• 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. 
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ZF + DC + AX4
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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
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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 $\aleph_\omega$
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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

SHOWING 1-4 OF 4 REFERENCES
Inequalities for Cardinal Powers
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
A Note on Cardinal Exponentiation