[FOM] How much of math is logic?

[Robbie Lindauer]

Neither is the successor function's existence demonstrated in FOL, so  
while the stronger "axiom of infinity" is not required for lots of  
things, the existence of a successor function is.

Similarly, the notion "is a member of" is not adequately defined in  
FOL, neither is the notion "powerset" nor is the notion  of a  
membership-minimal element.  Nor is the notion of a union, nor of a  
subset, nor of an unordered pair. ETC.

There is a REASON the axioms of ZFC are axioms, that is, because they  
can't be PROVED and in a way attempt to define what is meant by "is a  
member of" with regards to sets.

If you've got a PROOF of them, that'd be interesting to see.  Are you  
proposing that you have a proof of them?

A PROOF of the axioms of ZFC from FOL would be something indeed.

The reason the Logicist programme (fails, is failing, may fail) is  
that in order to strengthen FOL to give you "a proof of the axioms of  
ZFC" you end up with some equally strong (and from a certain point of  
view, equally odoriferous).

Robbie Lindauer


> 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"? This seems such a trivial quibble, it can't possibly be a
> serious objection to the logicist project.
>