×

On the role of the Baire category theorem and dependent choice in the foundations of logic. (English) Zbl 0567.03023

The principle of dependent choice is shown to be equivalent to: the Baire category theorem for Čech-complete spaces (or for complete metric spaces); the existence theorem for generic sets of forcing conditions; and a proof-theoretic principle that abstracts the ”Henkin method” of proving deductive completeness of logical systems. The Rasiowa-Sikorski lemma is shown to be equivalent to the conjunction of the ultrafilter theorem and the Baire category theorem for compact Hausdorff spaces.

MathOverflow Questions:

BCT equivalent to DC

MSC:

03E25 Axiom of choice and related propositions
06E10 Chain conditions, complete algebras
54E52 Baire category, Baire spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.2307/1968839 · Zbl 0017.42803 · doi:10.2307/1968839
[2] General topology (1977)
[3] Boolean-valued models and independence proofs in set theory (1977) · Zbl 0371.02028
[4] Annali di Matematica Pura ed Applicata 3 pp 1– (1899) · JFM 30.0359.01 · doi:10.1007/BF02419243
[5] Axiomatic set theory (1973)
[6] Fundamenta Mathematicae 37 pp 193– (1950)
[7] DOI: 10.1007/BFb0062859 · doi:10.1007/BFb0062859
[8] DOI: 10.1073/pnas.10.5.168 · doi:10.1073/pnas.10.5.168
[9] The completeness of the first-order functional calculus 14 pp 159– (1949)
[10] Grundzüge der Mengenlehre (1914) · JFM 45.0123.01
[11] Axiomatic set theory 13 pp 83– (1971)
[12] DOI: 10.1007/BF00136118 · doi:10.1007/BF00136118
[13] Bulletin de l’Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques 9 pp 163– (1961)
[14] Models of ZF-set theory 223 (1971)
[15] Fundamenta Mathematicae 56 pp 325– (1965)
[16] Review of Rasiowa and Sikorski [1950] 17 pp 72– (1952)
[17] Fundamenta Mathematicae 81 pp 315– (1974)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.