If we assume the axiom of choice, then every two cardinal numbers are comparable. In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary infinite set, we will give all the possible relationships between them, where possible means that the relationship is consistent with the axioms of set theory. Further we investigate the relationships between some other cardinal numbers in specific permutation models and give some results provable without using the axiom of choice.
We prove that the Generalized Continuum Hypothesis holds above a supercompact cardinal assuming the Ultrapower Axiom, an abstract comparison principle motivated by inner model theory at the level of supercompact cardinals.
We show that various tameness assertions about abstract elementary classes imply the existence of large cardinals under mild cardinal arithmetic assumptions.
We introduce the resurrection axioms, a new class of forcing axioms, and the uplifting cardinals, a new large cardinal notion, and prove that various instances of the resurrection axioms are equiconsistent over ZFC with the existence of an uplifting cardinal.
We argue against Foreman's proposal to settle the continuum hypothesis and other classical independent questions via the adoption of generic large cardinal axioms.
The Axiom of Full Reflection at a measurable cardinal has been conjectured to be equiconsitent with the existence of a coherent sequence of measures with a repeat point. However we prove that the Axiom of Full Reflection at a measurable cardinal is surprisingly equiconsistent just with the existence of a measurable cardinal. We also generalize the result to larger cardinals as strong or supercompact. Hence we can conclude that the Axiom of Full Reflection at large cardinals weaker than measurable, e.g. as n-Mahlo, does push the consistency strength up, but does not push the consistency strengt…
We show that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis. The proof is by interpolation and uses the Mapping Reflection Principle.
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.
This paper presents the main results in my Ph.D. thesis. In what follows several proofs of SCH are presented introducing a family of covering properties which implies both SCH and the failure of various forms of square. These covering properties are also applied to investigate models of strongly compact cardinals or of strong forcing axioms like MM or PFA.
We show that Shelah's Eventual Categoricity Conjecture follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC with $LS(K)$ below a strongly compact cardinal $κ$ is $< κ$ tame and applying the categoricity transfer of Grossberg and VanDieren. These techniques also apply to measurable and weakly compact cardinals and we prove similar tameness results under those hypotheses. We isolate a dual property to tameness, called \emph{type shortness}, and show that it follows simila…
Recently the second author introduced combinatorial principles that characterize supercompactness for inaccessible cardinals but can also hold true for small cardinals. We prove that the proper forcing axiom PFA implies these principles hold for $ω_2$. Using this, we argue to show that any of the known methods for forcing models of PFA from a large cardinal assumption requires a strongly compact cardinal. If one forces PFA using a proper forcing, then we get the optimal result that a supercompact cardinal is necessary.
In other work we have outlined how, building on ideas of Welch and Roberts, one can motivate believing in the existence of supercompact cardinals. After making this observation we strove to formulate a justification for large-cardinal axioms of greater strength, and arrived at a motivation for a new large-cardinal property, which we define here and prove to be equivalent to the property of being a Vopěnka scheme cardinal. Making use of this result, one can also show that a theory $B_0(V_0)$ described in a previous paper of Victoria Marshall implies the existence of a Vopěnka scheme cardinal $κ…
The inner model problem for supercompact cardinals, one of the central open problems in modern set theory, asks whether there is a canonical model of set theory with a supercompact cardinal. The problem is closely related to the more precise question of the equiconsistency of strongly compact cardinals and supercompact cardinals. This dissertation approaches these two problems abstractly by introducing a principle called the Ultrapower Axiom which is expected to hold in all known canonical models of set theory. By investigating the consequences of the Ultrapower Axiom under the hypothesis that…
We introduce and axiomatize the notion of a reflective cardinal, use it to give semantics to higher order set theory, and explore connections between the notion of reflective cardinals and large cardinal axioms.
We will consider a number of new large-cardinal properties, the $α$-tremendous cardinals for each limit ordinal $α>0$, the hyper-tremendous cardinals, the $α$-enormous cardinals for each limit ordinal $α>0$, and the hyper-enormous cardinals. For limit ordinals $α>0$, the $α$-tremendous cardinals and hyper-tremendous cardinals have consistency strength between I3 and I2. An $ω$-enormous cardinal has consistency strength greater than I0, and also all the large-cardinal axioms discussed in the second part of Hugh Woodin's paper on suitable extender models, not known to be inconsistent with ZFC an…
The purpose of this paper is to present some results which suggest that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. What will be proved is that a form of simultaneous reflection follows from the Set Mapping Reflection Principle, a consequence of PFA. While the results fall short of showing that MRP implies SCH, it will be shown that MRP implies that if SCH fails first at kappa then every stationary subset of S_{kappa^+}^omega = {a < kappa^+ : cf(a) = omega} reflects. It will also be demonstrated that MRP always fails in a generic extension by Prikry forcing.
A central theme in set theory is to find universes with extreme, well-understood behaviour. The case we are interested in is assuming GCH and has a strong forcing axiom of higher order than usual. Instead of "for every suitable forcing notion for~$λ$" we shall say "for every such family of forcing notions, depending on stationary $S\subseteq λ$, for some such stationary set we have\dots". Such notions of forcing are important for Abelian group theory, but this application is delayed for a sequel.
This paper contributes to the theory of large cardinals beyond the Kunen inconsistency, or choiceless large cardinal axioms, in the context where the Axiom of Choice is not assumed. The first part of the paper investigates a periodicity phenomenon: assuming choiceless large cardinal axioms, the properties of the cumulative hierarchy turn out to alternate between even and odd ranks. The second part of the paper explores the structure of ultrafilters under choiceless large cardinal axioms, exploiting the fact that these axioms imply a weak form of the author's Ultrapower Axiom. The third and fin…
William Tait and Peter Koellner have written on the topic of which reflection principles are intrinsically justified. Phillip Welch and Sam Roberts have recently sought to motivate much stronger reflection principles. This work also had some similarities with previous work of Victoria Marshall's on reflection principles, as may be seen from considerations which we will seek to articulate in what follows. The statement of our three philosophical theses shall be given in the Introduction. We will present a set of technical results in support of three philosophical theses which will be articulate…
The relationship between the large cardinal notions of strong compactness and supercompactness cannot be determined under the standard ZFC axioms of set theory. Under a hypothesis called the Ultrapower Axiom, we prove that the notions are equivalent except for a class of counterexamples identified by Menas. This is evidence that strongly compact and supercompact cardinals are equiconsistent.
One of the numerous characterizations of a Ramsey cardinal kappa involves the existence of certain types of elementary embeddings for transitive sets of size κsatisfying a large fragment of ZFC. We introduce new large cardinal axioms generalizing the Ramsey elementary embeddings characterization and show that they form a natural hierarchy between weakly compact cardinals and measurable cardinals. These new axioms serve to further our knowledge about the elementary embedding properties of smaller large cardinals, in particular those still consistent with V=L.
We introduce bounded category forcing axioms for well-behaved classes $Γ$. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe $H_{λ_Γ^+}$ modulo forcing in $Γ$, for some cardinal $λ_Γ$ naturally associated to $Γ$. These axioms naturally extend projective absoluteness for arbitrary set-forcing--in this situation $λ_Γ=ω$--to classes $Γ$ with $λ_Γ>ω$. Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms, but can be forced under mild large cardinal assumptions o…
We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by $Loc({\rm ZFC})$, says that every set belongs to a transitive model of ZFC. LZFC consists of $Loc({\rm ZFC})$ plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All $Π_2$ consequences of ZFC are provable in ${\rm LZFC}$. LZFC strongly extends Kripke-Platek (KP) set theory minus $Δ_0$-Collection and minus $\in$-induction scheme. ZFC+``there is an inaccessible cardinal'' proves the consistency of LZFC. In LZ…
We recently formulated a new large-cardinal axiom of strength intermediate between a totally indescribable cardinal and an $ω$-Erdős cardinal, positing the existence of what we called an "extremely reflective cardinal", and we showed that the property of being extremely reflective was in fact equivalent to the property of being remarkable, and we sought to argue that this axiom should be seen as intrinsically justified. This built on related earlier work in which the notion of an $α$-reflective cardinal was formulated. Then Welch and Roberts put forward a family of reflection principles, Welch…
In this series of papers, we advance Ramsey theory of colorings over partitions. In this part, a correspondence between anti-Ramsey properties of partitions and chain conditions of the natural forcing notions that homogenize colorings over them is uncovered. At the level of the first uncountable cardinal this gives rise to a duality theorem under Martin's Axiom: a function $p:[ω_1]^2\rightarrowω$ witnesses a weak negative Ramsey relation when $p$ plays the role of a coloring if and only if a positive Ramsey relation holds over $p$ when $p$ plays the role of a partition. The consistency of posi…
The bounded proper forcing axiom BPFA is the statement that for any family of aleph_1 many maximal antichains of a proper forcing notion, each of size aleph_1, there is a directed set meeting all these antichains. A regular cardinal kappa is called {Sigma}_1-reflecting, if for any regular cardinal chi, for all formulas phi, ``H(chi) models `phi ' '' implies ``exists delta < kappa, H(delta) models `phi ' '' We show that BPFA is equivalent to the statement that two nonisomorphic models of size aleph_1 cannot be made isomorphic by a proper forcing notion, and we show that the consistency strength…
We present a coherent collection of finite mathematical theorems some of which can only be proved by going well beyond the usual axioms for mathematics. The proofs of these theorems illustrate in clear terms how one uses the well studied higher infinities of abstract set theory called large cardinals in an essential way in order to derive results in the context of the natural numbers. The findings raise the specific issue of what consitutes a valid mathematical proof and the general issue of objectivity in mathematics in a down to earth way. Large cardinal axioms, which go beyond the usual axi…
We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $λ$, if $λ^{++}$ is not a Mahlo cardinal in Gödel's constructible universe, then $2^λ= λ^+$ entails the existence of a $λ^+$-complete $λ^{++}$-Souslin tree.
We give a brief survey on the interplay between forcing axioms and various other non-constructive principles widely used in many fields of abstract mathematics, such as the axiom of choice and Baire's category theorem. First of all we outline how, using basic partial order theory, it is possible to reformulate the axiom of choice, Baire's category theorem, and many large cardinal axioms as specific instances of forcing axioms. We then address forcing axioms with a model-theoretic perspective and outline a deep analogy existing between the standard Łoś Theorem for ultraproducts of first order s…
Zermelo's Axiom of Separation is: Exist x: Forall y: (y in x <==> y in a & E(y)) with definite(E) and parameter a. Thoralf Skolem suggested to characterize the terminus "definite" by "the property E should be representable by a FOL formula". But that is trivial. "definite" must mean more. The author claims that "definite" means "in accordance with the theory of definitions of logic". In this case the theorem of Cantor is no longer a theorem, but a undecidable sentence, and has to be established explicitly as axiom. This is not done by the community, but it is made a silent assumption that we c…
I explore two separate topics: the concept of jointness for set-theoretic guessing principles, and the notion of grounded forcing axioms. A family of guessing sequences is said to be joint if all of its members can guess any given family of targets independently and simultaneously. I primarily investigate jointness in the case of various kinds of Laver diamonds. In the case of measurable cardinals I show that, while the assertions that there are joint families of Laver diamonds of a given length get strictly stronger with increasing length, they are all equiconsistent. This is contrasted with …
Andrew Wiles' Proof of Fermat's Last Theorem, As Expected, Does Not Require a Large Cardinal Axiom. A Discussion of Colin McLarty's "The Large Structures of Grothendieck Founded on Finite-Order Arithmetic"
Our original aim was, in Abelian group theory to prove the consistency of: lambda is strong limit singular and for some properties of abelian groups which are relatives of being free, the compactness in singular fails. In fact this should work for R-modules, etc. As in earlier cases part of the work is analyzing how to move between the set theory and the algebra. Set theoretically we try to force a universe which satisfies G.C.H. and diamond holds for many stationary sets but, for every regular uncountable lambda, in some sense anything which "may" hold for some stationary set, does hold for s…
While many inner model theoretic combinatorial principles are incompatible with large cardinal axioms, on some rare occasions, large cardinals actually imply that the structure of the universe of sets is analogous to the canonical inner models. This note provides two new instances of this phenomenon.
This dissertation surveys several topics in the general areas of iterated forcing, infinite combinatorics and set theory of the reals. There are two parts. In the first half I consider alternative versions of the Cichoń diagram. First I show that for a wide variety of reduction concepts there is a Cichoń diagram for effective cardinal characteristics relativized to that reduction. As an application I investigate in detail the Cichoń diagram for degrees of constructibility relative to a fixed inner model of ZFC. Then I study generalizations of cardinal characteristics to the space of functions …
Assuming an abstract comparison principle called the Ultrapower Axiom, which is motivated by the comparison process of inner model theory and generalizes the statement that the Mitchell order is linear on normal ultrafilters, we characterize strongly compact cardinals in terms of supercompactness: a strongly compact cardinal is either supercompact or a limit of supercompact cardinals. Assuming the Ultrapower Axiom and the GCH, we also prove a local result that roughly states that every countably complete ultrafilter factors as a finite iteration of supercompact ultrafilters.
Assuming an inaccessible cardinal kappa, there is a generic extension in which MA + 2^{aleph_0} = kappa holds and the reals have a Delta^2_1 well-ordering.
We introduce a category whose objects are stationary set preserving complete boolean algebras and whose arrows are complete homomorphisms with a stationary set preserving quotient. We show that the cut of this category at a rank initial segment of the universe of height a super compact which is a limit of super compact cardinals is a stationary set preserving partial order which forces $MM^{++}$ and collapses its size to become the second uncountable cardinal. Next we argue that any of the known methods to produce a model of $MM^{++}$ collapsing a superhuge cardinal to become the second uncoun…
We show that in the theory ZF + DC + for every cardinal λ, the set of infinite subsets of λ is well-ordered (i.e., Shelah's AX4), the θ-function measuring the surjective size of the powersets P(κ) can take almost arbitrary values on any set of uncountable cardinals. This complements our results from [FK16], where we prove that in ZF (without DC), any possible behavior of the θ-function can be realized; and answers a question of Shelah in [She16], where he emphasizes that ZF + DC + AX4 is a reasonable theory, where much of set theory and combinatorics is possible.
The resurrection axioms are forcing axioms introduced recently by Hamkins and Johnstone, developing on ideas of Chalons and Velicković. We introduce a stronger form of resurrection axioms (the \emph{iterated} resurrection axioms $\textrm{RA}_α(Γ)$ for a class of forcings $Γ$ and a given ordinal $α$), and show that $\textrm{RA}_ω(Γ)$ implies generic absoluteness for the first-order theory of $H_{γ^+}$ with respect to forcings in $Γ$ preserving the axiom, where $γ=γ_Γ$ is a cardinal which depends on $Γ$ ($γ_Γ=ω_1$ if $Γ$ is any among the classes of countably closed, proper, semiproper, stationar…
Assuming the P-ideal dichotomy, we attempt to isolate those cardinal characteristics of the continuum that are correlated with two well-known consequences of the proper forcing axiom. We find a cardinal invariant $\mathfrak{x}$ such that the statement that $\mathfrak{x} > ω_{1}$ is equivalent to the statement that 1, $ω$, $ω_{1}$, $ω\times ω_{1}$, and ${\left[ω_{1}\right]}^{< ω}$ are the only cofinal types of directed sets of size at most ${\aleph}_{1}$. We investigate the corresponding problem for the partition relation $ω_{1} \rightarrow (ω_{1}, α)^2$ for all $α< ω_{1}$. To this effect, we i…
Since being isolated by Viale and Weiss in 2009, the Guessing Model Property has emerged as a particularly prominent and powerful consequence of the Proper Forcing Axiom. In this paper, we investigate connections between variations of the Guessing Model Property and cardinal arithmetic, broadly construed. We improve upon results of Viale and Krueger by proving that a weakening of the Guessing Model Property implies Shelah's Strong Hypothesis. We also prove that, though the Guessing Model Property is known not to put an upper bound on the size of the continuum, it does imply that $2^{ω_1}$ is a…
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 sat…
A celebrated argument of F. Hartogs (1915) deduces the Axiom of Choice from the hypothesis of comparability for any pair of cardinals. We show how each of a sequence of seemingly much weaker hypotheses suffices. Fixing a finite number $k>1$, the Axiom of Choice follows if merely any family of $k$ cardinals contains at least one comparable pair.
