TITLE: The ultrafinitary equivalent of the Peano axioms?
QUESTION [5 upvotes]: What are the axioms for ultrafinitary number theory? I have in the mind the school of thought that there is a largest number -- the width of the known universe as a multiple of the diameter of a hydrogen atom, or something like that. 
I have been playing around with finite chains of elements with a beginning and an end, and trying to come up with something the equivalent of induction.
EDIT:
Brainstorming discussion at my thread, "The axioms for ultrafinitist number theory?" at the sci.math and sci.logic newsgroups. 
See my tentative suggestions in my Follow-up below.

REPLY [4 votes]: There is no firmly established set of axioms for ultrafinitary number theory. This is one of the main problems for the field. For example, intuitionism was originally viewed as confusing and mostly incomprehensible (and, to some extent, the program pursued by Brouwer did have those properties). But once good formalizations of intuitionistic logic were developed, so that it was more clear what was going on, things became much more understandable. Now the study of intuitionistic logic is a mainstream topic in mathematical logic, because it has important applications even to classical mathematics. 
It is not clear, yet, whether any similarly successful formalization of ultrafinitism will be found.  Part of the reason for this is methodological. In normal logic, we can work with a metatheory that is simply finitary, rather than ultrafinitary (see below). But if someone really wants to have an ultrafinitary system of number theory, presumably they also want an ultrafinitary metatheory. This means that they can no longer talk about proofs of arbitrary finite length, for example.
There are some additional comments on this at https://mathoverflow.net/questions/44208/is-there-any-formal-foundation-to-ultrafinitism
An important clarification: "finitary" number theory is usually used to mean systems such as primitive recursive arithmetic (PRA), in which the axioms are all in particularly concrete forms. Systems such as PRA still have infinitely many natural numbers, they simply limit the ability to use "infinitary" methods on them. PRA is the standard finitary system used in proof theory. "Ultrafinitary" number theory refers to systems in which the set of "natural numbers" is, in some sense, finite.