Re: On Ultrafinitism


MoeBlee wrote:
Rupert wrote:
Edward Nelson, on the other hand, has done
some interesting research about certain weak axiomatic theories in
arithmetic, which may embody his stance. See his book "Predicative
Arithmetic".

Thanks. I just downloaded it for free as a PDF file. And the first
chapter of his unfinished book on IST too. Do you have any other
recommendations?


I'm afraid I don't know all that much about it. There's a famous essay
by Yessenin-Volpin called "The Ultraintuitionistic Criticism and the
Antitraditional Program for Foundations of Mathematics", in
"Intuitionism and Proof Theory, Proceedings of the Conference at
Buffalo 1968, North-Holland, Amsterdam, 1970." Some people like that
essay but it doesn't have enough of what I can recognize as
mathematical content for my taste. He announced that he had a
consistency proof for ZF with any finite number of inaccessible
cardinals, but apparently that proof is hard to get hold of, which is a
shame.

These might be interesting:

http://www.springerlink.com/content/v76473730365861x/

http://www.turpion.org/php/paper.phtml?journal_id=rm&paper_id=2870

http://math.ucsd.edu/~sbuss/ResearchWeb/nelson/paper.pdf

I can't actually find the one I was looking for, which talked about
"Nelson's Program".

What do you think of the system Shaughan Lavine gives in his
'Understanding The Infinite'?

I haven't seen it.

(I've read some high praise for this
book.) I only saw the book briefly, so I didn't have a chance to ponder
his system. But ab initio in my thinking about this, I can't imagine
how an ultrafinitist can give a truly rigorous axioimatization in which
there is largest natural and block us from adding 1 to that largest
natural (the paradox of the heap, as it were).

I don't think they would go about it that way. They could use a very
weak arithmetic in which only functions of very low computational
complexity could be proved total. Then only fairly small numbers could
be feasibly defined. They could even add an axiom saying that e.g. the
exponential function is not total, which would give rise to a theory
with nonstandard models. Nelson developed an alternative foundation for
probability theory using such an arithmetic.

But, of course, I need
to study the systems to see for myself, since otherwise my questions
are uninformed.



And, aside from that one chapter by Nelson, do you know of a good
explanation (rigorous but not so difficult that it starts with advanced
concepts).of axioms for IST?

Sorry, no.

I saw a book by Alain Robert but it's not
what I'd like; it's more of sketchbook of ideas than it is a tight
theorem by theorem treatment. And some of the other books I found in
the QA299 section of the library jump right into more advanced stuff
way too fast for me.

MoeBlee

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