[FOM] Bucknerism = Nominalism with Plural Quantification?

[Harvey Friedman]

Reply to Buckner 7:25PM 5/1/03.

>
>My position roughly this.  I have a pretty well finished story about
>reference, predication and quantification.  I would like to publish this one
>day (even quite soon) but spotted about a year ago that the underlying idea
>seemed to contradict the assumptions underlying ZFC.  To put it crudely, if
>ordinary mathematics is not possible without ZFC, we must assume ZFC is
>necessary (since, clearly, ordinary mathematics is possible).  So I have an
>interest in seeing whether ordinary mathematics is possible using a more
>constrained set of assumptions, otherwise about 20 years of very painstaking
>work, a life's work indeed, is ruined.  This is why for the last year I have
>been trying to understand set theory.

The good news is that very weak fragments of ZFC suffice for a 
tremendous amount of mathematics. And we have a good science of just 
what fragments are needed for what.

However, it is not yet clear that that science quite addresses some 
of your concerns. In particular, the most well studied fragments of 
ZFC all have variables ranging over infinite sets of natural numbers, 
and some set existence axioms that allow you to "construct" infinite 
sets of natural numbers.

However, an awful lot of mathematics can be done in fragments of ZFC 
in which every set is finite. But a lot of mathematics has to be 
seriously modified in order to be done in such a fragment. Results 
suggest that some mathematics can never be modified in any reasonable 
way in order to be done in such a fragment.

It is necessary for me to get a really clear picture of what is and 
what is not allowed in order to evaluate the effect of your position 
on f.o.m. General philosophical descriptions and descriptions from 
the philosophy of language, alone, are not enough for me.
.
>
>
>>  Which of the following assertions are meaningful?
>>
>>  1. for all integers x,y,z > 1, x^3 + y^3 not= z^3.
>
>for ANY integer x,y,z > 1, x^3 + y^3 not= z^3.
>
>>  2. for all integers x there exists a prime p > x.
>
>for ANY integer x there exists a prime p > x.
>
>>  3. for all integers x, there are some primes greater than x.
>
>>  4. There are some integers > 1 such that every integer > 1 is
>>  divisible by at least one of them.
>
>Assume "some integers" means "some finite bunch of integers", rather than
>"some kinds of integers", where a kind of integer is specified by something
>like "- is an even number greater than 103" or something like that.  I
>assume the former is meant.

Since you are satisfied by 2, we don't have to talk about 3.

With 4, I was deliberately trying to bring in the plural 
quantification "there are some integers > 1 such that ..." where this 
allows or forces one into infinite many.

I thought your idea might be that this sort of plural quantification 
is OK, but "there is a set of integers > 1 such that ..." is not.

I should point out that there cannot be a finite set of integers > 1 
such that every integer > 1 is divisible by at least one of them, 
simply because there are infinitely many primes.
>
>
>And how about "any integer > 0 is the product of some unique set of primes".
>Certainly meaningful and, I believe, provable using a minimal set of axioms
>+ plural quantification.  (My homework for the May bank holiday).

How does plural quantification come in here, according to your view? 
Any very elementary book on number theory has a treatment of the 
fundamental theorem of arithmetic, not paying any attention to plural 
quantification. Is your idea that you want finite sets of integers, 
in a treatment of the fundamental theorem of arithmetic, replaced by 
plural quantification over the integers?  Is your position that you 
accept only what might be called "finite plural quantification"?

It does appear that you are accepting quantification over natural 
numbers (not quantfication over sets of natural numbers) without 
reservation, given your answer to 2. Please confirm this. I.e., you 
accept "for all natural numbers x,...", and "there exists natural 
numbers x,...", but not "for all sets of natural numbers S,...", and 
not "there exist sets of natural numbers S,..." You seem to accept 
"for all finite sets of natural numbers S,..." and "there exists 
finite sets of natural numbers S,...", but only if it is rephrased so 
as to not mention a set.  ???