arXiv subjects

Joel David Hamkins

Publications and source records attributed to Joel David Hamkins.

At least 37 records · Page 2Linked to original sources

arXiv2020

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

We show that Kelley-Morse set theory does not prove the class Fodor principle, the assertion that every regressive class function $F:S\to\text{Ord}$ defined on a stationary class $S$ is constant on a stationary subclass. Indeed, it is relatively consistent with KM for any infinite $λ$ with $ω\leqλ\leq\text{Ord}$ that there is a class function $F:\text{Ord}\toλ$ that is not constant on any stationary class. Strikingly, it is consistent with KM that there is a class $A\subseteqω\times\text{Ord}$, such that each section $A_n=\{α\mid (n,α)\in A\}$ 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 $σ$-closed.

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arXiv2020

Topological models of arithmetic

Ali Enayat had asked whether there is a nonstandard model of Peano arithmetic (PA) that can be represented as $\langle\mathbb{Q},\oplus,\otimes\rangle$, where $\oplus$ and $\otimes$ are continuous functions on the rationals $\mathbb{Q}$. We prove, affirmatively, that indeed every countable model of PA has such a continuous presentation on the rationals. More generally, we investigate the topological spaces that arise as such topological models of arithmetic. The reals $\mathbb{R}$, the reals in any finite dimension $\mathbb{R}^n$, the long line and the Cantor space do not, and neither does any Suslin line; many other spaces do; the status of the Baire space is open.

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arXiv2020

The Sigma_1-definable universal finite sequence

We introduce the $Σ_1$-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 $Σ_1$-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.

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arXiv2020

Modal model theory

We introduce the subject of modal model theory, where one studies a mathematical structure within a class of similar structures under an extension concept that gives rise to mathematically natural notions of possibility and necessity. A statement $φ$ is possible in a structure (written $\Diamondφ$) if $φ$ is true in some extension of that structure, and $φ$ is necessary (written $\Boxφ$) if it is true in all extensions of the structure. A principal case for us will be the class Mod(T) of all models of a given theory T---all graphs, all groups, all fields, or what have you---considered under the substructure relation. In this article, we aim to develop the resulting modal model theory. The class of all graphs is a particularly insightful case illustrating the remarkable power of the modal vocabulary, for the modal language of graph theory can express connectedness, $k$-colorability, finiteness, countability, size continuum, size $\aleph_1$, $\aleph_2$, $\aleph_ω$, $\beth_ω$, first $\beth$-fixed point, first $\beth$-hyper-fixed-point and much more. A graph obeys the maximality principle $\Diamond\Boxφ(a)\toφ(a)$ with parameters if and only if it satisfies the theory of the countable random graph, and it satisfies the maximality principle for sentences if and only if it is universal for finite graphs.

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arXiv2020

Bi-interpretation in weak set theories

In contrast to the robust mutual interpretability phenomenon in set theory, Ali Enayat proved that bi-interpretation is absent: distinct theories extending ZF are never bi-interpretable and models of ZF are bi-interpretable only when they are isomorphic. Nevertheless, for natural weaker set theories, we prove, including Zermelo-Fraenkel set theory $\text{ZFC}^-$ without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of $\text{ZFC}^-$ that are bi-interpretable, but not isomorphic---even $\langle H_{ω_1},\in\rangle$ and $\langle H_{ω_2},\in\rangle$ can be bi-interpretable---and there are distinct bi-interpretable theories extending $\text{ZFC}^-$. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails.

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arXiv2020

Choiceless large cardinals and set-theoretic potentialism

We define a potentialist system of ZF-structures, that is, a collection of possible worlds in the language of ZF connected by a binary accessibility relation, achieving a potentialist account of the full background set-theoretic universe $V$. The definition involves Berkeley cardinals, the strongest known large cardinal axioms, inconsistent with the Axiom of Choice. In fact, as background theory we assume just ZF. It turns out that the propositional modal assertions which are valid at every world of our system are exactly those in the modal theory S4.2. Moreover, we characterize the worlds satisfying the potentialist maximality principle, and thus the modal theory S5, both for assertions in the language of ZF and for assertions in the full potentialist language.

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arXiv2020

Representing Ordinal Numbers with Arithmetically Interesting Sets of Real Numbers

For a real number $x$ and set of natural numbers $A$, define $x \ast A := \{ x a \bmod 1: a\in A\}\subseteq [0,1).$ We consider relationships between $x$, $A$, and the order-type of $x\ast A$. For example, for every irrational $x$ and order-type $α$, there is an $A$ with $x\ast A \simeq α$, but if $α$ is a well order, 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.

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arXiv2019

The Rearrangement Number

How many permutations of the natural numbers are needed so that every conditionally convergent series of real numbers can be rearranged to no longer converge to the same sum? We define the \emph{rearrangement number}, a new cardinal characteristic of the continuum, as the answer to this question. We compare the rearrangement number with several natural variants, for example one obtained by requiring the rearranged series to still converge but to a new, finite limit. We also compare the rearrangement number with several well-studied cardinal characteristics of the continuum. We present some new forcing constructions designed to add permutations that rearrange series from the ground model in particular ways, thereby obtaining consistency results going beyond those that follow from comparisons with familiar cardinal characteristics. Finally, we deal briefly with some variants concerning rearrangements by a special sort of permutation and with rearranging some divergent series to become (conditionally) convergent.

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arXiv2019

Set-Theoretic Blockchains

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.

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arXiv2018

Inner-model reflection principles

We introduce and consider the inner-model reflection principle, which asserts that whenever a statement $φ(a)$ 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 $φ(a)$ true in $V$ is also true in some non-trivial 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 $Π_2$-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. We also consider some questions concerning the expressibility of the principles.

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arXiv2018

The modal logic of set-theoretic potentialism and the potentialist maximality principles

We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on those of Hamkins, Leibman and Löwe, including the use of buttons, switches, dials and ratchets. Among the potentialist conceptions we consider are: rank potentialism (true in all larger $V_β$); Grothendieck-Zermelo potentialism (true in all larger $V_κ$ for inaccessible cardinals $κ$); transitive-set potentialism (true in all larger transitive sets); forcing potentialism (true in all forcing extensions); countable-transitive-model potentialism (true in all larger countable transitive models of ZFC); countable-model potentialism (true in all larger countable models of ZFC); and others. In each case, we identify lower bounds for the modal validities, which are generally either S4.2 or S4.3, and an upper bound of S5, proving in each case that these bounds are optimal. The validity of S5 in a world is a potentialist maximality principle, an interesting set-theoretic principle of its own. The results can be viewed as providing an analysis of the modal commitments of the various set-theoretic multiverse conceptions corresponding to each potentialist account.

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arXiv2018

Open class determinacy is preserved by forcing

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 $\text{ETR}_Γ$ for a fixed class well-order $Γ$ 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.

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arXiv2018

The universal finite set

We define a certain finite set in set theory $\{x\midφ(x)\}$ 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 $φ$ has complexity $Σ_2$, so that any affirmative instance of it $φ(x)$ is verified in any sufficiently large rank-initial segment of the universe $V_θ$; the set is empty in any transitive model and others; and if $φ$ defines the set $y$ in some countable model $M$ of ZFC and $y\of z$ for some finite set $z$ in $M$, then there is a top-extension of $M$ to a model $N$ in which $φ$ defines the new set $z$. Thus, the set shows that no model of set theory can realize a maximal $Σ_2$ 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.

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arXiv2018

A model of the generic Vopěnka principle in which the ordinals are not Mahlo

The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vopěnka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a $Δ_2$-definable class containing no regular cardinals. In such a model, there can be no $Σ_2$-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.

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arXiv2018

The subseries number

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.

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arXiv2017

ZFC proves that the class of ordinals is not weakly compact for definable classes

We prove that the class of all ordinals Ord is not weakly compact with respect to definable classes. Specifically, in any model of ZFC, the definable tree property fails for Ord, in that there is a definable Ord tree with no definable cofinal branch; the definable partition property fails, in that there is a definable 2-coloring of pairs from a certain definable proper class, with no definable homogeneous proper class; and the definable compactness property fails for $\mathcal{L}_{\infty,ω}$, in that there is a definable theory in this logic all of whose set-sized subtheories are satisfiable, but which has no definable proper class model. In addition, we prove that the definable diamond principle $\Diamond_{\rm Ord}$ holds if and only if there is a definable well-ordering of the universe. And we prove that the common theory of all spartan models of Gödel-Bernays set theory, those having only definable classes, is $Π^1_1$-complete.

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arXiv2017

The set-theoretic universe $V$ is not necessarily a class-forcing extension of HOD

In light of the celebrated theorem of Vopěnka (1972), proving in ZFC that every set is generic over 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 $\langle\text{HOD},\in,A\rangle$. 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 counterexamples to the intermediate model property, namely, a class-forcing extension $V\subseteq W\subseteq V[G]$ with an intermediate transitive inner model $W$ that is neither a class-forcing extension of $V$ nor a ground model of $V[G]$ by any definable class forcing notion with definable forcing relations.

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arXiv2017

Boolean ultrapowers, the Bukovsky-Dehornoy phenomenon, and iterated ultrapowers

We show that while the length $ω$ iterated ultrapower by a normal ultrafilter is a Boolean ultrapower by the Boolean algebra of Prikry forcing, it is consistent that no iteration of length greater than $ω$ (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 Prikry 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 Bukovsky-Dehornoy phenomenon, and we develop a sufficient criterion (the existence of a simple skeleton) for when a version of this phenomenon holds in the context of Boolean ultrapowers. Assuming that the canonical generic filter over the Boolean ultrapower model has what we call a continuous representation, we show that the Boolean model consists precisely of those members of the intersection model that have continuously and eventually uniformly represented codes.

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