# Tagged Questions

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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### Kurepa trees and inaccessible cardinals in $L$

Given a regular uncountable cardinal $\kappa$ we say that a $\kappa-$tree is $\kappa-$Kurepa if it has at least $\kappa^+$ branches. If $\kappa=\omega_1$ we simply say that $T$ is Kurepa. In this ...
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### Club class of inaccessibles

I am currently looking at what Drake calls the Axiom Schema F, "Every normal function defined for all ordinals has a regular fixed point". In ZFC+(Axiom F), does it hold that there is a club class of ...
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### Implications of existence of two inaccessible cardinals?

Many years ago in an oral exam I was asked, what could be concluded from the existence of an inaccessible cardinal in ZFC? I knew that would provide a model for ZFC and imply the consistency of ZFC. ...
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### Reference: Mahlo cardinals remain Mahlo in L

The following is stated on Wikipedia for Mahlo cardinals. Unfortunately, it's not sourced. Where can I find details? I wasn't able to google any articles dealing with Mahlo cardinals in $L$. Since ...
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### How does Ulam's argument about large cardinals work?

I am looking for either a reference, a proof, or a suitable proof sketch that can explain Ulam's original argument about measure theory and measurable cardinals. Here is the result I am looking for: ...
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### Together with the algebra of cardinal numbers, is there analysis of cardinal numbers? [closed]

Let $C$ be the collection of all cardinal numbers. Is there any norm, inner-product, metric (other than discrete metric), topology(other than discrete, co-finite topology) on $C$, which is very useful?...
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### What is the *exact* consistency strength of $MK$?

It's well known that the existence of an inaccessible cardinal proves $Con(MK)$. Joel Hamkins has a nice blog post (http://jdh.hamkins.org/km-implies-conzfc/) that explains what you get out of $MK$, ...
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### Proving ZFC is consistent

I've heard from a friend that we can actually prove the consistency of ZFC if we assume at least one inaccessible cardinal exists. How is this carried out, precisely? Googling doesn't help and my ...
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### Is it possible to define a group structure on arbitrary set? [duplicate]

Is it possible to define a group structure on arbitrary set? It is obvious for finite sets and also sets with cardinality |Q| and |R| and also we don't know that is there other cardinality betwen them ...
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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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### 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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### Measurable cardinals are weakly-compact?

First off, this isn't homework, but I'm doing research into large cardinal stuff so I wanna understand these theorems. I'm given this information to work with: a cardinal $\mu$ is measurable if ...
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### about the smallest $k$ that $V_k$ is a model of ZFC

Let $k$ to be the smallest ordinal that $V_k$ is a model of ZFC. I know that $k$ need not to be inaccessible cardinal,and $k$ has confinality $\omega$. Then how big is $k$? How to write down $k$ in ...
### Inaccessibility of $\omega^V_1$ in $L$ under determinacy
I'm looking for the proof of the following fact: Assume axiom of determinacy. Then $\omega^V_1$ is a strongly inaccessible cardinal in $L$. I have seen this result mentioned in few places (e.g. ...