Riff,
How graciously you conceed! We can call this point the "Ronald Regan"
proof. So that's clear: evolution doesn't favour intelligence, honour,
quality, truth, or anything else but survival, which is what makes it a kind
of tautology: what survives survives.
Now, as to Godel...
I really don't think your neat expression:
>>> "Any formal system of logic MAY be either complete OR consistent, but NOT
>>> BOTH."
can be a complete representation of Godel theorums about logic, mainly
because it's expressed in English, not formal logic.... You say:
> I had long assumed this was Hofstadter's translation of Godel's (sorry, I've
> misplaced my "umlaut". Ouch.) Second Theorem or infamous "Incompleteness
> Theorem". As I understand it, this was designed/intended to punch holes in
> Russel & Whitehead's "Principia Mathematica". Hofstadter described Godel as
> a Metalogician or "Metamathematician", and provided a brief but brilliantly
> accessible view of the spirit of Godel's work.
Well, from what I dimly understood of it, Godel's work is the very paradigm
of inaccessibility. So whatever Hofstadter made accessible, probably
wasn't Godel. An entry on Godel in Flew's Dictionary of Philosophy
describes the two theorums in outline, without the use of the formal logic
in which they are and must properly be expressed, and therefore has to jump
straight over the argument to the conclusions. Flew says that the first
theorum:
"...states that in any formal system S of arithmetic, there will be a
sentence P of the language S, such that if S is consistent, neither P nor
it's negation can be proved in S"
- This is called an "incompleteness" theorum because it states that not
every proposition can be proved: our *demonstrations* are "incomplete". I
don't think you have been using "incomplete" in this way, because you have
been talking about a system S which has propositions P for 'completely'
everything, which is just fine by Godel. In other words, you're worried
about the completness or incompleteness of the system, and how this would
impact on consistency, and Godel's worried about the completeness or
incompleteness of proofs within that system - two quite different worries.
The first theorum would allow that systems can be consistent and complete in
your sense of comprehensiveness, just not that they can be complete in the
Godelian sense of Proof, which has a quite specific application to
mathematical logic.
On the second theorum (a corrollary), Flew gives this precis:
"... the consistency of a formal system of arithmetic cannot be proved by a
means formalizable within that system"
Now this observation (based on detailed formal logic, and having application
to formal logic) is symultaneously the most revolutionary thing you can say,
or utterly unsurprising, depending on how you look at it. But in this
theorum, contrary to what you say in your attribution to Godel, it's not
specifically the consistency of the the system which is under threat, but
the provability of that consistency. You can't (as you seem to) interpret
Godel as saying that this system must be inconsistent, if it's
comprehensive.
One simple way of looking at this would be to say that the the system of
mathematics is thoroughly consistent, and can be proved, but that the way we
'prove' this is by sucessfully carrying on as we aways have done with apples
and pears. The system can be 'demonstrated' thus, in a sense, only not (and
this is Godel's point) by some nifty manipulation of algebra, which wouldn't
have any real meaning but for the existence of the apples and stuff.
Now all this is quite a different kettle of fish from the worries you and I
have about whether the attempt to rationalise everything can over-reach
itself.....
Or is it?
A puzzled elephant.
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