[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  

I should perhaps also repeat my old point once again:

There are two different senses of "an axiom of infinity" in the logic  
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 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

E-mail: panu.raatikainen at helsinki.fi


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