[FOM] Buckner 3?

[Harvey Friedman]

Reply to Buckner 5:17PM  5/17/03.

>Harvey:
>>There appears to be a minority of philosophers of mathematics and
>>philosophical logicians who properly understand the issues and
>>nonissues regarding first order/second order axiomatizations for
>>f.o.m.
>
>I'm with the majority.

I'm sorry but I don't have the time to help you with this issue 
unless I see some very clear explicit way to do so. I didn't see any 
such clear opportunity for doing this by reading your posting.

Furthermore, there are probably a significant number of subscribers 
on the FOM email list whose less than complete understanding of these 
issues would be much easier to address.

>But isn't the difficulty that there is something in
>between, and we have failed to recognise this?

There is no difficulty, as there is a huge literature on various 
"logics" in between first and second order "logic". If you are 
familiar with this literature, then the efficiency of communication 
would greatly increase.
>
>
>I think Harvey's idea is that we can get from plural logic to set
>theoretical logic by means of a translation or interpretation from one
>symbolic language to another.  Let me repeat that is impossible.

Sometimes I have more than one idea, although I try to consolidate 
them into a very few.

I can assure you that what I have been telling you is not impossible. 
(Of course, I make errors in details from time to time). Perhaps you 
have some difficulties in understanding the notion of interpretation.

In particular, in my posting Buckner 1,2, I claimed that the 
following interpretations exist:

1. From ACA0 into PA'.
2. From WKL into PA.
3. From PA into COA (count arithmetic).
4. From COA into PA.
5. From WKL into COA.

Also related conservative extension results:

5 follows from 2,3.

ACA0 supports a great deal of real analysis directly, and WKL 
somewhat less but still a great deal. COA avoids induction in favor 
of (the results of) counting.

Buckner 1 accepts PA and PA'. Buckner 2 accepts COA.

If you doubt whether these interpretations exist (1-5), then you may 
be having trouble with the standard concept of "interpretation" 
(unless I have made some sort of technical error in 1-5).

I anticipated a Buckner 3 that would not accept COA.

For example, Buckner 3 might complain that natural numbers do not 
exist, so that direct quantification over natural numbers is not 
acceptable.

There is no clear explicit formulation of Buckner 3 in your posting.

Of course, there was no clear explicit formulation of Buckner 1,2 in 
your earlier postings, either. But I could see how to make Buckner 
1,2 clear and explicit rather easily.

>... Plural logic has to be stronger than PA, since it
>supports predicates and allows us to quantify over any finite set
>imaginable.  Surely that is strong enough for decent mathematics?
>
>

I would like to confirm with you, your statement that "plural logic 
is stronger than PA". In PA, we quantify over all natural numbers, 
and there are no sets at all.

PA is obviously not strong enough for any direct treatment of real 
analysis, because real numbers cannot be directly treated in any 
reasonably direct way.

However, WKL can be interpreted in PA,COA, and WKL does support a 
direct treatment of a substantial amount of real analysis. And WKL is 
interpretable in PA,COA.

By thinking through what an interpretation is, you will clearly see 
the great power of f.o.m. in clarifying and addressing a huge range 
of issues including the ones that you raise. Do you agree with this?