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6
votes
1answer
152 views

$\Delta^1_2$-well ordering vs $\Delta^1_3$

It is a classical result that if $0^{\sharp}$ exists, then there is a model of $ZFC$ in which there is a $\Delta^1_3$ well ordering of reals but no $\Delta^1_2$-well ordering. My question is: Is ...
4
votes
1answer
164 views

Two complementing consequences of supercompactness

I would like to know if the following two consequences of having a supercompact cardinal are orthogonal: 1) On one hand being supercompact is equivalent to being "A ineffable for all A" ...
7
votes
2answers
257 views

Relation between $\neg \square(\kappa)$ and the tree property at $\kappa$.

If $\kappa$ is an inaccessible cardinal then the tree property at $\kappa$ is equivalent to weak compactness of $\kappa$, which implies that $\square(\kappa)$ fails---that is, that every coherent ...
13
votes
2answers
607 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 ...
7
votes
1answer
294 views

A question about the comparability of large cardinals.

Are there any examples of two large cardinal axioms $AX$ and $AY$, in the language of first order $ZFC$, which satisfy the following conditions. Each of them defines a unique cardinal number - ...
2
votes
0answers
328 views

large cardinal tree properties as properties of sheaves

As follows from this talk Large Properties for Small Cardinals, p.7,p.4 http://www2.dm.unito.it/paginepersonali/viale/SEMINARS-TORINO/Fontanella-Torino-19.1.2012.pdf, the definitions of weakly compact ...
3
votes
0answers
216 views

Internal large cardinal embeddings. How deep can we go?

Suppose, in V, that $\kappa$ is a large cardinal, say $\kappa$ is supercompact, and $j:V \to M$ is an ultrapower embedding witnessing the large cardinal properties of $\kappa$. From here, we can take ...
11
votes
1answer
409 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 ...
1
vote
0answers
184 views

a partial order not dense iff a measurable exists

For $\kappa>0$ a regular cardinal, let $Ht_\kappa$ denote the following partial quasi-order: (i) elements(objects) of $Ht_\kappa$ are classes X of sets of size $\kappa$ with the property that, ...
8
votes
1answer
348 views

Can $\omega_1$ be supercompact?

Is "ZF + $\omega_1$ is supercompact" consistent relative to "ZFC + there is a supercompact cardinal"? In particular, if $\delta$ is supercompact, does it remain so in $V(\mathbb{R} \cap V[G])$ where ...
11
votes
2answers
795 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 ...
5
votes
2answers
268 views

Measures that are not OD

Is anything known about the consistency strength of the statement: "There is a normal measure (on a cardinal) that is not ordinal-definable"? In particular, is it consistent relative to the ...
19
votes
2answers
773 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 ...
4
votes
2answers
672 views

Large cardinals without the ambient set theory?

In an attempt to understand a bit better large cardinals, I have been thinking along the following lines, which could be summarized under the slogan Talk about cardinals without the (ambient) ...
6
votes
8answers
2k views

ULTRAINFINITISM, or a step beyond the transfinite

Cantor has, in the immortal words of D. Hilbert, given all of us a paradise (or perhaps, I would rather say, a great vacation spot), the TRANSFINITE. $\aleph_0, \aleph_1,\aleph_2\dots$ the lists ...
6
votes
2answers
440 views

Large cardinal axioms and total recursive functions

Are there known relationships between large cardinal axioms (say Mahlo or Woodin cardinals) and total recursive functions (over the natural numbers) of the type: $ZFC$ + large cardinal axiom $\vdash$ ...
5
votes
1answer
405 views

An Extender is a Generalization of an Ultrafilter?

I'm not sure whether this belongs here or on math stackexchange, but I'll give a try here. I've heard that an extender is a generalization of an ultrafilter. This is not to say that an ultrafilter ...
6
votes
1answer
289 views

Vopenka's Theorem on L(A) and Large Cardinals from Weaker Assumptions?

I know that sometime ago Vopenka proved this: Theorem: Assume there is a strongly compact cardinal. Then for any set $A$, $V \neq L(A)$. Can we get by with a consistency-wise strictly weaker ...
19
votes
0answers
660 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 ...
9
votes
1answer
333 views

What can we learn about an elementary embedding from the image of the ordinals?

If $j : V \rightarrow M$ is an elementary embedding, what can we learn in $M$ from $j''ORD$? That is, what is $M[j''ORD]$? In particular, Is it $M[j''ORD]$ equal to all of $V$? If not, do we ...
3
votes
0answers
231 views

On Successive Regular Cardinals With No Ladders

Definition: Let $\kappa$ be an $\aleph$ cardinal, we say that $\langle f_\alpha\colon\alpha\to\kappa\mid\alpha<\kappa^+\rangle$ is a ladder if every $f_\alpha$ is injective. Equivalently this is ...
3
votes
2answers
531 views

Set Cardinality Game - Can a player with numbers in R win over a player with numbers in N as each of them in one turn has to present a new number?

Let there be 2 players, p$\mathbb{N}$ and p$\mathbb{R}$. They are playing the Set Cardinality Game where p$\mathbb{N}$ has to present some number n that has not been used in the game so far by either ...
10
votes
3answers
924 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. ...
14
votes
3answers
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 ...
6
votes
1answer
340 views

GCH+ Kurepa Families

I have a couple of questions about known theorems for GCH+Kurepa families. Definition first: Let $\kappa$ be a infinite cardinal. A $\kappa^+$ Kurepa family is a family $F$ of subsets of $\kappa^+$ ...
14
votes
2answers
955 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 ...
13
votes
0answers
359 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
813 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 ...
4
votes
2answers
272 views

End extensions of models which do not preserve axioms

Assuming the axiom of choice there is a neat way defining inaccessible cardinals as uncountable, regular, strong limit cardinals. Without the axiom of choice we have several notions of ...
7
votes
1answer
385 views

Why “adding” a single extender cannot give an L-like inner model for say, a strong cardinal?

The constructible universe $L$ is too thin for large cardinals greater than measurable. To build $L$-like inner models for large cardinal, it is natural to think about "adding" the evidences into the ...
4
votes
1answer
581 views

Some questions from the paper “Forcing the failure of CH by adding a real” by Shelah and Woodin

1-Let $P=Add(\omega_1, \kappa)$, and let $D$ be a dense open subset of $P$. Then there is a dense subset $S$ of $D$ such that for every $f \in S$ and any $g \in P$, if $domf=domg$ and the set $\{ ...
13
votes
1answer
792 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 ...
10
votes
1answer
280 views

Do indiscernibility embeddings exist for an initial segment of an inner model of many measurable cardinals?

Background I am interested in elementary embeddings from a model of set theory into itself. One way of producing such elementary embeddings is when the model is generated by indiscernibles; this idea ...
5
votes
0answers
319 views

$\Pi_0^1$-weakly indescribable cardinals are exactly the regulars

Hi, I'm not sure if I should ask here or over at math.stackexchange.com, but I think here it's a bit more fitting. This question stems from a homework problem: Definition: Given some class of ...
7
votes
0answers
419 views

A proposed axiom of Laver (updated)

A few months back, I posted a question asking about a proposed axiom of Laver's and I, unfortunately, left out a critical piece. Here is the full axiom: (L) Some elementary embedding ...
7
votes
1answer
501 views

Are there large cardinals for $n$-elementarity?

In July, Asaf Karagila asked three questions about elementary substructures of the universe of sets. The latter two were answered, the upshot being that the hypothesis $V_\kappa \prec V$ doesn't alone ...
6
votes
0answers
252 views

PFA and the compactness/incompactness of $\omega_2$

There are many consequences of the Proper Forcing Axiom (PFA) which essentially say that $\omega_2$ has a lot of compactness. For instance, Matteo Viale and Christoph Weiss have a few papers in ...
5
votes
1answer
367 views

Characterizing elementary embeddings of $L$ and $L_\alpha$ under 0#

Suppose 0# exists. It is clear that every order preserving map from the indiscernibles to the indiscernibles gives an elementary embedding from $L$ to $L$. Furthermore, following lemmas 18.7 and 18.8 ...
17
votes
2answers
2k 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: ...
4
votes
2answers
176 views

Relation between indiscernibles for $L$ and for $L[A]$

It is known that $L\models 2^\kappa=\kappa^+$, and that for a set of ordinals $A$ we know that $L[A]\models \exists\lambda\forall\kappa>\lambda(2^\kappa=\kappa^+)$. In this sense, there is some ...
5
votes
2answers
345 views

Ultrapowers by normalized ultrafilters

Suppose $j\colon V\to M$ is an elementary embedding and $\kappa$ is the critical point of $j$, then $\kappa$ is measurable, and we can define the ultrafilter $U$ over $\kappa$ as: $$A\in U\iff ...
3
votes
2answers
245 views

Dual covering theorem

Jensen's covering theorem states that if $0^\sharp$ doesn't exist, then every uncountable set of ordinals can be covered by a constructible set of the same cardinality. Now consider the following ...
9
votes
3answers
733 views

How elementary can we go?

It is a theorem of A. Levy, if $\kappa$ is an inaccessible cardinal, then $V_\kappa\prec_{\Sigma_1} V$ namely $V_\kappa$ is an elementary submodel when considering only $\Sigma_1$ sentences. One ...
10
votes
1answer
345 views

Consistency strengths related to the perfect set property

I want a model of $\mathrm{MA}_{\sigma\mathrm{-centered}}+\neg\mathrm{CH}$ in which every set of reals in $L(\mathbb{R})$ has the perfect set property. In terms of consistency strength, it is known ...
6
votes
1answer
639 views

Large cardinals and constructible universe

We know that if $V=L$ holds, then $|\cal{P}(\omega)|=|\cal{P}(\omega)\cap \textrm{L}|=\aleph_1$ whereas, in the presence of a measurable cardinal (in fact, even Ramsey) $|\cal{P}(\omega)\cap ...
9
votes
2answers
577 views

Why is this theorem (about $L(P(\omega_1))^V$ and $L(P(\omega_1))^{V[G]}$) nice?

I was recently told that the following (due to M. Viale) is a nice theorem: Suppose there are arbitrarily large supercompacts, and $\mathrm{MM}$ holds in $V$. Let $G$ be generic for a proper ...
5
votes
1answer
323 views

Adding large sets by countable conditions preserving the GCH

Let $\kappa$ be an inaccessible cardinal. Is there any forcing notion $P$ with the following properties: 1-$P$ preserves GCH and the strong inaccessibility of $\kappa$, 2-$P$ adds a subset of ...
3
votes
1answer
241 views

$< \aleph_1-$support Product of Cohen forcings

Suppose $\kappa$ is an inaccessible cardinal, and let $P$ be the $< \aleph_1-$support product of $Add(\alpha^{++}, 1)$ for singular cardinals $\alpha < \kappa.$ 1- Does this forcing preserve ...
11
votes
4answers
672 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 = ...
14
votes
2answers
668 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 ...