theorem by situating it in the rich hierarchy of theories between GBC and KM, displayed in part in the figure above, with the class forcing theorem highlighted in blue. It turns out that the class forcing theorem is equivalent over GBC to an attractive collection of several other natural set-theoretic assertions; it is a robust axiomatic principle.

The main theorem is naturally part of the emerging subject we call the reverse mathematics of second-order set theory, a higher analogue of the perhaps more familiar reverse mathematics of second-order arithmetic. In this new research area, we are concerned with the hierarchy of second-order set theories between GBC and KM and beyond, analyzing the strength of various assertions in second-order set theory, such as the principle ETR of elementary transfinite recursion, the principle of $\Pi^1_1$-comprehension or the principle of determinacy for clopen class games. We fit these set-theoretic principles into the hierarchy of theories over the base theory GBC. The main theorem of this article does exactly this with the class forcing theorem by finding its exact strength in relation to nearby theories in this hierarchy.

Main Theorem. The following are equivalent over Gödel-Bernays set theory.

1. The atomic class forcing theorem: every class forcing notion admits forcing relations for atomic formulas $$p\forces\sigma=\tau\qquad\qquad p\forces\sigma\in\tau.$$
2. The class forcing theorem scheme: for each first-order formula $\varphi$ in the forcing language, with finitely many class names $\dot \Gamma_i$, there is a forcing relation applicable to this formula and its subformulas
$$p\forces\varphi(\vec \tau,\dot\Gamma_0,\ldots,\dot\Gamma_m).$$
3. The uniform first-order class forcing theorem: every class forcing notion $\P$ admits a uniform forcing relation $$p\forces\varphi(\vec \tau),$$ applicable to all assertions $\varphi$ in the first-order forcing language with finitely many class names $\mathcal{L}_{\omega,\omega}(\in,V^\P,\dot\Gamma_0,\ldots,\dot\Gamma_m)$.
4. The uniform infinitary class forcing theorem: every class forcing notion $\P$ admits a uniform forcing relation $$p\forces\varphi(\vec \tau),$$ applicable to all assertions $\varphi$ in the infinitary forcing language with finitely many class names $\mathcal{L}_{\Ord,\Ord}(\in,V^\P,\dot\Gamma_0,\ldots,\dot\Gamma_m)$.
5. Names for truth predicates: every class forcing notion $\P$ has a class name $\newcommand\T{{\rm T}}\dot\T$ and a forcing relation for which $1\forces\dot\T$ is a truth-predicate for the first-order forcing language with finitely many class names $\mathcal{L}_{\omega,\omega}(\in,V^\P,\dot\Gamma_0,\ldots,\dot\Gamma_m)$.
6. Every class forcing notion $\P$, that is, every separative class partial order, admits a Boolean completion $\mathbb{B}$, a set-complete class Boolean algebra into which $\P$ densely embeds.
7. The class-join separation principle plus $\ETR_{\Ord}$-foundation.
8. For every class $A$, there is a truth predicate for $\mathcal{L}_{\Ord,\omega}(\in,A)$.
9. For every class $A$, there is a truth predicate for $\mathcal{L}_{\Ord,\Ord}(\in,A)$.
10. For every class $A$, there is an $\Ord$-iterated truth predicate for $\mathcal{L}_{\omega,\omega}(\in,A)$.
11. The principle of determinacy for clopen class games of rank at most $\Ord+1$.
12. The principle $\ETR_{\Ord}$ of elementary transfinite recursion for $\Ord$-length recursions of first-order properties, using any class parameter.

We prove the theorem by establishing the complete cycle of indicated implications. The red arrows indicate more difficult or substantive implications, while the blue arrows indicate easier or nearly immediate implications. The green dashed implication from statement (12) to statement (1), while not needed for the completeness of the implication cycle, is nevertheless used in the proof that (12) implies (4). The proof of (12) implies (7) also uses (8), which follows from the fact that (12) implies (9) implies (8).

• V. Gitman, J. D. Hamkins, P. Holy, P. Schlicht, and K. Williams, “The exact strength of the class forcing theorem,” ArXiv e-prints, 2017. (manuscript under review)
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