An appendix explains the axiom of choice
and its equivalents.
There is one more parallel we have to draw regarding Marx's construction of the money form (form IV) and Badiou's usage of the axiom of choice
. (21) The first step is to once more return to the simple or accidental form of value, one with a one-to-one correspondence (i:i) between two commodities.
Let us briefly recall three of them: the irrationality of [square root of]2, the intermediate value theorem in the calculus, and the axiom of choice
in set theory.
(20) Their objections may be rooted in the fact that one of the more significant axioms for Badiou, the "axiom of choice
," was already controversial among mathematicians.
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]).
One of the disputes regarding the axioms of ZFC arises over the "C" in the acronym, which refers to the axiom of choice
. Loosely speaking, this axiom stipulates that, given any collection of non-empty bins, it is possible to select one item from each of them.
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.