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On BPI Restricted to Boolean Algebras of Size Continuum

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Hall E., Keremedis K.

Bulletin of the Polish Academy of Sciences. Mathematics

2013

tom 61

nr 1

9-21

(i) The statement P(ω) = "every partition of ℝ has size ≤ |ℝ|" is equivalent to the proposition R(ω) = "for every subspace Y of the Tychonoff product $2^{𝓟(ω)}$ the restriction 𝓑|Y = {Y ∩ B: B ∈ 𝓑} of the standard clopen base 𝓑 of $2^{𝓟(ω)}$ to Y has size ≤ |𝓟(ω)|". (ii) In ZF, P(ω) does not imply "every partition of 𝓟(ω) has a choice set". (iii) Under P(ω) the following two statements are equivalent: (a) For every Boolean algebra of size ≤ |ℝ| every filter can be extended to an ultrafilter. (b) Every Boolean algebra of size ≤ |ℝ| has an ultrafilter.

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1 Bulletin of the Polish Academy of Sciences. Mathematics

1 Hall E.

1 Keremedis K.

1 2013

Wyniki wyszukiwania

On BPI Restricted to Boolean Algebras of Size Continuum