3
votes
1answer
48 views

A question about second-order logic and inaccessible cardinals.

Let $\kappa$ denote an inaccessible cardinal, and suppose $T \in V_\kappa$ is a second-order theory. Now consider some mathematical structure $X \in V_\kappa$. Then I think it is clear that $X \models ...
2
votes
0answers
102 views

How large or small can the gap in Shelah's main gap theorem be, up to consistency?

Shelah's main gap theorem in model theory says: For each first order complete theory $T$ in a countable language if $I(T,\kappa)$ denotes the number of its models of size $\kappa$ then one of ...
4
votes
0answers
63 views

How to think about iterated ultrapowers?

I would like to gain some basic intuition about iterated ultrapowers. I am perfectly happy with accepting the construction and can see that it fits into a fundamental role in many places (for example, ...
1
vote
1answer
37 views

Model for replacement

If $R_{k} $ is a model for replacement, is necesarly k a strong limit cardinal? And if k is a regular cardinal, are then equivalent the sentences: k is a strongly inaccesible cardinal if and only if ...
3
votes
0answers
124 views

Formalizing model-theoretical large cardinals in a formal system for ZFC

I work with the Metamath formal system, which is expressive enough to define ZFC and prove some nontrivial stuff, but my current goal is to define large cardinals, and I'm hitting a wall somewhere ...
1
vote
0answers
86 views

Tarski's axiom implies a proper class of inaccessible cardinals

I'm trying to prove this theorem, but is it even true? A Tarski's class is a set $T$ such that: For each $y\in T$, ${\cal P}(y)\subseteq T$ and ${\cal P}(y)\in T$, and for each $A\subseteq T$ such ...
3
votes
1answer
125 views

Is it possible to prove $|V_\kappa|=\kappa$ for strongly inaccessible $\kappa$ without AC?

First, let me note that my definition of "strongly inaccessible" is that a nonzero ordinal $\kappa$ is strongly inaccessible if ${\rm cf}(\kappa)=\kappa$ and $\forall\alpha<\kappa, {\cal ...
2
votes
1answer
143 views

Is the expressive power of infinitary logic language $L(\infty,\infty)$ larger, the same or smaller than that of ZFC+large cardinal axioms?

In a previous question I learned that the power of statements of the form $\Pi_m^n$ or $\Sigma_m^n$ for arbitrary positive $m$ and $n$, is smaller than that of ZFC. For instance, the GCH cannot be ...
4
votes
0answers
77 views

What are the “ordinary” (e.g. arithmetic) consequences of the universe axiom?

Consider the first-order theory obtained by adjoining to ZFC (or, better, Mac Lane set theory) the Grothendieck–Verdier (or Bourbaki?) universe axiom: For each set $x$ there exists a Grothendieck ...
3
votes
4answers
178 views

Are all large cardinal axioms expressible in terms of elementary embeddings?

An elementary embedding is an injection $f:M\rightarrow N$ between two models $M,N$ of a theory $T$ such that for any formula $\phi$ of the theory, we have $M\vDash \phi(a) \ \iff N\vDash \phi(f(a))$ ...
2
votes
1answer
119 views

$V_k$ being a model of ZFC whenever $k$ is strongly inaccessible

ZFC implies that the $V_k$ is a model of ZFC whenever $k$ is strongly inaccessible.. So if $k$ is weakly inaccessible, it can't be a model of ZFC? Why is it like this? And ZF implies that ...
1
vote
0answers
76 views

Ultrapowers by extenders of potential premice

I have a problem with an argument in Fine structure and iteration trees by Mitchell and Steel. Let $E$ be a $(\kappa, \lambda)$-extender. Let $\dot E^{\mathcal{M}}$ the a unary predicate with is ...
4
votes
2answers
257 views

Which are “big theorems” of descriptive set theory?

Question: If one were to fully understand 10 theorems in DST, or 15,20,25,30 theorems, which ones would be the most important to understand in order to work towards an understanding of descriptive set ...
3
votes
1answer
154 views

How many weak/strong limit cardinals exist under different assumptions?

I am trying to hastily answer the question in the title because I need it for another problem. I have been consulting multiple sources and am confused about the standard meanings of the following ...
4
votes
3answers
483 views

Why is Kunen inconsistency at the top of Cantor's upper attic?

Motivation: I have reproduced part of page 396 and 397 from Handbook of Mathematical Logic below: So if we start with a concept of number and play the game of naming the largest one, does Kunen ...
6
votes
1answer
337 views

Inaccessible cardinals and well-founded models of ZFC

This was a problem in an exam I took last semester, but I never got the chance to ask my professor how to solve it. Here goes: Let $\kappa$ be an inaccessible cardinal. Then $V_\kappa$ is a ...
11
votes
1answer
439 views

“Nice” well-orderings of the reals

I have a question which I believe could be easily resolved if I happened to look at the right source - hence my asking it here as opposed to at MathOverflow. I've tried googling it, but I haven't been ...
5
votes
1answer
98 views

Basic constructibility question

I'm currently reading J. D. Hamkins' paper "Unfoldable cardinals and the GCH," and I've run across a comment that I think I ought to find trivial, but I don't. On page 1187, he says that ...