[FOM] How much of math is logic?

praatika@mappi.helsinki.fi praatika at mappi.helsinki.fi
Wed Feb 28 08:44:26 EST 2007

Quoting joeshipman at aol.com:

> I shouldn't have used the word "exactly". Raatikainen's first type of 
> axiom of infinity is not relevant to what I was trying to say, since 
> ZFC does not actually contain such an "axiom" -- the fact that the 
> collection of axioms of ZFC (not counting the standard AxInf) implies 
> the infinity of the domain is neither here nor there, because this 
> "domain" is not something I need to assert the existence of in order to 
> prove those sentences which don't require the actual AxInf.

Wrong! Even the very weak fragment of set theory, consisting only in the
three axioms, 

(i) extensionality, 
(ii) the existence of empty set, and 
(iii) the existence of the union x U {y} for sets x and y, 

is mutually interpretable with the Robinson arithmetic Q (Tarski &  Szmielew
1950), and hence it (or, the conjunction of the axioms) constitutes an axiom
of infinity in my first sense. 

And even this weak fragment fails to be logically true in the sense of
logically true Shipman favors, i.e., true in all models. 

> Responding to others: will it make you happy if instead of saying I can 
> derive all theorems of PA from pure logic, I say I can derive all 
> theorems of PA from pure logic plus the statement "the empty set 
> exists"? 

But you cannot! Cf. above. 

Best, Panu

Panu Raatikainen

Academy Research Fellow, The Academy of Finland
Docent in Theoretical Philosophy, University of Helsinki

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

e-mail: panu.raatikainen at helsinki.fi

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