I don't know why NF is inconsistent with AC. I've read that it seems
to be a problem with AC as it applies to "big" sets, which NF has
plenty of (a universal set, for example). The fact that the universe
is not well-ordered also explains why the axiom of infinity follows
in NF.
The weird thing about it is that NFU, which only differs from NF
by allowing urelements, is consistent with the axiom of choice,
and fails to prove the axiom of infinity.
Re: Some basic set theory questions ... with me that getting the existence of the empty set from ... using infinity.... This debate concerns whether the Axiom of Infinity is ... required to prove that the empty set exists. ... (sci.math)
Re: Ultimate debunking of Cantors Theory ... without the axiom of infinity.... laid the foundations of the geometry associated with his name. ...mathematics lays it foundations with the axioms of set theory. ... (sci.math)
Re: Infinities and infinitesimals ... An `infinitesimal' is the reciprocal of an infinity, namely the decay rate of a function from natural numbers to rational numbers. ... I might add that these two beliefs are "sort-of" consistent, because infinity only exists in set theory because there is an axiom which states it does exist. ... Now, if a "real" mathematician wanted to prove that "infinity doesn't exist", they would produce a proof that the axiom of infinity was inconsistent with the other axioms of ZF. ... I don't know why you bothered; Peano arithmetic itself doesn't require the axiom of infinity, so why bother producing a non-standard construction of the natural numbers at all? ... (sci.math)
Re: Infinities and infinitesimals ... and somehow infinity doesn't exist. ... often called the "axiom of infinity") which states it does exist. ... if a "real" mathematician wanted to prove that "infinity doesn't ... so why bother producing a non-standard construction of the natural ... (sci.math)
Re: Some basic set theory questions ... with me that getting the existence of the empty set from ... using infinity.... This debate concerns whether the Axiom of Infinity is ...MoeBlee obviously adheres to 3. ... (sci.math)
