Large cardinals are such cardinals whose existence cannot be proved within ZFC, and requires stronger axioms to be added to $ZFC$, they are often used to measure the consistency strength of a certain statement in the language of set theory.
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What are large cardinals for?
I've heard large cardinals talked about, and I (think I) understand a little about how you define them, but I don't understand why you would bother.
What's the simplest proof or whatever that ...
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Forcing Classes Into Sets
I am still studying the topics in forcing and did not yet study much about forcing with a class of conditions.
I know from Jech's Set Theory that you can force that the class of ordinals in the world ...
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When does $V=L$ becomes inconsistent?
In a wonderful course I'm taking with Magidor we are finishing the proof of the Covering Theorem for $L$.
The theorem, in a nutshell, says that $V$ is very close to being $L$ if and only if $0^\#$ ...
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On the large cardinals foundations of categories
(This was cross-posted to MathOverflow.)
It is well-known that there are difficulties in developing basic category theory within the confines of $\sf ZFC$. One can overcome these problems when ...
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336 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 ...
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279 views
Real-measurable cardinals that are not measurable ones
I'm reading Jech's Set Theory, and in the chapter about measurable cardinals there is a theorem that if $\kappa$ is real-measurable but not measurable then it is $\le 2^{\aleph_0}$ and so and so. ...
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$A^\#$ and inner models
For a set of ordinals $A$ we say that $A^\#$ exists if there exists a closed and unbounded class of indiscernibles, $I\subseteq\operatorname{Ord}$, for $L[A]$. Formally, if such class exists we define ...
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Proving $V_{\kappa}$ is a model of ZFC for inaccessible $\kappa$
Prove that if $\kappa$ is an inaccessible cardinal, then $V_{\kappa}$ satisfies all the axioms of ZFC.
How is this done for the axiom of choice and for regularity?
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What is the largest cardinal which can inject into $\mathbb{R}$ in ZF?
This question takes place in ZF. Assume some mild large cardinals; then it is consistent (in fact, it follows from AD, the consistency of which follows from mild large cardinals) that there are very ...
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184 views
Shelah Cardinals and Modern Set Theory
Do Shelah cardinals play an essential role in any modern set theory results or was the concept basically made obsolete by Woodin cardinals?
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142 views
Equivalent ways to describe the Mitchell order
For a measurable cardinal $\kappa$, we define an ordering over $\kappa$-complete ultrafilters as follows:
Suppose $W,U$ are both $\kappa$-complete free ultrafilters over $\kappa$, we say that $U\lhd ...
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256 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 ...
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Does asserting the existence of large cardinals allow one to prove the existence of new subsets of $\mathbb{R}$?
ZFC proves the existence of certain subsets of $\mathbb{R}$. Suppose ZFC is consistent, and we adjoin a large cardinal axiom to ZFC, obtaining ZFC'. Assume ZFC' is also consistent. It it possible to ...
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What's the latest on Laver tables?
A couple of years ago, I was astonished and delighted to learn about Laver tables, a sequence (indexed on $n$) of Cayley-like tables for a binary operation $\star$ on numbers $i,j\leq 2^n$ that ...
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91 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 ...
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232 views
Can forcing push the continuum above a weakly inacessible cardinal?
There is a famous quote of Paul Cohen which reads $\lt\lt$ A point of view which the author feels may eventually come to be accepted is that the continuum hypothesis is obviously false ...
The ...
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mahlo and hyper-inaccessible cardinals
Wikipedia states that a Mahlo cardinal is hyper-inaccessible, hyper-hyper-inaccessible, etc. Is this a characterisation of Mahlo? If not what about "alpha = hyper^alpha-inaccessible" with the obvious ...
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180 views
Strong cardinals and reflection
I'm new to all this large cardinal thing and I have trouble in proving the following:
If $\kappa$ is a $\gamma$-strong cardinal, for some large enough $\gamma$, then $\kappa$ is ...
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152 views
Silver indiscernibles and definable injections
This is a follow up to my previous question on Silver indiscernibles.
Background: Suppose that $0^\#$ exists, $\alpha<\lambda$ are limit ordinals, $i_\alpha$ is the $\alpha$th Silver ...
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133 views
Weakly Compact Cardinals and the Extension Property
From $\textit{The Higher Infinite}$ by Kanamori, in his proof (page 39 - 41) that $\kappa$ is weakly compact if and only if $\kappa$ satisfies the Extension Property, the proof does something that I ...
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A reference that forcing with a poset of cardinality less than $\kappa$ preserves supercompact cardinals
I'm looking for a reference for the fact that if $\kappa$ is supercompact and $Q$ is a poset of cardinality less than $\kappa$, then $V^Q\models \check{\kappa} \ \text{is supercompact}$.
Apparently ...
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3answers
327 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 ...
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Relation between inacessible cardinals and CH
I know that the two statements “There exists an inaccessible cardinal” and “Continuum Hypothesis” are both independent of ZFC. Now, are those two statements independent of each other?
That older ...
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119 views
Effective Well ordering of reals
Is there an effective (constructive) well order on reals ?
I know several questions were already asked on this topic, and the answers were very good to this well known problem. My question is more ...
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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 ...
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Silver indiscernibles and constructibility
We know that if $0^\#$ exists then it's not in $L$. For an infinite ordinal $\alpha$, denote by $I_\alpha$ the initial segment of length $\alpha$ of Silver indiscernibles.
Question: For which ...
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Normal ultrafilters and Stationary sets
If $\kappa$ is a measurable cardinal, and $\mathcal{U}$ is a normal ultrafilter which is $\kappa$-complete then $\mathcal{U}$ extends the club filter (i.e. every club is a member of $\mathcal{U}$).
...
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The real cofinality of singular cardinals in $L$ under $0^\#$
Suppose that $0^\#$ exists, is there a relatively simple way to show that for any ordinal $\lambda$, if $\lambda$ is a singular cardinal in $L$ then its real cofinality is $\omega$?
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Consistency of $\operatorname{cf}(\omega_1)=\operatorname{cf}(\omega_2)=\omega$
It is known that if $\operatorname{cf}(\omega_1)=\operatorname{cf}(\omega_2)=\omega$ then $0^\#$ exists.
I am familiar with the Feferman-Levy model in which $\omega_1$ is singular, which has the same ...
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Generalizing $0^\#$
Background and motivation:
The following theorem is due to Silver:
If there exists a Ramsey cardinal then:
For every $\aleph_0 < \kappa < \lambda$, $L_{\kappa}$ is an elementary ...
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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 ...
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Good Reference for Large Cardinals/Homotopy Theory
I'm planning on attending a conference in Barcelona in September called "Large Cardinal Methods in Homotopy Theory" and want to try to be as prepared as I possibly can. Are there good references for ...
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99 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 ...
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Intuition behind extenders
What's the best way to think about extenders? For instance if we have a normal $\kappa$-complete ultrafilter on $\kappa$, call it $D$, $M$ the ultrapower given by $D$, and if we look at $j: V \to M$, ...
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1answer
221 views
Normal ultrafilters on measurable cardinals
Let $\kappa$ be a measurable cardinal, we say that $\mathcal{D}$ is a normal ultrafilter iff whenever $g\in\kappa^\kappa$ such that $g<_\mathcal{D} Id$, we have some $\alpha<\kappa$ such that ...
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1answer
62 views
Existence of a probability measure under which a subset of $[0,1]$ cannot be approximated by a Borel set
Is there some probability measure, $p$, on $\left(\left[0,1\right],2^{\left[0,1\right]}\right)$ relative to which some $D\subseteq\left[0,1\right]$ cannot be approximated by a Borel set, i.e. if $E$ ...
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1answer
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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 ...
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1answer
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Which infinite cardinals can be defined using partition relations?
Several types of infinite cardinals are easily defined in terms of partition relations. For instance, if $\kappa > \omega$ then
$\kappa$ is weakly compact if $\kappa$ satisfies ...
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1answer
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Measurable cardinal existence condition
We know that a cardinal $k > \omega$ is measurable if there is a measure function $\mu :2^k\mapsto \{0, 1\}$ that satisfies the following 3 conditions:
1.$<k$ -additive: for every set I of ...
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Can any large cardinal axiom be reached by a recursive sequence of unbounded sequences and fixed points?
The lowest large cardinals can be seen as a simple recursive rule that defines an unbounded sequence, then defines a fixed point of the sequence, then an unbounded sequence of fixed points, a fixed ...
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Are large cardinals bi-interperable with type theory?
"Sets in Types, Types in Sets" establishes that the calculus of constructions is bi-interpretable with ZFC + infinitely many inaccessibles.
Are there any more results like this higher up the large ...
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How is Mahlo cardinal used?
I would like to know how Mahlo cardinals are used - as such examples may help me understand why they were created and so on.
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115 views
example of weakly inaccessible cardinal that is not a strongly inaccessible cardinal
As I study through inaccessible cardinals, I find many examples that show some cardinal being both weakly inaccessible and strongly inaccessible.
So, can anyone show me the example of weakly ...
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Is the class of all ordinals independent of set theory?
I am little confused with this. In any model of set theory (or is only in ZFC?), we start with an initial set, and then by transfinite recursion we build the universe of sets, which is a proper class. ...
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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 $M\vDash \phi(a) \ \iff N\vDash \phi(f(a))$ ...
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Weakly-compact cardinals
I am reading Jech's Set Theory, in particular the chapter about large cardinals. After discussing measurable cardinals he moves on to weakly-compact cardinals, which have been discussed far earlier in ...
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The tree property for non-weakly compact $\kappa$
In my previous question, Weakly-compact cardinals, I was asking about weakly-compact cardinals and equivalent definitions to the basic one, which is $\kappa \to (\kappa)^2_2$.
One of which was that ...
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1answer
128 views
Did large cardinals exist before 1963?
I'm curious to know the history of the interaction between large cardinals and traveling to (creating) universes through forcing. The question arose because I understand that Peano Arithmatists ...
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Strongly inaccessible cardinals and certain kinds of universes
Write $\mathrm{tc}(*)$ for transitive closure.
For all cardinals $\kappa$, let $H(\kappa)$ denote the collection of all sets $X$ such that
$|X|<\kappa$, and
For all $x \in \mathrm{tc}(X)$, it ...
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Kunen's result and $L(\mathbb{R})$
Recall Kunen's result: If $j:V \to M$ is a nontrivial elementary embedding then $V \ne M$.
Assume $AD$ is true in $L(\mathbb{R})$. Let $j : L(\mathbb{R}) \to M$ be a nontrivial elementary embedding. ...
