• Corpus ID: 118389077

# PCF without choice

@article{Shelah2005PCFWC,
title={PCF without choice},
author={Saharon Shelah},
journal={arXiv: Logic},
year={2005}
}
• S. Shelah
• Published 11 October 2005
• Mathematics
• arXiv: Logic
We mainly investigate model of set theory with restricted choice, e.g., ZF + DC + "the family of countable subsets of lambda is well ordered for every lambda" (really local version for a given lambda). In this frame much of pcf theory can be generalized. E.g., there is a class of regular cardinals, and we can prove cardinal inequality.
9 Citations
PCF arithmetic without and with choice
We deal with relatives of GCH which are provable. In particular, we deal with rank version of the revised GCH. Our motivation was to find such results when only weak versions of the axiom of choice
PCF ARITHMETIC WITHOUT AND WITH CHOICE SH938
We deal with relatives of GCH which are provable. In particular we deal with rank version of the revised GCH. Our motivation was to find such results when only weak versions of the axiom of choice
Splitting stationary sets from weak forms of Choice
• Chemistry, Mathematics
Math. Log. Q.
• 2009
Working in the context of restricted forms of the Axiom of Choice, we consider the problem of splitting the ordinals below λ of cofinality θ into λ many stationary sets, where θ < λ are regular
Generic I0 at ℵω
In this paper we introduce a generic large cardinal akin to I0 , together with the consequences of ℵω being such a generic large cardinal. In this case ℵω is Jónsson, and in a choiceless inner model
An Easton-like Theorem for Zermelo-Fraenkel Set Theory with the Axiom of Dependent Choice
• Mathematics
• 2018
We show that in the theory ZF + DC + for every cardinal {\lambda}, the set of infinite subsets of {\lambda} is well-ordered (i.e., Shelah's AX4), the {\theta}-function measuring the surjective size
ZF + DC + AX4
It is proved that for a sequence δ¯=⟨δs: s∈Y⟩,cf(δ s) large enough compared to Y, the pcf theorem can be proved with minor changes (in particular, using true cofinalities not the pseudo ones).
AXIOM I0 AND HIGHER DEGREE THEORY
A generic absoluteness theorem is obtained in the theory of I0, from which an analogue of Perfect Set Theorem for “projective” subsets of Vλ+1, and the Posner–Robinson follows as a corollary is obtained.
Generic I0 at $\aleph_\omega$
A generic large cardinal akin to I0 is introduced, and in a choiceless inner model many properties hold that are in contrast with PCF in ZFC.
More on the revised GCH and the black box

## References

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Silver [7] recently proved that, if GCH holds below , then it holds at 8 [Lo2lo]wr for some p < 03. (Here (wo, is an ordinal power; in the rest of this paper, only cardinal exponentiation is used.)
The covering lemma for K
• Mathematics
Ann. Math. Log.
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A Note on Cardinal Exponentiation