[FOM] Potential and Actual Infinity
Panu Raatikainen
panu.raatikainen at helsinki.fi
Sat Mar 14 05:01:49 EDT 2015
Timothy Y. Chow" <tchow at alum.mit.edu>:
> Joseph Shipman wrote:
>
>> This discussion seems to be making too much of a simple point. In
>> Peano Arithmetic, which can be equivalently formalized by taking ZF
>> and replacing the axiom of Infinity with its negation, there is no
>> actual infinity.
>
> Yes, this is what I said in my initial post, and I think I was the
> first one to bring up the term "potential infinity" in the current
> discussion.
This is certainly one possible way to interpret what a commitment to
"actual infinity" is.
But surely it is not the common sense intended by, e.g., finitists and
constructivists, who use the notion so much:
they suggest that an unlimited use of classical logic to more complex,
quantified formulas in the context of arithmetic (where the domain of
quantification is necessarily infinite) already commits one to "actual
infinity".
I should perhaps also repeat my old point once again:
There are two different senses of "an axiom of infinity" in the logic
literature.
On the one hand, in logic, it often 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 the 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.
All the Best
Panu
--
Panu Raatikainen
Ph.D., Adjunct Professor in Theoretical Philosophy
Theoretical Philosophy
Department of Philosophy, History, Culture and Art Studies
P.O. Box 24 (Unioninkatu 38 A)
FIN-00014 University of Helsinki
Finland
E-mail: panu.raatikainen at helsinki.fi
http://www.mv.helsinki.fi/home/praatika/
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