From: Lee Daniel Crocker (lcrocker@mercury.colossus.net)
Date: Mon Oct 13 1997 - 11:00:27 MDT
>> Geoff Smith:
>> Does G=F6del prove that mathematics is infinitely complex, or just
>> incomprehensibly so?
>> Anders Sandberg:
>> I think infinitely complex is the likely choice. I'm not sure I can
>> prove it (metamathematics is not my strong suit), but given the
>> results by Chaitin about the random and non-algorithmic structures
>> in the natural numbers it seems that the complexity of mathematics
>> is infinite, not just incomprehensible.
>
> I suspect the actual answer is "both". The kinds of propositions which
> meet Godel's criterion are lengthy, involuted and not the kind of thing
> human minds find easy to process. Note that the use of computers to help
> mathematical proofs have already produced very lengthy complex proofs
> which require teams of mathematicians to verify.
The only objective meaning of "infinitely complex" I can think of is
axiomatically true: "mathematics" is simply "the set theorems expressible
in mathematical terms". The cardinality of that set is that same as the
cardinality of largest set of symbols in the language. We can never
actually write down more than Aleph-null theorems (the number of integers),
but since mathematics includes real numbers, of which there are Aleph-one,
the cardinality of mathematical theorems is at least that.
Godel really has nothing to do with complexity at all. It is nothing more
than the proof that of that infinite number of mathematical theorems,
the set of "true" theorems and the set of "provable" ones differ.
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