Spectra of uniformity

Hayut, Yair; Karagila, Asaf

doi:10.14712/1213-7243.2019.008

Cited by 4 publications

(4 citation statements)

References 7 publications

(11 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following Lemmas 3.3-3.6 are known (see [4]), but we present proofs for the completeness. Again, we do not need AC for proving these lemmas.…”

Section: ℵ ω+1 Can Be the Least Elemet Of Umentioning

confidence: 99%

“…Let P be the poset from the previous section, and let Col(ℵ ω+1 , < κ) be the standard ℵ ω+1 -closed Levy collapsing poset which force κ to be ℵ ω+2 . Take a (V, Col(ℵ ω+1 , < κ))-generic H. In V [H], by Hayut-Karagila's symmetric collapse argument ( [4]), we can find a symmetric extension M of V in which the following hold:…”

Section: ℵ ω Can Be the Least Cardinal Not In Umentioning

confidence: 99%

“…Theorem 1.1 (Hayut-Kagagila [4]). Relative to a certain large cardinal assumption, it is consistent that ZF+" ℵ 0 , ℵ ω ∈ U but ℵ n+1 / ∈ U for every n < ω".…”

Section: Introductionmentioning

confidence: 99%

“…

In [4], Hayut and Karagila asked some questions about uniform ultrafilters in a choiceless context. We provide several answers to their questions.

…”

mentioning

confidence: 99%

See 3 more Smart Citations

Choiceless Löwenheim–Skolem Property and Uniform Definability of Grounds

Usuba

2021

Springer Proceedings in Mathematics &Amp; Statistics

View full text Add to dashboard Cite

show abstract

“…The following Lemmas 3.3-3.6 are known (see [4]), but we present proofs for the completeness. Again, we do not need AC for proving these lemmas.…”

Section: ℵ ω+1 Can Be the Least Elemet Of Umentioning

confidence: 99%

Section: ℵ ω Can Be the Least Cardinal Not In Umentioning

confidence: 99%

“…Theorem 1.1 (Hayut-Kagagila [4]). Relative to a certain large cardinal assumption, it is consistent that ZF+" ℵ 0 , ℵ ω ∈ U but ℵ n+1 / ∈ U for every n < ω".…”

Section: Introductionmentioning

confidence: 99%

“…

In [4], Hayut and Karagila asked some questions about uniform ultrafilters in a choiceless context. We provide several answers to their questions.

…”

mentioning

confidence: 99%

See 2 more Smart Citations

Choiceless Löwenheim–Skolem Property and Uniform Definability of Grounds

Usuba

2021

Springer Proceedings in Mathematics &Amp; Statistics

View full text Add to dashboard Cite

show abstract

Combinatorial properties and dependent choice in symmetric extensions based on Lévy collapse

Banerjee

2022

Arch. Math. Logic

View full text Add to dashboard Cite

We work with symmetric extensions based on Lévy collapse and extend a few results of Apter, Cody, and Koepke. We prove a conjecture of Dimitriou from her Ph.D. thesis. We also observe that if V is a model of $$\textsf {ZFC}$$ ZFC , then $$\textsf {DC}_{<\kappa }$$ DC < κ can be preserved in the symmetric extension of V in terms of symmetric system $$\langle {\mathbb {P}},{\mathcal {G}},{\mathcal {F}}\rangle $$ ⟨ P , G , F ⟩ , if $${\mathbb {P}}$$ P is $$\kappa $$ κ -distributive and $${\mathcal {F}}$$ F is $$\kappa $$ κ -complete. Further we observe that if $$\delta <\kappa $$ δ < κ and V is a model of $$\textsf {ZF}+\textsf {DC}_{\delta }$$ ZF + DC δ , then $$\textsf {DC}_{\delta }$$ DC δ can be preserved in the symmetric extension of V in terms of symmetric system $$\langle {\mathbb {P}},{\mathcal {G}},{\mathcal {F}}\rangle $$ ⟨ P , G , F ⟩ , if $${\mathbb {P}}$$ P is ($$\delta +1$$ δ + 1 )-strategically closed and $${\mathcal {F}}$$ F is $$\kappa $$ κ -complete.

show abstract

Dependent Choice, Properness, and Generic Absoluteness

Asperó

Karagila

2020

The Review of Symbolic Logic

View full text Add to dashboard Cite

We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to $\mathsf {DC}$ -preserving symmetric submodels of forcing extensions. Hence, $\mathsf {ZF}+\mathsf {DC}$ not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in $\mathsf {ZF}$ , and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in $\mathsf {ZF}+\mathsf {DC}$ and $\mathsf {ZFC}$ . Our results confirm $\mathsf {ZF} + \mathsf {DC}$ as a natural foundation for a significant portion of “classical mathematics” and provide support to the idea of this theory being also a natural foundation for a large part of set theory.

show abstract

Spectra of uniformity

Cited by 4 publications

References 7 publications

Choiceless Löwenheim–Skolem Property and Uniform Definability of Grounds

Choiceless Löwenheim–Skolem Property and Uniform Definability of Grounds

Combinatorial properties and dependent choice in symmetric extensions based on Lévy collapse

Dependent Choice, Properness, and Generic Absoluteness

Contact Info

Product

Resources

About