All Questions
Tagged with cardinals large-cardinals
61
questions
5
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1
answer
113
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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
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56
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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
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
2
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0
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142
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Does worldly cardinal exist if $\mathsf{ZFC}$ is consistent?
A worldly cardinal is a cardinal $\kappa$ such that $V_\kappa$ is a model of $\mathsf{ZFC}$. Please forgive me if this is very silly, but if $\mathsf{ZFC}$ is consistent (so there exists a model of $\...
- 1,923
0
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0
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277
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The Axioms of 'Fictional Googology'
There have been questions on whether proper classes have cardinality (some say yes). However, I have my own axioms about it.
Firstly, we define the 'cardinality' of a proper class as the conglomerate ...
- 239
0
votes
0
answers
187
views
What is the power set of an inaccessible cardinal?
I've been reading about set theory and the difference between small and large cardinals.
since taking the power set of small cardinals (alephs) allows us to create larger cardinals/alephs
I know that ...
- 11
0
votes
0
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119
views
"Proof" that "ZFC + there exists an inaccessible cardinal" is consistent
I have a proof that this theory is consistent using this theory itself. I want to know what's wrong with this proof:
"ZFC + there exists an inaccessible caridnal" proves that "ZFC is ...
- 1,437
0
votes
1
answer
95
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Are there any proposed operations to actually construct an inaccessible set?
When we postulate the smallest infinite set, we define it using an interative process involving iterations on the empty set.
When we postulate the existence of a set of continuum cardinality, we again ...
- 1,437
1
vote
1
answer
60
views
elementary embeddings $j$ in set theory with $V$ and $M$
I'm confused by a variety of elementary non-trivial elementary embedings $j$ we might have.
There are 9 "syntactical" possiblities;Here $M$ is a transitive model. I'll name them with a wish ...
- 3,978
1
vote
2
answers
117
views
Quick question about inaccessible cardinals
I am trying to find a source that states, it is consistent with ZFC that weakly inaccessible cardinal does not exist. Can I please get some sources? It was quite hard for me to find such.
Indeed, I ...
- 782
3
votes
3
answers
90
views
Compact cardinal cannot be successor?
This is a follow-up question to $\kappa$ is compact $\implies$ $\kappa$ is regular. The definition I'm using for "compact" is the same as there.
I am trying to show if $\kappa$ is compact, ...
- 1,847
2
votes
1
answer
49
views
Axiom schema of Replacement analogue for Von Neumann stage
I am working without Choice for the time being, so "cardinal" means "well-founded cardinal."
Let $\kappa$ be a strongly inaccessible cardinal. I want to show that the Axiom Schema ...
- 1,847
2
votes
1
answer
184
views
Name of infinite cardinals which has nonprincipal $\sigma$-complete ultrafilters?
The book "General Topology" by Engelking defines non-measurable cardinals as cardinals admitting no nonprincipal $\sigma$-complete ultrafilters. And then it claims that the discrete space of ...
- 1,292
0
votes
1
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71
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For a measurable cardinal $κ$, show that $cf(γ)≠κ$ implies $j_U(γ)=\sup\{j_U(δ):δ<γ\}$ ($U$ $κ$-complete ultrafilter, $j_U$ associated embedding) [closed]
For :
$κ$ a measurable cardinal,
$U$ a $κ$-complete ultrafilter over $κ$
$j_U$ the elementary embedding of $V$ into the ultrapower of $V$ to $U$
How to show that : If $\operatorname{cf}(γ)≠κ$ ...
0
votes
0
answers
213
views
Books on infinite sets
I'm a senior undergraduate Pure Math student. I'm looking for books/ problems about large/infinite cardinals and infinite sets. Unsolved or challenging problems are very much welcome.
It shouldn't be ...
- 373