All Questions
Tagged with axiom-of-choice large-cardinals
36
questions
7
votes
0
answers
180
views
Large cardinals beyond choice and HOD(Ord^ω)
Are Reinhardt and Berkeley cardinals (and even a stationary class of club Berkeley cardinals) consistent with $V=\mathrm{HOD}(\mathrm{Ord}^ω)$ ?
It seems natural to expect no, but I do not see a proof....
- 6,365
14
votes
1
answer
1k
views
Are Berkeley cardinals easier to refute in ZFC than Reinhardt cardinals?
Kunen showed that Reinhardt cardinals are inconsistent in ZFC. But his proof is a bit technical for a non-set-theorist to follow. Berkeley cardinals are stronger than Reinhardt cardinals. You can ...
- 62.6k
4
votes
0
answers
221
views
stating large cardinal axioms in ZF
Can I ask whether there is a good reference for how to state the standard large cardinal axioms in the context of $ZF$? My concern is that it seems that the usual proof that embeddings defined from ...
- 2,055
5
votes
1
answer
515
views
Large cardinals in ZF + DC + AD
The Axiom of Dependent Choice (DC) is often considered to be an "intuitive and non-controversial" version of choice used in the proofs of many theorems in Analysis. Similarly, the Axiom of ...
- 381
4
votes
0
answers
254
views
Can we have full choice prior to Reinhardt cardinals?
Working in $\sf ZF + Reinhardt \ cardinal$, can we have full choice over all stages $V_{\alpha < \kappa}$ where $\kappa$ is the Reinhardt cardinal, i.e., the critical point of the elementary ...
- 10.2k
1
vote
0
answers
74
views
Can this method let choiceless large cardinals be smaller than cardinals compatible with choice?
Recall question "Can we have this sequence where choice fails and returns?"
Can that theory be extended with requiring the $\mathcal V_n$'s to fulfill a choiceless large cardinal extension ...
- 10.2k
6
votes
1
answer
321
views
Can we have this sequence where choice fails and returns?
Can we have a sequence of transitive sets $\langle\mathcal V_0, \mathcal V_1, \mathcal V_2,...\rangle$, all modeling $\sf ZF$, such that $\mathcal P(V_n) \subset \mathcal V_{n+1}$, and the cardinality ...
- 10.2k
8
votes
1
answer
1k
views
Is there a form of choice that can elude Kunen's inconsistency theorem?
When it is said that Kunen inconsistency theorem proves that given $\sf ZFC$ no elementary embedding can exist from the universe to itself. Most references quote full choice in stating that result, ...
- 10.2k
8
votes
1
answer
416
views
Compatibility of $\mathsf{SVC}$ and Reinhardtness
Can we prove the consistency of $\mathsf{ZF+SVC}$ + "There is a Reinhardt cardinal?" (Preferably from the consistency of $\mathsf{ZF}$ with a Reinhardt cardinal, but using a stronger ...
- 2,910
8
votes
1
answer
258
views
Some relevant questions about the consistency strength of singularity of $\omega_1$ and $\omega_2$
The following question was asked years ago on MSE, but let me recap it:
Question: Is there anything currently known about the exact consistency strength of "$\mathsf{ZF}$ + both $\omega_1$ and $\...
- 2,910
1
vote
0
answers
138
views
Can Reinhardt cardinals be compatible with Choice in absence of Extensionality?
Is the proof of existence of Reinhardt (and higher) cardinals violating Choice dependent on Extensionality in an essential manner?
What I mean is if we work in $\sf ZFA$ would it be possible to have a ...
- 10.2k
5
votes
0
answers
222
views
Which very large cardinals are preserved under Woodin's forcing for $\mathsf{AC}$?
Woodin showed that we can force $\mathsf{AC}$ if there is a proper class of supercompact cardinals while preserving supercompacts, by forcing Easton-support iteration of $\operatorname{Col}(\kappa,<...
- 2,910
9
votes
0
answers
2k
views
What are the known implications of “There exists a Berkeley cardinal”?
Inspired by this question: What are the known implications of "There exists a Reinhardt cardinal" in the theory "ZF + j"?
Definitions:
$\delta$ is Berkeley iff for every $\alpha\...
- 1,123
4
votes
1
answer
397
views
Does $H\vDash AC$
The set $H_\kappa$ of sets hereditarily of cardinality less than $\kappa$ is defined as $H_\kappa=\{x||tc(x)|\lt\kappa\}$. What if we define the set $H=H_{Ord}$ of sets hereditarily of cardinality ...
- 1,123
34
votes
1
answer
3k
views
Is the theory Flow actually consistent?
Recently the paper
Adonai S. Sant'Anna, Otavio Bueno, Marcio P. P. de França, Renato Brodzinski, Flow: the Axiom of Choice is independent from the Partition Principle, arXiv:2010.03664
appeared on ...
- 731