[FOM] How much of math is logic?
joeshipman@aol.com
joeshipman at aol.com
Tue Feb 27 15:25:07 EST 2007
-----Original Message-----
From: tchow at alum.mit.edu
**************
Shipman:
>> 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?
Chow:
>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.
**********
I would elucidate it, except that it has already been done by Russell
and others.
********
Shipman:
>> 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.
Chow:
>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.
***********
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.
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.
-- JS
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