[FOM] Infinity and the "Noble Lie"

[praatika@mappi.helsinki.fi]

> > "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 
sense.] 


Best, Panu


Panu Raatikainen
Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy

Department of Philosophy
P.O. Box 9
FIN-00014 University of Helsinki
Finland

E-mail: panu.raatikainen at helsinki.fi

http://www.helsinki.fi/collegium/eng/Raatikainen/raatikainen.htm