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7
votes
2answers
551 views

What is the consistency strength of the failure of square, in terms of large cardinals

In Jech one can find a lower bound for the consistency strength of PFA in terms of large cardinals. I don't have my copy of Jech in front of me at the moment, but as I recall the presentation of this ...
6
votes
1answer
546 views

Nonstandard models of PA of large cardinal size

It is easy to overlook the fact that the existence of a given large cardinal provides us with a true arithmetical statement that would otherwise be false if the large cardinal notion were not ...
4
votes
3answers
1k views

Nonessential use of large cardinals

In Awfully sophisticated proof for simple facts, we are asked for examples of complex proofs of simple results. To quote from the questioner's post, we are asked for proofs that are akin to "nuking ...
7
votes
2answers
663 views

failure of $\square(\kappa)$ at an inaccessible $\kappa$

How can we force the failure of $\square(\kappa)$ at an inaccessible $\kappa$, where $\square(\kappa)$ is defined as follows: There is a sequence $(C_i:i< \kappa)$ such that: (1) $C_{i+1} = ...
6
votes
1answer
244 views

What does the partially ordered class of cardinals look like in L(R)?

Assuming the existence of enough large cardinals (I'm not sure whether I mean in the original V or in L(R), do whatever is standard), is the partially ordered class of cardinals order-isomorphic to ...
11
votes
1answer
514 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 ...
6
votes
1answer
1k views

Aleph 0 as a large cardinal

The first infinite cardinal, $\aleph_0$, has many large cardinal properties (or would have many large cardinal properties if not deliberately excluded). For example, if you do not impose ...
1
vote
3answers
1k views

Large Cardinals Imply a Model of ZFC

I've run across the statement, "The existence of a strongly inaccessible cardinal implies the consistency of ZFC" in several places (Cohen's Set Theory and the Continuum Hypothesis p. 80, for one). ...
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 ...
3
votes
1answer
416 views

Strong Cardinals and Supercompact Cardinals

This is exercise 20.5 out of Jech: Let $\lambda \geq \kappa$ and let $U$ be a normal measure on $P_{\kappa}(\lambda)$. The ultraproduct $\mathrm{Ult} _U \{ (V _{\lambda _x},\in) : x \in P ...
28
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 ...
1
vote
3answers
365 views

Normal measures and Elementary Embeddings

This is a question from Jech's Set Theory (Ex. 17.12) which I'm reading at the moment and pretty much stuck on. If $D$ is a normal measure on $\kappa$ and $\{ \aleph_\alpha \colon > ...
3
votes
1answer
431 views

Question about John Steel's “The derived model theorem”

In John Steel's paper "The derived model theorem", http://math.berkeley.edu/~steel/papers/dm.ps John Steel asserts that it is clear that $\mathrm{Hom}^{Y}_{\kappa}$ is closed downward under ...
5
votes
2answers
525 views

Vopenka's Principle at Small Cardinals

I'm trying to understand Vopěnka's Principle, which is a large cardinal axiom. One version of the principle is that there does not exist a proper class of directed graphs such that there are no ...
9
votes
2answers
1k views

Large cardinals

Looking at the chart of cardinals in Kanamori's book, one realizes that all large cardinals are implied by stronger ones and imply weaker ones. For instance measurable implies Jonsson which implies ...
7
votes
5answers
721 views

Measurable cardinals under Axiom of Determinacy

I seem to remember reading somewhere that ZF+AD proves that omega-1 and omega-2 are measurable cardinals. Is that right? If so, can someone [point me to or give here] a [sketch or proof] of these ...
6
votes
3answers
674 views

Tractability of forcing-invariant statements under large cardinals

It is usual to mention theorems of the kind: Th. Assume there is a proper class of Woodin cardinals, $\mathbb{P} $ is a partial order and $G \subseteq \mathbb{P}$ is V-generic, then $V \models \phi ...
16
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 ...
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 ...
5
votes
1answer
283 views

The closure of a generic ultrapower

Let $I$ be a normal ideal on $P_{\kappa} (\lambda)$. Let $V$ denote our ground model. Now we force with the $I$-positive sets, and if $G$ is the resulting generic filter, it can be shown that $G$ is a ...
3
votes
1answer
442 views

A question about supercompact cardinal numbers

Is it possible to tell if a cardinal number C is supercompact by looking only at the properties of subsets of C and paying no attention to sets whose cardinal numbers are greater than C? This can ...
7
votes
1answer
389 views

A question on ultrapower

Suppose $\kappa_0$ is a measurable cardinal and $\mu_0$ is a normal measure on $\kappa_0$. $M_1$ is the transitive collapse of $Ult(V,\mu_0)$, $j_{0,1}:V\rightarrow{M_1}$ is the elementary embedding ...
8
votes
2answers
647 views

Model of ZF + $\neg$C in which Solovay's Theorem on stationary sets fails?

It is a theorem of Solovay that any stationary subset of a regular cardinal, $\kappa$ can be decomposed into a disjoint union of $\kappa$ many disjoint stationary sets. As far as I know, the proof ...
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 ...
3
votes
1answer
397 views

What notions of universe does predicative type theory admit?

Palmgren (1997), On universes in type theory, discusses work of several theorists that provide what we might call a family of Large Universe Axioms (LUAs) for predicative type theory, culminating in ...
16
votes
1answer
682 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. ...