PCF arithmetic without and with choice

@article{Shelah2009PCFAW,
  title={PCF arithmetic without and with choice},
  author={Saharon Shelah},
  journal={Israel Journal of Mathematics},
  year={2009},
  volume={191},
  pages={1-40}
}
  • S. Shelah
  • Published 19 May 2009
  • Mathematics
  • Israel Journal of Mathematics
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 are assumed, but some of the results give us additional information even in ZFC. We also start to deal with pcf for pseudo-cofinality (in ZFC with little choice). 
5 Citations
Pcf without Choice Sh835
We mainly investigate models of set theory with restricted choice, e.g., ZF + DC + the family of countable subsets of λ is well ordered for every λ (really local version for a given λ). We think that
ZF+DC+AX_4
We consider mainly the following version of set theory: "ZF + DC and for every lambda, lambda^{aleph_0} is well ordered", our thesis is that this is a reasonable set theory, e.g. on the one hand it
ZF + DC + AX4
TLDR
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).
LIST OF PUBLICATIONS
1. Sh:a Saharon Shelah. Classification theory and the number of nonisomorphic models, volume 92 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam-New
LIST OF PUBLICATIONS
1. Sh:a Saharon Shelah. Classification theory and the number of nonisomorphic models, volume 92 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam-New

References

SHOWING 1-10 OF 12 REFERENCES
PCF without choice
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
PCF and infinite free subsets in an algebra
This contains several results connected with pcf theory, a powerful theory devised by the present author to obtain deep results in cardinal arithmetic [ S. Shelah , Cardinal arithmetic. Oxford
Splitting stationary sets from weak forms of Choice
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
Set theory without choice: not everything on cofinality is possible
TLDR
It is proved in ZF+DC that if $\mu=|{\cal H}(\mu)|$ and \(\mu>\cf(\mu)>\aleph_0$ then $\ mu ^+$ is regular but non measurable, in contrast with the results on measurability for $\Mu=\ aleph_\omega$ due to Apter and Magidor.
On long increasing chains modulo flat ideals
  • S. Shelah
  • Chemistry, Mathematics
    Math. Log. Q.
  • 2010
We prove that, e.g., in (ω 3)(ω 3) there is no sequence of length W4 increasing modulo the ideal of countable sets (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
More on the revised GCH and the black box
The generalized continuum hypothesis revisited
We can reformulate the generalized continuum problem as: for regular κ<λ we have λ to the power κ is λ, We argue that the reasonable reformulation of the generalized continuum hypothesis, considering
Saharon Shelah. PCF with little choice
  • Saharon Shelah. PCF with little choice
PCF without choice. Archive for Mathematical Logic, submitted. math.LO/0510229
  • PCF without choice. Archive for Mathematical Logic, submitted. math.LO/0510229
A Note on Cardinal Exponentiation
...
...