Thanks for the description of the axiom of choice. I vaguely recollect that is about the way Prof. Grimm (his real name) explained it to us. I didn't remember, however, that it is a theorem if the collection is countable. I am still somewhat confused as to how it applies to my original post.
You offered a "Proof that the number of integers is infinite", but you didn't say what system of integers you are assuming. If you assume the commonly accepted system of integers, then, of course there is an infinite number of integers.
But, if you assume a largest integer, such as in my proposed system of Practical Numbers, then your proof fails at step 2,"Add 1 to the largest, you get a larger one, contradiction." You cannot add 1 to the largest integer because the sum is not defined. So, rather than "get[ting] a larger one", you don't get a number at all. There is no contradiction.
I realize that my proposed Practical Number System is not a Field or a Number System in the strict sense that those structures are defined in mathematics. To wit, not all pairs of Practical Numbers have defined sums or products. But "for all Practical purposes", they do. We simply have to avoid conditions that would cause "overflow". So, I do not consider failing to conform to the rules of formal Number Systems or Fields to be a serious loss.
With your help in clearing the old cobwebs in my brain, I think I can recall the gist of Prof. Grimm's lecture. The proof of the "axiom of choice" for countable sets involves some kind of procedure, like Cantor's diagonal procedure or something else, that actually constructs the choice set by automatically defining integers successively and sticking them into the choice set.
It is the assumption of this automatic procedure which can run ad-infinitum without conscious attention that has never sat well with me. I may have erred in my use of the axiom of choice in explaining my proposal, but I still stand behind what I said.
Since I am proposing a different set of assumptions for the foundations of my proposal, it is not a matter of proving or disproving what I am proposing. The only question is whether or not a fruitful system of mathematics could result from these assumptions.