[FOM] How much of math is logic?

[praatika@mappi.helsinki.fi]

Quoting joeshipman at aol.com:

> ... but there are alternative 
> formulations of number theory which are built up from pure logic, going 
> back to Russell. It is possible to define a mapping from sentences of 
> arithmetic to statements of logic, in such a way that the axioms and 
> theorems of Peano Arithmetic map to logical validities. 

It is possible to effectively map (almost) any r.e. set to any other r.e.
set, but that does not make an arbitrary r.e. set of formulas a set of
logically valid sentences. (I am simplifying a bit, but nothing hinges on that)

> However, this route does not seem to transcend Peano Arithmetic; as I 
> understand it, arithmetical statements whose proof (in ZF) requires the 
> use of the axiom of Infinity will not be reachable in the versions of 
> this setup that correspond in some way to "first order logic" ...

Let me first repeat what I have said here in FOM already earlier: 
There are two different senses of "an axiom of infinity". 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 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.

Now I don't think that simple type theory and such can provide us much
mathematics without the addition of an axiom of infinity in the first sense.
And the claim that such an assumption is logically true is quite controversial. 
 
> (By the way, I use "logical truth" and "logical validity" 
> interchangeably to mean sentences true in all models; in the case of 
> first-order logic there is an enumeration of such sentences, but I'm 
> not restricting my attention to first-order logic.)

So, an axiom of infinity (in the first sense), which is false in any finite
model, is not logically true. 

> My second point is that apart from the Axiom of Infinity, which is an 
> existence postulate that is necessary because pure first-order-logic 
> does not entail such a rich ontology, it can be argued that the axioms 
> of ZF (or of an equivalent system like VNBG) appear to deserve the 
> status "logical" rather than merely "mathematical". This supports the 
> slogan "mathematics is just (first-order) logic plus the axiom of 
> infinity".

Almost all axioms (well, not extensionality) of ZFC are set existence
axioms, and even a few of them constitute together an axiom of infinity in
the first sense. Hence, they are not logically true. 
 
> My third point, that there does not seem to be any interesting open 
> question outside of set theory which is not equivalent to the validity 
> of a sentence of second-order logic with standard semantics, supports 
> the slogan "almost all math is just (second-order) logic".

Well, this takes us back to the old dispute on whether second-order logic
really is logic. I think not, but I would not like to open that worm can again. 



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
Finland

e-mail: panu.raatikainen at helsinki.fi

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