From: Tomaz Kristan (tomaz.kristan@gmail.com)
Date: Wed Aug 18 2004 - 12:16:22 MDT
Just a side note.
Goedel's incompleteness asserts, that you will sooner or later bump to
an unprovable theorem inside the infinite arithmetics.
>From there, you may add this unprovable theorem to the system as a new
axiom, or you may equally add it's negation to the system as a new
axiom. Not both, however.
Either way, you'll have a new, extended system, which will eventually
bump to a new unprovable theorem. And so on, ad infinitum.
The chance is, however, that already this infinite arithmetics as we
know it today, is paradoxical. This possibility is encapsulated inside
Goedel's theorem.
Staying infinity atheist on the other hand, makes those dilemmas
irrelevant. The question, whether or not every Goodstein's sequence
stops, may be undecidable. In any case, there are only finite number
of cases to worry about.
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