Re: "God of the Gaps"

Steve Anonsen/GPS (Steve_Anonsen/GPS.GPS@gps.com)
14 Mar 96 17:44:36 EDT

Allan Harvey wrote:
[ secondary quote is from a mathematician friend of Allan's ]
> >Suppose you have
>>any collection of axioms that include the axioms that postulate the
>>existence of the natural numbers. Then there are statements about the
>>natural numbers that can be neither proved false nor true with these
>>axioms.
>
>There would not seem to be any way to get from this the existence of things
>you "know" to be true but can't prove. In fact, in the context of
>mathematics and logic, the concept of "knowing" something to be true without
>rigorously proving it seems entirely meaningless.
>
The key is that it can't be proven _with the axioms of that system_; if you are
somehow outside that system, you may be able to decide the theorem (by having
additional axioms, for example). But, there is no formal system with the
axiomatic strength of the natural numbers that is complete in the sense that
all theorems are decidable _within that system_ -- I believe that is Godel's
theorem.

> Glenn wrote:
>>OK, let me explain this. First, Goedel's theorem says that within any logical
>>system, there are statements that you can KNOW to be true but you can't prove
>>to be true. There are technical limitations on the type of logical system.
>>Goedel's theorem does not apply to arithmetic with only +/- operations, (See
>>Tipler, Physics of Immortality p 193 (I think, my book is at home and this is
>>from memory)).
>
>Well, maybe we need to find a mathematician on the list to referee (since
>Glenn didn't like the opinion of the one I consulted). But I'm pretty sure
>the statement of Godel's theorem above is just plain wrong. Let me repeat
>my friend's version:
[ snipped, as it is stated above ]

I believe Glenn is agreeing with Allan's mathematician friend here about the
strength of the formal system: it must have axioms that postulate the existence
of the natural numbers and the operators thereupon.

However, with Allan I'm having trouble understanding how the undecidability of
theorems in formal systems bears on the question of SETI. Penrose has used
Godel to postulate that the current type of computing systems will never
reproduce human thought. That makes some sense to me, but I'm not seeing the
connection of Godel and SETI. Could you please elaborate Glenn?

Steve Anonsen
Erstwhile but no longer math student