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Results tagged with large-cardinals
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user 26705
1
vote
A "Completion" of $ZFC^-$
Overnight the following occurred to me...
The answer to Question 2 is negative (with an asterisk), and so the same is true of Question 1. Namely, let $T$ be a set of $\Pi_2$ sentence with $ZFC^- \cup …
- 2,146
6
votes
2
answers
636
views
A "Completion" of $ZFC^-$
Let $T_0$ be the set theory axiomatized by $ZFC^-$ (that is $ZFC$ without powerset) + every set is countable + $\mathbb{V}=\mathbb{L}$.
Question 1: Suppose $\phi$ is a sentence of set theory. Must t …
- 2,146
16
votes
Accepted
Can an ultrapower be undone by forcing?
For set-forcing, the answer is no, see the following article
Joel David Hamkins, Greg Kirmayer, and Norman Lewis Perlmutter, Generalizations of the Kunen inconsistency, Ann. Pure Appl. Logic 163 (201 …
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8
votes
1
answer
333
views
Consistency Strength of "HC is elementary in V[G]"
Let $P$ be the Levy-collapse of the ordinals, so $P$ is a class forcing notion that makes every ordinal countable.
Note that since $P$ is weakly homogeneous, for any formula $\phi(\overline{a})$ wit …
- 2,146
7
votes
2
answers
373
views
A Weakening of the Tree Property
If $f$ and $g$ are two functions, define $f \sim g$ if they differ only finitely often on their common domain.
The following property of a large cardinal arose from a problem in model theory. I am in …
- 2,146