[FOM] Must every proper class contain an infinite subset?
Johan Belinfante
belinfan at math.gatech.edu
Wed Mar 1 15:35:38 EST 2006
This only applies to theories with proper classes, so it has no
relevance to ZFC, etc...
To be definite, consider this to be a question about Goedel's 1939
theory of classes.
Question: Must every proper class contain an infinite subset? If so,
does the proof
require the global axiom of choice and/or the axiom of regularity
(Goedel's Axioms D, E)?
Writing P(x) for the class of all subsets of x, FINITE for the class
of all finite sets,
and V for the class of all sets, this boils down to the question
whether the following
implication holds for all x:
subclass(P(x), FINITE) => member(x, V).
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