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Results tagged with large-cardinals
Search options questions only
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user 48041
8
votes
2
answers
388
views
The consistency of $\Sigma_1$-elementary embeddings $j\colon V_{\lambda+2}\to V_{\lambda+2}$...
One way to stratify the large cardinal hierarchy between I3 and I1 is by using second-order elementary embeddings. We may view $j\colon V_{\lambda+1}\to V_{\lambda+1}$ as a second-order embedding $j\c …
- 3,042
5
votes
1
answer
128
views
On a question about ordinals $\xi$ satisfying $j_0(\xi)=j_1(\xi)$ for an $I_3$-embedding $j$
Let $j\colon V_\lambda\to V_\lambda$ be an $I_3$ embedding with the critical sequence $\kappa_n$. Define $j_0=j$, $j_1 = j[j]=\bigcup_{\alpha<\lambda} j(j\upharpoonright V_\alpha)$. My question is the …
- 3,042
5
votes
1
answer
174
views
Finding many subsets of $V_{\lambda+2}$ stable under $j:V\prec V$
Working over $\mathsf{ZF}$ with an embedding $j:V\prec V$ with a critical point $\kappa$. Take $\lambda=\sup_{n<\omega}j^n(\kappa)$. (You may assume $\mathsf{DC}_\lambda$ if you need, but I am not sur …
- 3,042
8
votes
1
answer
430
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 assumption is als …
- 3,042
9
votes
0
answers
271
views
Can we have a 'universal class' for elementary embeddings $j\colon V\to V$
Work over $\mathsf{GB}$, Gödel-Bernays set theory (without choice). My question is the following:
Question. Is the following statement consistent with $\mathsf{GB}$? There is a universal class for el …
- 3,042
5
votes
1
answer
189
views
Can a generic ultrafilter over $\mathrm{NS}^+_{\omega_1}$ witness $\omega_1$ is Ramsey-like?
Suppose that $\kappa$ is an appropriate large cardinal (preferably a Woodin cardinal, but possibly something stronger) and let $G$ be a $\operatorname{Col}(\omega_1,<\kappa)$-generic filter over $V$. …
- 3,042
9
votes
1
answer
273
views
Some relevant questions about the consistency strength of singularity of $\omega_1$ and $\om...
The following question was asked years ago on MSE, but let me recap it:
Question: Is there anything currently known about the exact consistency strength of "$\mathsf{ZF}$ + both $\omega_1$ and $\omeg …
- 3,042
10
votes
0
answers
230
views
Absoluteness of the core model under a proper class of completely Jónsson cardinals
Example 2.4.2 of Larson's The stationary tower (based on Woodin's lecture) describes how we can absorb a generic filter over a set forcing in a definable inner model into a generic extension by $\math …
- 3,042
5
votes
0
answers
228
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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,<V_ …
- 3,042