From newshub.ccs.yorku.ca!torn!cs.utexas.edu!zaphod.mps.ohio-state.edu!rpi!scott.skidmore.edu!psinntp!psinntp!scylla!daryl Sat Oct 24 20:44:26 EDT 1992
Article 7335 of comp.ai.philosophy:
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>From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Human intelligence vs. Machine intelligence
Message-ID: <1992Oct19.025334.8881@oracorp.com>
Organization: ORA Corporation
Date: Mon, 19 Oct 1992 02:53:34 GMT
Lines: 62

In article <Bw6qFJ.IoD@cs.bham.ac.uk>, Aaron Sloman writes:

>(ii) If, as Go"del proved, G and not-G are both not derivable in T
>(if T is consistent), then if T is consistent then so are both
>    T + G,
>and
>    T+ not-G.
>That means both will have models. That means that in some models
>of T G will be true and in some models G will be false, if T is
>consistent.
>
>This seems to me to undermine anyone's claim to be able to *know*
>that G is true, whether it is Penrose (who says he does and
>computing machines can't) or his critics (some of whom say they do
>and computing machines can also).

It is true that both T + G and T + not-G will be consistent, and
therefore, they both will have models. However, a model of
T + not-G will not be the natural numbers.

How can I say that? Because the natural numbers are defined to be the
*smallest* set that obeys the axioms of Peano Arithmetic. In other words,
the set of natural numbers N has the property that:

    1. N is a model of PA.
    2. For any proper subset N' of N, N' is *not* a model of PA.

If you take away a single number from the set of natural numbers, and
leave the operations + and * alone, then the result will not obey the
axioms of PA.

Nonstandard models of PA do not have this property. The numbers of
nonstandard models look something like this:

0, 1, 2, ... a - 3, a - 2, a - 1, a, a + 1, a + 2, ...

That is, there are an *infinite* number of terms that are greater than
0 and smaller than a (where a is some hyperfinite number). A set like
this might obey all the axioms of PA, but it is clearly not the
smallest set that obeys the axioms, since the set of finite numbers
0, 1, ... is a subset which obeys PA.

Now, let's turn to the question of whether G or not-G is true of the
natural numbers. G is sentence of the form: "There does not exist a
number which codes the proof of such and such". (Where, by clever
coding, such and such turns out to be G itself.) The fact that G is
independent of PA means that we can construct models of PA with G
true, and we can also construct models of PA with G false. However,
not-G says "There *does* exist a number which codes the proof of such
and such". Therefore, any model of not-G will have to throw in extra
numbers, namely a number coding the proof of such and such. On the
contrary, a model of G will not have to throw in such numbers, since G
says that such numbers don't exist. Therefore, for any model of not-G,
there is a *smaller* model of G (which can be constructed by throwing
away the extra numbers that are supposed to code proofs of such and
such). Therefore, while G is true in some models of PA, and not-G is
true in other models of PA, only G can be true in the smallest model of
PA (which is the so-called standard model).

Daryl McCullough
ORA Corp.
Ithaca, NY


