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Results tagged with axiom-of-choice
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user 165275
An important and fundamental axiom in set theory sometimes called Zermelo's axiom of choice. It was formulated by Zermelo in 1904 and states that, given any set of mutually disjoint nonempty sets, there exists at least one set that contains exactly one element in common with each of the nonempty sets. The axiom of choice is related to the first of Hilbert's problems.
11
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What can be preserved in mathematics if all constructions are carried out in ZF?
One of the standard texts which presents functional analysis only based on ZF+DC is the monograph (consisting of 3 volumes) Henry G. Garnir, Marc de Wilde, and Jean Schmets, Analyse Fonctionnelle.
Als …
- 1,869
5
votes
BCT equivalent to DC
Yet another formulation of Blair's proof is in M. Väth, Topological Analysis, DeGruyter 2012.
- 1,869
4
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Simpler proofs using the axiom of choice
For an orthomorphism $A$ in a (real or complex) Banach lattice $X$ the formula $\lvert A\rvert x=\lvert Ax\rvert$ for $x\ge0$ can be obtained relatively quickly if one uses the fact that $X$ is order …
1
vote
Axiom of Countable Choice and meager sets
What is your definition of meager in (ZF)? Being a countable union of nowhere dense sets? In that case, the given argument about the failure of (UMM) in (ZF) is not clear: It is not obvious (to me) th …
- 1,869