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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