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revised The axiom $I_0$ in the absence of $AC$
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17h
asked The axiom $I_0$ in the absence of $AC$
23h
awarded  Nice Answer
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revised A specific Model of ZFC
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2d
revised A specific Model of ZFC
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Apr
27
revised Embedding property of weakly compact cardinals
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Apr
27
comment Embedding property of weakly compact cardinals
In Hauser, it is directly proved $\Pi^1_1$-indescribability implies the embedding property.
Apr
27
comment Embedding property of weakly compact cardinals
Another natural question that can be added is who (and where) proved the equivalence between weak compactness and embedding property. I think the extension property is due to Keisler.
Apr
27
comment Embedding property of weakly compact cardinals
Your welcome. Maybe the best thing is to look at Hauser paper I mentioned above and simplify his argument for the case of weakly compact cardinals. I gave the above as it was the simplest way (for me) to give such a characterization.
Apr
27
answered Embedding property of weakly compact cardinals
Apr
27
comment Embedding property of weakly compact cardinals
We require the model $N$ to have one more property which is called $\Sigma^m_{n-1}$-correct.
Apr
27
comment moving up a consequence of PFA
In such a model there are no $\aleph_2$-Souslin trees, so at least if $CH$ holds (which is the case in our model), yes we need some large cardinals.
Apr
27
comment Embedding property of weakly compact cardinals
What do you mean?
Apr
26
comment Embedding property of weakly compact cardinals
Also the book ``Set theory an introduction to large cardinals'' by Drake gives the equivalence of weak compactness and $\Pi^1_1$-indescribability. Putting these results together gives you the required result.
Apr
26
comment Embedding property of weakly compact cardinals
In the paper ``Indescribable cardinals and elementary embeddings'' an equivalence of the type you asked is proved for $\Pi^m_n$-indescribable cardinals.
Apr
26
revised moving up a consequence of PFA
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Apr
26
comment moving up a consequence of PFA
Thanks, in fact we were working on a different problem, but later it was realized that our methods work to give the above result. My motivation for the above comes from Foreman's maximality principle.
Apr
26
comment moving up a consequence of PFA
Yes, we essentially did the same thing.
Apr
26
comment moving up a consequence of PFA
Your question follows from the above mentioned theorem. If the forcing adds no new subsets to $\aleph_1,$ then forcing adds a new subset to $\aleph_2$ so the theorem applies.
Apr
26
revised moving up a consequence of PFA
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