[FOM] How much of math is logic?

[joeshipman@aol.com]

Heck (excerpted from a much longer and more interesting post):
>If you approach this via set theory, then, with the logic again being
<(either full or \Pi^1_1) second-order logic, the weak axiom of infinity
>can take the form of null set, adjunction, and extensionality, as has
>already been mentioned. Adding a strong axiom of infinity to this set 
of
>axioms doesn't give a very strong set theory, even in full second-order
>logic, since you don't have power set.

I don't really think this "weak axiom of infinity"  is an axiom of 
infinity in an ontologically problematic sense. You never have to treat 
an infinite object as a definite mathematical entity, it is only in a 
metatheory that you can say there are infinitely many objects. I am 
perfectly comfortable with null set and adjunction and extensionality 
being regarded as "logical" notions (especially since extensionality 
can be treated as a definition rather than an axiom, which simply 
allows you to call things equal rather than always having to say that 
they have the same members).

Your point about a strong axiom of infinity not adding much strength in 
the absence of a power-set axiom is the kind of thing I was driving at. 
If you deny the strong axiom of infinity AxInf, then all the other 
axioms of ZFC are true in the hereditarily finite sets (and can be 
proven from Extensionality, Emptyset, Adjunction, and ~AxInf). But they 
do not necessarily seem as "logical" and subject-matter-free as 
Emptyset and Adjunction. I think you can make a good case for all of 
the axioms of ZFC-Inf in a weak second-order logic, and that the 
ontological commitment to the actual infinite is really the only way in 
which mathematics essentially transcends logic. But I'd like to see 
some contrary arguments, which would necessarily entail statements of 
the form "ZFC theorem X is not a logical consequence of the existence 
of an actually infinite set".

I would imagine the PowerSet and Replacement axioms are the ones that 
would be the hardest to defend as "logical" (I am ignoring the 
Foundation axiom because it is not actually used in ordinary 
mathematics). However, all the arguments I can think of against their 
being "logical" assume an infinite set. I don't see how someone who 
denies AxInf could have a problem with treating PowerSet and 
Replacement as rules of logic.

-- JS

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