All Questions
Tagged with embeddings large-cardinals
8 questions
4
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0
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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 ...
- 2,125
2
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How to express Kunen's inconsistency, Reinhardt and Wholeness axioms, by single sentences?
Working in $\sf NBG, $ can we express the property of a class being set theoretically definable, by a single sentence? Like for example, the following way:
$$\operatorname {std}(X) \iff \exists x_1 \...
- 11.3k
4
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0
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271
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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 ...
- 11.3k
1
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0
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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 ...
- 11.3k
2
votes
1
answer
357
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Can we interpret Reinhardt cardinals this way?
To the language of set theory add a primitive unary predicate $\operatorname {Universe}$ and a primitive total unary function $j$. Add all axioms of $\sf ZF$ in the language of this theory, i.e. the ...
- 11.3k
1
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2
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274
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Can we iteratively reflect on self elementary embeddable stages of the cumulative hierarchy?
Is it possible to iterate elementary embeddability and reflect on those stages that are elementary embeddable to themselves?
The following is a formal capture of that idea:
To the language of $\sf ZF$...
- 11.3k
8
votes
1
answer
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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, ...
- 11.3k
3
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1
answer
292
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If we add stratified\acyclic replacement to the wholeness axiom, would that increase its consistency strength?
If we add to the wholeness axiom, the axiom of stratified\acyclic replacement, what would be the consistency state and strength of the resulting theory?
The wholeness axiom $\sf WA$, introduced by ...
- 11.3k