[FOM] Infinity and the "Noble Lie"
praatika at mappi.helsinki.fi
Sat Jan 7 02:21:37 EST 2006
> > "Are you prepared to say that the question of the "truth" of an
> > arithmetical statement proved using the axiom of infinity is also
> > ridiculous?"
I wonder whether everyone here is using the notion of "an axiom of
infinity" in the same sense.
Often, in logic, it means any sentence which forces the domain to be
infinite (also the standard axioms of successor are together an axiom of
infinity in this sense). In set theory, on the other hand, the axiom of
infinity is the axiom which says that there is an infinite set. The axioms
of ZFC without this axiom* already make domain infinite, but it is this
axiom which gives ZFC its extreme power. It is much stronger assumption
than an axiom of infinity in the first sense.
(There may be also other senses...)
[*This is an infinite set of axioms, but one can give a finite
conservative extension of them. The latter (or, a conjunction of them) is
an axiom of infinity in the first sense, but obviously, not in the second
Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy
Department of Philosophy
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FIN-00014 University of Helsinki
E-mail: panu.raatikainen at helsinki.fi
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