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Results tagged with set-theory
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user 5697
forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
4
votes
1
answer
185
views
Theories and indiscernible propositions
Are there known examples of statements which are strong from a proof-theoretic standpoint but which are indistinguishable by one set of axioms (or proof system) yet distinct according to a stronger se …
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2
votes
0
answers
207
views
Logical relationships between weakenings of AC
What are the known logical implications between weak choice principles like $DC_\kappa$", the ultrafilter theorem for sets of size $\kappa$" (by which I mean every filter over a set A of size $\kappa$ …
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3
votes
2
answers
512
views
Antichains and the Knaster Property
This may be a naive question, but I'll pose it.
Is there an example of a notion of forcing $\mathbb{P}$ that has the $\kappa$-c.c. which is not also $\kappa$-Knaster Property also is not "factorabl …
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9
votes
1
answer
437
views
Stationary sets in HOD
My questions concern the following quote from “The HOD Dichotomy”, page 8.
"… notice that $\ cof(\omega)\cap\lambda$ belongs to $HOD$ even though it might mean
something else there. Also, $\{S\subse …
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9
votes
1
answer
1k
views
Critical points of rank-into-rank embeddings
$\DeclareMathOperator{\crit}{\operatorname{crit}}$A rank-into-rank embedding is a non-trivial elementary embedding from a rank initial segment of $V$ into itself: $j:V_\delta\prec V_\delta$. Define th …
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2
votes
Can one force there to be an elementary embedding $j:V_{\lambda}\rightarrow V_{\lambda}$ for...
Given Victoria's wonderful comment, I wonder if the following small forcing example is relevant to your question. The argument is from Laver's article "Certain very large cardinals are not created in …
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10
votes
1
answer
482
views
Generic Extensions and $L(V_{\lambda+1})$
Suppose $\lambda$ is a strong limit cardinal of cofinality $\omega$ and for $A$ a transitive set, define $L(A)$ in the usual fashion by setting
$$L_0(A)=A;$$
$$L_{\alpha+1}(A) = L_\alpha (A)\cup \mat …
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7
votes
1
answer
494
views
Elementary Embeddings and Relative Constructibility
Suppose $$j:M\prec N$$ is a non-trivial elementary embedding. Under what conditions on the sets (classes?) $M$ and $N$ (or even the critical point of $j$) does $j$ extend to an elementary embedding $$ …
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4
votes
2
answers
555
views
Set forcing and ultrapowers
The following is a result of Woodin (the proof is found after Theorem 5 of "Generalizations of the Kunen Inconsistency" by J.D.Hamkins, G.Kirmayer and N.L.Perlmutter):
(Woodin) Let $V[G]$ be a set-f …
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5
votes
1
answer
590
views
Solovay's Theorem on Partitions of Stationary Sets and Weak Choice Principles
There is a weak choice principle called $DC_\lambda$ which holds in $L(V_{\lambda+1})$ under the assumption of a non-trivial elementary embedding $$j:L(V_{\lambda+1})\prec L(V_{\lambda+1})$$ and it is …
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3
votes
Very Large Cardinal Axioms and Continuum Hypothesis
There are some candidate axioms that are beginning to surface on the internet that appear to have large cardinal characteristics and could potentially settle questions like the CH. If they are consist …
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3
votes
3
answers
573
views
Infinite Partitions of the Primes and Sums of Reciprocals (Revised)
I have revised my original post. The questions I asked there were not well-put or even thought through. I don't want to delete, however, since some of the comments may be of interest to other MO users …
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6
votes
What is the definition of a large cardinal axiom?
While I think I agree with Tim Chow and Joel Hamkins in some of their comments above regarding a single formal definition of what it is to be a large cardinal, I want to suggest that a large cardinal …
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