[bibtex key=”GroszekHamkins2019:The-implicitly-constructible-universe”]
Abstract. We answer several questions posed by Hamkins and Leahy concerning the implicitly constructible universe
[bibtex key=”GroszekHamkins2019:The-implicitly-constructible-universe”]
Abstract. We answer several questions posed by Hamkins and Leahy concerning the implicitly constructible universe
[bibtex key=BlassBrendleBrianHamkinsHardyLarson2020:TheRearrangementNumber]
Abstract. 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 show that the minimum number of permutations needed for this purpose, which we call the rearrangement number, is uncountable, but whether it equals the cardinal of the continuum is independent of the usual axioms of
set theory. 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 permutations and with rearranging some divergent series to become (conditionally) convergent.
This project started with Michael Hardy’s question on MathOverflow, How many rearrangements must fail to alter the value of a sum before you conclude that none do? I had proposed in my answer that we should think of the cardinal in question as a cardinal characteristic of the continuum, the rearrangement number, since we could prove that it was uncountable and that it was the continuum under MA, and had begun to separate it from other familiar cardinal characteristics. Eventually, the research effort grew into the collaboration of this paper. What a lot of fun!
Colloquium talk at Vassar | Lecture notes | talk at CUNY | the original MathOverflow question
The lecture notes are for an introductory talk on the topic I had given at the Vassar College Mathematics Colloquium.
[bibtex key=EnayatHamkins2018:Ord-is-not-definably-weakly-compact]
In ZFC the class of all ordinals is very like a large cardinal. Being closed under exponentiation, for example, Ord is a strong limit. Indeed, it is a beth fixed point. And Ord is regular with respect to definable classes by the replacement axiom. In this sense, ZFC therefore proves that Ord is definably inaccessible. Which other large cardinal properties are exhibited by Ord? Perhaps you wouldn’t find it unreasonable for Ord to exhibit, at least consistently with ZFC, the definable proper class analogues of other much stronger large cardinal properties?
Meanwhile, the main results of this paper, joint between myself and Ali Enayat, show that such an expectation would be misplaced, even for comparatively small large cardinal properties. Specifically, in a result that surprised me, it turns out that the class of ordinals NEVER exhibits the definable proper class analogue of weak compactness in any model of ZFC.
Theorem. The class of ordinals is not definably weakly compact. In every model of ZFC:
The proof uses methods from the model theory of set theory, including especially the fact that no model of ZFC has a conservative
Theorem. The definable
We close the paper by proving that the theory of the spartan models of Gödel-Bernays set theory GB — those equipped with only their definable classes — is
Theorem. The set of sentences true in all spartan models of GB is
[bibtex key=Hamkins:The-Vopenka-principle-is-inequivalent-to-but-conservative-over-the-Vopenka-scheme]
Abstract. The Vopěnka principle, which asserts that every proper class of first-order structures in a common language admits an elementary embedding between two of its members, is not equivalent over GBC to the first-order Vopěnka scheme, which makes the Vopěnka assertion only for the first-order definable classes of structures. Nevertheless, the two Vopěnka axioms are equiconsistent and they have exactly the same first-order consequences in the language of set theory. Specifically, GBC plus the Vopěnka principle is conservative over ZFC plus the Vopěnka scheme for first-order assertions in the language of set theory.
The Vopěnka principle is the assertion that for every proper class
In contrast, the first-order Vopěnka scheme makes the Vopěnka assertion only for the first-order definable classes
Because the Vopěnka principle is a second-order assertion, it does not make sense to refer to it in the context of ZFC set theory, whose first-order language does not allow quantification over classes; one typically retreats to the Vopěnka scheme in that context. The theme of this article is to investigate the precise meta-mathematical interactions between these two treatments of Vopěnka’s idea.
Main Theorems.
It follows that the Vopěnka principle VP and the Vopěnka scheme VS are not equivalent, but they are equiconsistent and indeed, they have the same first-order consequences.
Corollaries.
These results grew out of my my answer to a MathOverflow question of Mike Shulman, Can Vopěnka’s principle be violated definably?, inquiring whether there would always be a definable counterexample to the Vopěnka principle, whenever it should happen to fail. I interpret the question as asking whether the Vopěnka scheme is necessarily equivalent to the Vopěnka principle, and the answer is negative.
The proof of the main theorem involves the concept of a stretchable set
[bibtex key=HamkinsKikuchi2016:Set-theoreticMereology]
Abstract. We consider a set-theoretic version of mereology based on the inclusion relation
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
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
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
Theorem. In any universe of set theory
Corollary. One cannot define
A counterpoint to this is provided by the following theorem, however, which identifies a weak sense in which
Theorem. Assume ZFC in the universe
That counterpoint is not decisive, however, in light of the question whether we really need
Theorem. For any two consistent theories extending ZFC, there are models
For example, we cannot determine in
Theorem. Set-theoretic mereology, considered as the theory of
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
Mereology on MathOverflow | Mereology on Stanford Encyclopedia of Philosophy | Mereology on Wikipedia
Some previous posts on this blog:
[bibtex key=Hamkins2016:UpwardClosureAndAmalgamationInTheGenericMultiverse]
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
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.
[bibtex key=EvansHamkinsPerlmutter2017:APositionInInfiniteChessWithGameValueOmega^4]
Abstract. We present a position in infinite chess exhibiting an ordinal game value of
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
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 rook towers are similar to the corresponding part of the previous
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.
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
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.
Read more about it in the article, which is available at the arxiv (pdf).
See also:
[bibtex key=GitmanHamkins2016:OpenDeterminacyForClassGames]
Abstract. The principle of open determinacy for class games — two-player games of perfect information with plays of length
See my earlier posts on part of this material:
[bibtex key=Hamkins2015:AMathematiciansYearInJapan]
Years ago, when I was still a junior professor, I had the pleasure to live for a year in Japan, working as a research fellow at Kobe University. During that formative year, I recorded brief moments of my Japanese experience, and every two weeks or so—this was well before the current blogging era—I sent my descriptive missives by email to friends back home. I have now collected together those vignettes of my life in Japan, each a morsel of my experience. The book is now out!
A Mathematician’s Year in Japan
Joel David Hamkins
Glimpse into the life of a professor of logic as he fumbles his way through Japan.
A Mathematician’s Year in Japan is a lighthearted, though at times emotional account of how one mathematician finds himself in a place where everything seems unfamiliar, except his beloved research on the nature of infinity, yet even with that he experiences a crisis.
Available on Amazon $4.49.
Please be so kind as to write a review there.
[bibtex key=FuchsGitmanHamkins2018:EhrenfeuchtsLemmaInSetTheory]
This is joint work with Gunter Fuchs and Victoria Gitman.
Abstract. Ehrenfeucht’s lemma asserts that whenever one element of a model of Peano arithmetic is definable from another, then they satisfy different types. We consider here the analogue of Ehrenfeucht’s lemma for models of set theory. The original argument applies directly to the ordinal-definable elements of any model of set theory, and in particular, Ehrenfeucht’s lemma holds fully for models of set theory satisfying
[bibtex key=FuchsGitmanHamkins2017:IncomparableOmega1-likeModelsOfSetTheory]
This is joint work with Gunter Fuchs and Victoria Gitman.
Abstract. We show that the analogues of the Hamkins embedding theorems, proved for the countable models of set theory, do not hold when extended to the uncountable realm of
In this article, we consider the question of whether the embedding theorems of my article, Every countable model of set theory embeds into its own constructible universe, which concern the countable models of set theory, might extend to the realm of uncountable models. Specifically, in that paper I had proved that (1) any two countable models of set theory are comparable by embeddability; indeed, (2) one countable model of set theory embeds into another just in case the ordinals of the first order-embed into the ordinals of the second; consequently, (3) every countable model of set theory embeds into its own constructible universe; and furthermore, (4) every countable model of set theory embeds into the hereditarily finite sets
The embedding theorems are expressed collectively in the theorem below. An embedding of one model
Theorem.
1. For any two countable models of set theory
2. Indeed, such an
3. Consequently, every countable model
4. Furthermore, every countable model of set theory embeds into the hereditary finite sets
One can begin to get an appreciation for the difference in embedding concepts by observing that ZFC proves that there is a nontrivial embedding
We leave it as a fun exercise to verify that
We find it interesting to note in contrast to the theorem above that there is no such embedding phenomenon in the the context of the countable models of Peano arithmetic (where an embedding of models of arithmetic is a function preserving all atomic formulas in the language of arithmetic). Perhaps the main reason for this is that embeddings between models of PA are automatically
Our main theorems are as follows.
Theorem.
1. If
2. The models in statement (1) can be constructed so that their ordinals order-embed into each other and indeed, so that the ordinals of each model is a universal
3. If
4. If there is a Mahlo cardinal, then in a forcing extension of
Note that the size of the family
[bibtex key=ChengFriedmanHamkins2015:LargeCardinalsNeedNotBeLargeInHOD]
Abstract. We prove that large cardinals need not generally exhibit their large cardinal nature in HOD. For example, a supercompact cardinal
In this article, we prove that large cardinals need not generally exhibit their large cardinal nature in HOD, the inner model of hereditarily ordinal-definable sets, and there can be a divergence in strength between the large cardinals of the ambient set-theoretic universe
Questions.
1. To what extent must a large cardinal in
2. To what extent does the existence of large cardinals in
For large cardinal concepts beyond the weakest notions, we prove, the answers are generally negative. In Theorem 4, for example, we construct a model with a supercompact cardinal that is not weakly compact in HOD, and Theorem 9 extends this to a proper class of supercompact cardinals, none of which is weakly compact in HOD, thereby providing some strongly negative instances of (1). The same model has a proper class of supercompact cardinals, but no supercompact cardinals in HOD, providing a negative instance of (2). The natural common strengthening of these situations would be a model with a proper class of supercompact cardinals, but no weakly compact cardinals in HOD. We were not able to arrange that situation, however, and furthermore it would be ruled out by Conjecture 13, an intriguing positive instance of (2) recently proposed by W. Hugh Woodin, namely, that if there is a supercompact cardinal, then there is a measurable cardinal in HOD. Many other natural possibilities, such as a proper class of measurable cardinals with no weakly compact cardinals in HOD, remain as open questions.
[bibtex key=HamkinsJohnstone2017:StronglyUpliftingCardinalsAndBoldfaceResurrection]
Abstract. We introduce the strongly uplifting cardinals, which are equivalently characterized, we prove, as the superstrongly unfoldable cardinals and also as the almost hugely unfoldable cardinals, and we show that their existence is equiconsistent over ZFC with natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing.
The strongly uplifting cardinals, which we introduce in this article, are a boldface analogue of the uplifting cardinals introduced in our previous paper, Resurrection axioms and uplifting cardinals, and are equivalently characterized as the superstrongly unfoldable cardinals and also as the almost hugely unfoldable cardinals. In consistency strength, these new large cardinals lie strictly above the weakly compact, totally indescribable and strongly unfoldable cardinals and strictly below the subtle cardinals, which in turn are weaker in consistency than the existence of
Definitions.
Remarkably, these different-seeming large cardinal concepts turn out to be exactly equivalent to one another. A cardinal
Theorem. The following theories are equiconsistent over ZFC.
[bibtex key=HamkinsYang:SatisfactionIsNotAbsolute]
Abstract. We prove that the satisfaction relation
of first-order logic is not absolute between models of set theory having the structure and the formulas all in common. Two models of set theory can have the same natural numbers, for example, and the same standard model of arithmetic , yet disagree on their theories of arithmetic truth; two models of set theory can have the same natural numbers and the same arithmetic truths, yet disagree on their truths-about-truth, at any desired level of the iterated truth-predicate hierarchy; two models of set theory can have the same natural numbers and the same reals, yet disagree on projective truth; two models of set theory can have the same or the same rank-initial segment , yet disagree on which assertions are true in these structures. On the basis of these mathematical results, we argue that a philosophical commitment to the determinateness of the theory of truth for a structure cannot be seen as a consequence solely of the determinateness of the structure in which that truth resides. The determinate nature of arithmetic truth, for example, is not a consequence of the determinate nature of the arithmetic structure
itself, but rather, we argue, is an additional higher-order commitment requiring its own analysis and justification.
Many mathematicians and philosophers regard the natural numbers
Feferman provides an instance of this perspective when he writes (Feferman 2013, Comments for EFI Workshop, p. 6-7) :
In my view, the conception [of the bare structure of the natural numbers] is completely clear, and thence all arithmetical statements are definite.
It is Feferman’s `thence’ to which we call attention. Martin makes a similar point (Martin, 2012, Completeness or incompleteness of basic mathematical concepts):
What I am suggesting is that the real reason for confidence in first-order completeness is our confidence in the full determinateness of the concept of the natural numbers.
Many mathematicians and philosophers seem to share this perspective. The truth of an arithmetic statement, to be sure, does seem to depend entirely on the structure
Nevertheless, in this article we should like to tease apart these two ontological commitments, arguing that the definiteness of truth for a given mathematical structure, such as the natural numbers, the reals or higher-order structures such as
We make our argument in part by proving that different models of set theory can have a structure identically in common, even the natural numbers, yet disagree on the theory of truth for that structure.
Theorem.
The proofs use only elementary classical methods, and might be considered to be a part of the folklore of the subject of models of arithmetic. The paper includes many further examples of the phenomenon, and concludes with a philosophical discussion of the issue of definiteness, concerning the question of whether one may deduce definiteness-of-truth from definiteness-of-objects and definiteness-of-structure.
[bibtex key=DaghighiGolshaniHaminsJerabek2013:TheFoundationAxiomAndElementarySelfEmbeddingsOfTheUniverse]
In this article, we examine the role played by the axiom of foundation in the well-known Kunen inconsistency, the theorem asserting that there is no nontrivial elementary embedding of the set-theoretic universe to itself. All the standard proofs of the Kunen inconsistency make use of the axiom of foundation (see Kanamori’s books and also Generalizations of the Kunen inconsistency), and this use is essential, assuming that
This is joint work with Ali Sadegh Daghighi, Mohammad Golshani, myself and Emil Jeřábek, which grew out of our interaction on Ali’s question on MathOverflow, Is there any large cardinal beyond the Kunen inconsistency?