[FOM] How much of math is logic?

Timothy Y. Chow tchow at alum.mit.edu
Tue Feb 27 09:50:02 EST 2007

Joe Shipman wrote:
> Responding to Raatikainen:
> >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)
> This is a silly objection, the point is that the mapping **can be 
> seen** to connect arithmetical statements to logical ones in a 
> truth-preserving way. What do you think Russell thought he was doing?

Why is it silly?  It seems to me that the burden is on you to elucidate 
what sort of things "can be seen" in this context.  Otherwise one can 
"sneak in" arbitrarily strong non-logical assumptions (that nevertheless 
"can be seen" to be true in some sense) and claim that all kinds of 
non-logical assertions "can be seen" to be "equivalent" to logical ones.

> This is exactly what I was trying to say -- I was careful to 
> distinguish the ZFC axiom of Infinity as something that was NOT 
> "logical", in order to argue that those parts of math which do not need 
> it CAN be thought of as simply logical.

It is *not* exactly what you were trying to say; you're talking about the 
second type of axiom of infinity, and he's talking about the first.


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