21) But Badiou courts controversy: "within ontology, the axiom of choice
formalizes the predicates of intervention" (BE 227); he offers the following example:
By [L], the existence of real closures for formally real fields depends on the Axiom of Choice
, while the existence of real closures for ordered fields is known to follow from ZF alone (see [San]).
In 1922 the logicians Ernst Zermelo and Abraham Fraenkel produced a collection of axioms that, together with another axiom called the axiom of choice
, serves as the basis for a large portion of mathematics (the theory of sets, which can model what one normally thinks of as arithmetic).
Among them are countable versions of axiom of choice
and real analysis, a partition of the real line into a continuum of many thick subsets, a relationship between absolutely non-measurable functions and Sierpinski-Zygmund type functions, additive properties of certain classes of pathological functions, and a combinatorial problem on translation-invariant extensions of the Lebesgue measure.
By the axiom of choice
, you may choose any menu item(s).
The book views the basic axiom of choice
of transitivity, vector dominance, and endogeniety of preferences through a new lens to explore the vacuum left in the sophisticated models for existence of multiple equilibrium and government intervention.
Both the generalized continuum hypothesis and the axiom of choice
are consistent with set theory.
It is generally accepted that the (presumably) non-contradictory Zermelo-Fraenkel set theory ZF with the axiom of choice
is the most accurate and complete axiomatic representation of the core of Cantor set theory.
It also focuses on broader background, with brief but representative discussions of naive set theory and equivalents of the Axiom of Choice
, quadratic reciprocity, and basic complex analysis.
The demonstration of the consistency of the axiom of choice
and the continuum hypothesis, another of his great contributions, was announced in 1938 in the Proceedings of the National Academy of Sciencies, and published in more complete form in 1939.
5) The Axiom of Choice
is existential, rather than constructivist.
My aim will be to show that there is a possible subjective figure, based on the independence of the Axiom of Choice
, which remains unexamined in both these works.