On 3/1/2013 7:05 AM, Bruno Marchal wrote:
I don't think that what you ask is possible, even if I am pretty sure that x + 0 = x, x + s(y) = s(x + y), etc.

I'm not at all sure that there is successor for every x.

Then you adopt ultrafinitism, and indeed comp does not make sense with such hypothesis, and UDA1-7 suggests that ultrafinitism might save physicalism, but step 8 put a doubt on this. The axiom that all natural numbers have a successor is used in basically all scientific paper though.

It is assumed, but I'm not sure it is used in an essential way. I recognize it difficult to do mathematics without it, but still it may be only a convenience.

OK. I don't see the problem with this. Convenience is a fuzzy notion. A brain too is convenient. Universes can be convenient. I am not sure to see your point.

In physics we sometimes get big numbers, like 10^88 or 10^120, but we never need 10^120 + 1. We make an axiom of succession and assume it applies to 10^120 like other numbers, but maybe that is because it easier than thinking of axioms to describe how we really calculate: 10^88 + 1 = 10^88. There's a book "Ad Infinitum" by Rotman that proposes something along these lines, but he writes like a French philosopher so I found it hard to tell whether his idea really works. But we do know that real computers work, and their mathematics are finite.


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