Questions tagged [large-cardinals]
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.
506
questions
5
votes
1
answer
113
views
Absoluteness of inaccessible cardinals
I'm studying large cardinals and I'm hoping to fully understand the proof that says ZFC is not able to prove the existence of inaccessibles (given ZFC is consistent, of course).
I've already fully ...
- 3,630
0
votes
0
answers
36
views
Aleph sequence fixed points and the least inaccessible [duplicate]
Using the properties of the least inaccessible cardinal (being a regular fixed point of the aleph function) I was able to prove that the least inaccessible $\kappa$ is the $\kappa$-th fixed point of ...
7
votes
1
answer
167
views
Use of (weak forms of) AC for elementary embeddings proof
I encounter this issue when going through equivalent characterizations of measurable cardinals. For completeness, let me reproduce the statement:
For ordinal $\kappa$, the following are equivalent.
...
- 105
1
vote
0
answers
46
views
What is $j_\mathcal{U}(\kappa)$ if $2^{\kappa}<j_\mathcal{U}(\kappa)<(2^{\kappa})^{+}$?
Let $\mathcal{U}$ a non-principal $\kappa$-complete ultrafilter on $\kappa$ and let $j_\mathcal{U}\colon V\to M$ its associated elementary embedding of the universe $V$ in $M=Ult_\mathcal{U}(V)$. In ...
0
votes
0
answers
56
views
On the Singular Cardinal Hypothesis
I'm trying to find the proof of this result.
If for each $\lambda\geq2^\omega$, $\lambda^\omega\le\lambda^+$, then the SCH holds.
I'm not sure where to look. So if you have any info about this, please ...
- 103
7
votes
1
answer
114
views
If $\mathsf{GCH}$ fails on a measure one set must it fail at $\kappa$?
Suppose $U$ is a normal measure on $\kappa$. It is well known that if $\{\alpha<\kappa:2^\alpha=\alpha^+\}\in U$ then $2^\kappa=\kappa^+$.
Question: does $\{\alpha<\kappa:2^\alpha=\alpha^{++}\}\...
- 998
2
votes
0
answers
48
views
Every weakly compact cardinal $\kappa$ is the $\kappa$th inaccessible cardinal
I'm currently reading Jech's Set Theory book, and in particular, while reading chapter 9, he mentions right after Lemma 9.9 that "We shall prove in Chapter 17 that every weakly compact cardinal $\...
- 329
6
votes
1
answer
130
views
A metrizable space is realcompact iff it has non-measurable cardinality?
A space is realcompact if its a closed subspace of an arbitrary product of real lines, with product topology.
A cardinal $\kappa$ is called measurable if there exists a (countably additive) $\{0, 1\}$-...
- 10.5k
2
votes
1
answer
59
views
How to show that the application of two elementary embeddings is an elementary embedding?
Let $\mathcal L$ be a language of first-order logic. Given two structures $\mathfrak M$, $\mathfrak N$ for $\mathcal L$ with domains $M$, $N$ respectively, an elementary embedding from $\mathfrak M$ ...
- 1,311
0
votes
0
answers
74
views
Is this property of Mahlo cardinals correct?
Quoting from a paper discussing large cardinals:
for any ρ that is a Mahlo cardinal, there must also exist ρ smaller cardinals (call them κ, with
all κ < ρ) that are k-inaccessible, hyper k-...
- 21
0
votes
1
answer
107
views
What is the relation between the first and second inaccessible cardinals? [closed]
I know how the first inaccessible transcends the other cardinals by being regular and strong limit but what does a second inaccessible mean.I had two possibilities in mind:
Either:
It is inaccessible ...
0
votes
1
answer
78
views
How to prove consistency with choice for large cardinal extensions?
How can we know if an extension of $\sf ZF$ by some large cardinal property that results in a consistency strength beyond $0^{\#}$ is compatible with choice or not?
I mean the easiest way to know if ...
- 4,631
1
vote
1
answer
75
views
Why is strongness of a cardinal a different notion than Reinhardtness?
A cardinal $\kappa$ is Reinhardt if there is a nontrivial elementary embedding $j:V\to V$ such that $\kappa$ is the critical point of $j$. Kunen proved (using choice) that this large cardinal axiom is ...
- 1,311
7
votes
1
answer
272
views
If there is a large cardinal, can GCH also hold?
Let $P$ be a statement saying there is large cardinal of some kind. For example, $P$ can be one of
There is a weakly inaccessible cardinal.
There is a Mahlo cardinal.
There is a weakly compact ...
- 1,821
2
votes
1
answer
82
views
If $\kappa$ is weakly inaccessible, then it is the $\kappa$th element of $\{\alpha: \alpha =\aleph_\alpha\}$
This is an exercise from Kunen:
Exercise I.13.17 If $\kappa$ is weakly inaccessible, then it is the $\kappa$th element of $\{\alpha: \alpha =\aleph_\alpha\}$. If $\kappa$ is strongly inaccessible, ...
- 4,827