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4
votes
1answer
255 views

Stationary many subsets of $\kappa^+$ whose order type is a cardinal and whose intersection with $\kappa$ is an inaccessible cardinal

Is anything known about the consistency strength of the following statement? $\kappa$ is a Mahlo cardinal and there is a stationary set of $a \in \mathcal{P}_\kappa(\kappa^+)$ such that $a \cap ...
7
votes
1answer
249 views

Applications of higher-order reflection principles

Let $L_3$ be the third-order language of set theory with identity on the first sort. Variables $x$ are first-order, $y$ are second-order, and $z$ are third-order. In the style of Lévy and Bernays, we ...
5
votes
1answer
207 views

Consistency of many Erdos cardinals

Does anyone have a reference for the consistency strength of saying that the Erdos cardinal $\kappa(\alpha)$ exists for all $\alpha$? Or of various weakenings about how far such cardinals extend into ...
4
votes
1answer
288 views

Models of Determinacy

Today we have that $L(\mathbb{R}) \models AD$ (assuming there are $\omega$ many Woodin cardinals and a measurable above them all). I was wondering what other models of $AD$ might look like and if it ...
4
votes
1answer
150 views

$\omega$-small and properly small premice.

Let $\mathcal M$ be a premouse. $\mathcal M$ is said to be $\omega$-small if and only if whenever we have that $\kappa=crit(E)$ for some extender $E$ on the sequence of $\mathcal M$, then $\mathcal ...
5
votes
1answer
156 views

Versions of large cardinals with target model in a generic extension

(I ran into this question while thinking about the (Strong) Inner Model Hypothesis: see http://www.jstor.org/stable/4093051 or http://arxiv.org/abs/0711.0680.) A measurable cardinal is a cardinal ...
5
votes
3answers
298 views

A question about Mitchell/Steel Fine Structure and Iteration Trees

In chapter 8 of Mitchell's and Steel's FSIT, they prove a central fine structural result, which basically states that if $\mathcal{M}$ is 1-small, $k$-sound, $k$-iterable premouse then the ...
10
votes
2answers
538 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 ...
7
votes
2answers
395 views

What can the degrees of constructibility be?

If $r, s\in\mathbb{R}$, we say $r$ is constructible relative to $s$ - and write $r\le_cs$ - if $r\in L[s]$. Modding out by the induced equivalence relation $\equiv_c$, we get a partial order, the ...
9
votes
1answer
290 views

Homogeneous Namba-like forcing

Let $\kappa \ge \aleph_3$ be a regular cardinal that is countably closed ($\alpha^\omega < \kappa$ for every $\alpha < \kappa$.) I'm mostly interested in the case that $\kappa$ is strongly ...
8
votes
1answer
257 views

Elementary Embeddings and Relative Constructibility

Suppose $$j:M\prec N$$ is a non-trivial elementary embedding. Under what conditions on the sets (classes?) $M$ and $N$ (or even the critical point of $j$) does $j$ extend to an elementary embedding ...
4
votes
1answer
303 views

if k is weakly inaccessible, then it is the k-th aleph fixed point

A cardinal $\kappa$ is weakly inaccessible iff $\kappa > \omega$, $\kappa$ is regular, and $\forall\lambda<\kappa(\lambda^+<\kappa)$ (here $\lambda^+$ is the successor cardinal) A cardinal ...
4
votes
1answer
194 views

Extending complete filters

Suppose $\kappa$ is a measurable cardinal and let $\mathcal{F}\subset\wp(\kappa)$ be a $\kappa$-complete non-principal filter. Can we extend $\mathcal{F}$ to a $\kappa$-complete ultrafilter? My ...
6
votes
1answer
155 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
166 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
272 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
621 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
304 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
330 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
223 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
412 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 ...
2
votes
0answers
187 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
359 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 ...
12
votes
1answer
834 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
270 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
795 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
682 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) ...
7
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
450 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
415 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
296 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
693 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
338 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
241 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
538 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
999 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
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 ...
6
votes
1answer
345 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
987 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
368 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
837 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
275 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
388 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
590 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
815 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
285 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
330 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
422 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
518 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
262 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 ...