The tag has no wiki summary.

learn more… | top users | synonyms

31
votes
7answers
5k views

Arguments against large cardinals

I started to learn about large cardinals a while ago, and I read that the existence, and even the consistency of the existence of an inaccessible cardinal, i.e. a limit cardinal which is additionally ...
25
votes
3answers
1k views

Latest stand of core model theory?

What is the "strongest" core model to this day? In particular, how far are we from a core model for supercompact cardinals? There are rumors of some notes from a workshop in 2004: ...
22
votes
1answer
636 views

Does an existence of large cardinals have implications in number theory or combinatorics?

Does an existence of large cardinals have implications in more down-to-earth fields like number theory, finite combinatorics, graph theory, Ramsey theory or computability theory? Are there any ...
22
votes
4answers
969 views

What is the definition of a large cardinal axiom?

In different books one can find different implicit definitions for a large cardinal axiom. My question is that which one of these definitions are more popular or standard amongst set theorists? Any ...
21
votes
0answers
623 views

categorical characterization of large cardinals

Question 1. Is there a categorical representation of Kunen's inconsistency result? Question 2. Is there a categorical characterization of very large cardinals (in particular for strong and ...
20
votes
2answers
811 views

Can there be an embedding j:V to L, from the set-theoretic universe V to the constructible universe L, when V not= L?

Main Question. Can there be an embedding $j:V\to L$ of the set-theoretic universe $V$ to the constructible universe $L$, if $V\neq L$? By embedding here, I mean merely a proper class isomorphism from ...
19
votes
0answers
706 views

Supercompact and Reinhardt cardinals without choice

A friend of mine and I ran into the following question while reading about proper forcing, and have been unable to resolve it: Definition. A cardinal $\kappa$ is supercompact if for all ordinals ...
17
votes
6answers
2k views

Reasons to believe Vopenka's principle/huge cardinals are consistent

There are a number of informal heuristic arguments for the consistency of ZFC, enough that I am happy enough to believe that ZFC is consistent. This is true for even some of the more tame large ...
17
votes
2answers
3k views

Recent claim that inaccessibles are inconsistent with ZF

Here it is mentioned that someone claims to have proven that there are no weakly inaccessibles in ZF. Question 1: What reasons are there to believe that weakly inaccessibles exist? Question(s) 2: ...
16
votes
2answers
720 views

Woodin's unpublished proof of the global failure of GCH

An unpublished result of Woodin says the following: Theorem. Assuming the existence of large cardinals, it is consistent that $\forall \lambda, 2^{\lambda}=\lambda^{++}.$ In the paper "The ...
16
votes
4answers
1k views

What are the known implications of “There exists a Reinhardt cardinal” in the theory “ZF + j”?

This is, alas, in large part a series of questions on unpublished work of Hugh Woodin; it's also quite frivolous if Reinhardt cardinals turn out inconsistent. Definitions: Call $\kappa$ an ...
16
votes
1answer
685 views

Logical endofunctors of Set?

What set-theoretic assumptions are necessary and sufficient to ensure the existence of a nontrivial (i.e. not isomorphic to the identity) endofunctor of the category Set which is logical (i.e. ...
15
votes
3answers
1k views

Any paradoxical theorems arising from large cardinal axioms?

If we accept the axiom of choice, we take the responsibility of living in a world in which, e.g., a ball in Euclidean 3-space is equiscindable to two isometric copies of itself (Banach-Tarski). So we ...
15
votes
2answers
2k views

Large cardinal axioms and Grothendieck universes

A cardinal $\lambda$ is weakly inaccessible, iff a. it is regular (i.e. a set of cardinality $\lambda$ can't be represented as a union of sets of cardinality $<\lambda$ indexed by a set of ...
15
votes
1answer
875 views

Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)

Gödel's constructible universe ($L$) is defined using definable power set operator in first order logic ($\mathcal{L}_{\omega ,\omega}$). One can produce such a universe in infinitary logics in ...
14
votes
6answers
2k views

What “forces” us to accept large cardinal axioms?

Large cardinal axioms are not provable using usual mathematical tools (developed in $\text{ZFC}$). Their non-existence is consistent with axioms of usual mathematics. It is provable that some of ...
14
votes
2answers
2k views

Should there be a true model of set theory?

As I understand it, there is a program in set theory to produce an ultimate, canonical model of set theory which, among other things, positively answers the Continuum Hypothesis and various questions ...
14
votes
4answers
1k views

A New Continuum Hypothesis (Revised Version)

Define $N_n$ as $n$ th natural number: $N_0=0, N_1=1, N_2=2, ...$. What happens after exponentiation? We have the following equation: $2^{N_n}=N_{2^{n}}$. (Which says: For all finite cardinal $n$ ...
14
votes
2answers
990 views

Why does inner model theory needs so much descriptive set theory (and vice versa)?

I am curious about how much descriptive set theory is involved in inner model theory. For instance Shoenfield's absoluteness result is based on the construction of the Shoenfield tree which ...
14
votes
2answers
728 views

Questions about $\aleph_1-$closed forcing notions

"Foreman`s maximality principle" states that every non-trivial forcing notion either adds a real or collapses some cardinals. This principle has many consequences including: 1) $GCH$ fails ...
13
votes
3answers
359 views

Is Prikry forcing minimal?

Let $M$ be a model of $\sf ZFC$ in which $\kappa$ is a measurable cardinal, and $\cal U$ is a normal measure on $\kappa$. We can define the Prikry forcing (the most simple one) as the poset: $$\Bbb ...
13
votes
2answers
434 views

Singularizing forcing of “small” cardinality?

Can there be a large cardinal $\kappa$ and a forcing of size $\kappa$ that makes $\kappa$ a singular cardinal? The motivation is that the standard Prikry forcing does not have a dense set of size ...
13
votes
2answers
626 views

Can measures be added by forcing?

The Lévy-Solovay theorem says that small forcings do not create measures. J.D. Hamkins has generalized this to a larger class of forcings called gap forcings. I would assume this cannot be ...
13
votes
1answer
832 views

Devlin's “Constructibility” as a resource

It is fairly well-known among set-theorists that Keith Devlin's 1984 book "Constructibility" has flaws in its initial development of fine structure theory. (See Lee Stanley's review 1 of the text for ...
13
votes
0answers
562 views

Souslin trees and weakly compact cardinals

In Souslin trees on the first inaccessible cardinal it is asked if it is consistent that there are no $\kappa-$Souslin trees at the least inaccessible cardinal $\kappa$. In this question I would like ...
13
votes
0answers
397 views

Some questions about $0^{\sharp}$ and forcing over $L$

1-Let $P\in L$ be a nontrivial set forcing or even a tame class forcing (tameness in the sense of Sy Friedman; see for example his Handbook paper), and let $G$ be $P-$generic over $L$. Then it is ...
13
votes
0answers
369 views

How to prove projective determinacy (PD) from I0?

Martin and Steel (in 1987?) showed that if there are infinite many Woodin cardinals then every projective set of reals is determined (PD). However, it is mentioned in many texts that in 1983/1984 ...
12
votes
1answer
848 views

What's the difference between ZFC+Grothendieck, ZFC+inaccessible cardinals and Tarski-Grothendieck set theory?

Say that "U" is the axiom that "For each set x, there exists a Grothendieck universe U such that x $\in$ U", where Grothendieck universes are defined in the usual way (or, if that's unclear, in this ...
12
votes
1answer
839 views

Large cardinal axiom: everything that happen once must happen an unbounded number of times

I remember reading something about a large cardinal axiom saying something like If some cardinal $\kappa$ has some property $P$, then there should be a proper class of cardinals with the property ...
12
votes
1answer
189 views

extending elementary embeddings from initial segments of V to all of V

Suppose $j:V_\lambda \rightarrow V_\eta$ is (elementary and) cofinal. Can $j$ be extended to all of $V$? (Subsidiary question: What conditions are there on an ultrafilter/extender/whatever so that ...
12
votes
1answer
341 views

capturing small sets in small factors

Suppose $\kappa$ is a regular cardinal and $P$ is a $\kappa$-c.c. partial order. I want to know when are small sets added by subforcings of size $<\kappa$. The following seems well-known: ...
11
votes
4answers
699 views

Is there a least-fixed-point formulation of inaccessible cardinals?

The infinity axiom can be formulated by defining a function $S$ as $$S(N) = \{0\} \cup \{n+1\\ |\\ n \in N\}$$ (FWIW, I'm assuming the von Neumann ordinals.) The axiom is then $$\exists I . I = ...
11
votes
4answers
535 views

Proving ZFC results using large cardinals

There are many $ZFC$ results that their proof uses forcing: The idea is that we force the statement to be true, and then using absoluteness (or other reasons) we conclude that the result is true in ...
11
votes
1answer
320 views

Is the inclusion version of Kunen inconsistency theorem true?

The relations $\in$ and $\subsetneq$ seem so similar in some sense. For example they are equal on ordinal numbers. So there is a natural question about their possible similar behaviors on the ...
11
votes
1answer
530 views

Can Vopenka's principle be violated definably?

One form of Vopenka's principle (a large cardinal axiom) states that no locally presentable category contains a full subcategory which is large (= a proper class) and discrete (= contains no ...
11
votes
1answer
414 views

Generic Extensions and $L(V_{\lambda+1})$

Suppose $\lambda$ is a strong limit cardinal of cofinality $\omega$ and for $A$ a transitive set, define $L(A)$ in the usual fashion by setting $$L_0(A)=A;$$ $$L_{\alpha+1}(A) = L_\alpha (A)\cup ...
10
votes
2answers
543 views

Kunen's inconsistency result

A well-known result of Kunen says that there is no non-trivial elementary embedding $j: V \rightarrow V.$ There are several proofs of this theorem (see Kanamori, The higher infinite). I wonder to know ...
10
votes
2answers
586 views

Questions about Prikry forcing and Cohen forcing

I have some questions. The first one is about the product of Prikry's forcing. Let $\kappa$ be a measurable cardinal, $U_1, U_2$ be normal measures on $\kappa$ and let $\mathbb{P}_{U_1}, ...
10
votes
2answers
417 views

Does there exist a supercompactness theorem?

Large cardinals such as weakly compact cardinals, measurable cardinals, strongly compact cardinals, and extendible cardinals all can be characterized in terms of a certain compactness theorem of ...
10
votes
1answer
377 views

How strong is the iterated consistency of ZFC?

Let $T_0$ be $\mathsf{ZFC}$ and, for $n\in\omega$, set $T_{n +1}=T_{n}+\mathrm{Con}(T_{n})$. Question 1: Is there a natural number $n$ such that $T_{n}$ is equiconsistent with $\mathsf{ZFC}+$ ...
10
votes
1answer
246 views

Consistency strength of projective determinacy (PD)

Let PD stand for projective determinacy, and consider the two claims: (1) For each n=1,2,..., Con(ZFC+PD) implies Con(ZFC + there are n Woodin cardinals) (2) Con(ZFC+PD) implies Con(ZFC + there are ...
10
votes
2answers
184 views

What is the strength of chains of 1-extendibles?

Let $X$ be a collection of cardinals such that if $\kappa,\lambda\in X$ and $\kappa<\lambda$, then there is a non-trivial elementary embedding $j:V_{\kappa+1} \to V_{\lambda+1}$ with $crit(j) = ...
10
votes
3answers
1k views

What sort of large cardinal can $\aleph_1$ be without the axiom of choice?

Assuming the axiom of choice it is very easy to see that $\aleph_1$ is a regular Joe of a successor cardinal. It is not very large in any way except the fact that it is the first uncountable cardinal. ...
10
votes
1answer
345 views

V=HOD & The Height of the Large Cardinal Tree

As we know the assumption $V=L$ adds a restriction on the height of the large cardinal tree. Also there is a strict border like $0^{\sharp}$ exists such that all large cardinal axioms which are ...
10
votes
2answers
530 views

What are Moschovakis cardinals?

The question is exactly that of the title: what are Moschovakis cardinals? Background. In a recent answer to the question, "Are there examples of statements that have been proven whose consistency ...
10
votes
1answer
300 views

Erdős cardinals and ineffable cardinals

In Cantor's Attic it is stated that an $\omega$-Erdős cardinal is a stationary limit of ineffable cardinals, and Jech book is given as a reference, but I cannot find this result in that book. I have ...
10
votes
1answer
273 views

What is the large cardinal strength of the assertion that every $\kappa$-complete filter on $\kappa$ extends to a $\kappa$-complete ultrafilter?

It is well-known that an uncountable regular cardinal $\kappa$ is strongly compact if and only if every $\kappa$-complete filter on any set extends to a $\kappa$-complete ultrafilter on that set. The ...
10
votes
2answers
261 views

Singular successors without large cardinals

Assuming the axiom of choice we have that successor cardinals are regular. However as one of the first examples of uses of forcing show, it is consistent relative to $\sf ZF$ that $\omega_1$ is ...
10
votes
1answer
422 views

Does Turing determinacy imply full determinacy?

The axiom of Turing determinacy is a weakening of the full axiom of determinacy, $AD$, in which only games with payoff sets which are $\equiv_T$-invariant are demanded to be determined. In "Turing ...
10
votes
1answer
413 views

Forcing mildly over a worldly cardinal.

A cardinal $\theta$ is worldly if $V_{\theta}$ is a model of ZFC. We could force to collapse $\theta$ to a successor cardinal, for example, and destroy the worldliness of $\theta$, but is there a ...