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Topic: Go Deep Blue!
Replies: 334   Last Post: Aug 22, 1997 11:23 AM

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 Matthew P Wiener Posts: 513 Registered: 12/8/04
Re: More on 0-1 Laws (was Re: 0-1 Laws
Posted: Jun 24, 1997 3:42 PM

In article <5omjn1\$l3e\$1@news.fas.harvard.edu>, kubo@abel.harvard.edu (Tal Kubo) writes:
>Matthew P Wiener <weemba@sagi.wistar.upenn.edu> wrote:

>>>>I guess we are assuming that as the X's come up, any inconsistent
>>>>ones are weeded out "with probability 1".

>>>"Guess" is exactly right. We're playing Name That Argument at this point.

>>Your juvenile sarcasm is as prevaricating as ever. I've only been
>>explicit on the above point several times.

>Repeating the words "weeded out" is not an explicit explanation.

Indeed not. What of it? Ilias seemed to be momentarily happy with the
terminology, and rather than congratulate him for his astuteness on such
a trivial matter, or even pass over it to help get the discussion itself
converging somewhere, you lied so as to score more sarcasm points.

Rather juvenile of you, is all.

>When a contradiction is reached, *which* axioms are removed?

The lineage dies out. It's evolution, OK? Not that complicated.

>Once a statement is removed, can it come back?

For sure. It's evolution, OK? Not that complicated.

>>>The assumptions appear to be the following:

>>> 1. The "alpha" theory A is a recursively axiomatized extension of PA.

>>A is "physical mathematics". You may think of it as PA. Whether it
>>is actually stronger or weaker is pretty much unimportant.

>0. Consider some minimal theory M (M = PA, say) that can carry out
> Goedel numbering and prove a certain 0-1 law about evolving theories.
> M is presumed insufficient for "physical mathematics". However,
> all definitions and relative consistency proofs discussed in the
> argument can be carried out in M.

I have no such M anywhere, although if you like to stick it in, be my
guest. It's not important.

>1a. The "alpha" theory A is any recursively axiomatized extension of M
> strong enough to be presumed sufficient for "physical mathematics".
> (Plugging in particular A's such as ZF we get statements supposedly
> useful for AI debates.)

The A is presumed just good enough for physical mathematics. Not ZF.
I've phrased things in terms of A only, allowing the possibility that
there may be a need for a little bit more--your M.

>1b. Natural selection "for" A is not modelled.

It's assumed. Being part of the model, I'd call that "modelled", myself.

> model built into the 0-1 law that M proves, forbids axioms of A
> from being removed (or removed with limit probability > 0) and M
> can prove this. This is why we get consistency assertions relative
> to A only: inconsistencies outside A are the only ones weeded out.

Now that your not trying to score silly sarcasm points, you suddenly
find "weeded out" sufficiently clear for the nonce. Golly.

>>> 2. The "omega" theory C is a definable, consistent, not necessarily
>>> r.e. extension of A. i.e., a definable set of sentences in the
>>> language of A.

>>Actually, I would expect physically reasonable models of evolution to
>>produce r.e. C's. Non-r.e. would mean that evolution is coding up the
>>halting problem or worse.

>Statements are being put in AND OUT of systems converging to C.
>Why should C be r.e., even extensionally?

Statements being put in and out? You mean, like in a priority argument?

Really, you would do better if you didn't try to think so hard about
something you obviously know just the vocabulary.

>>This does not follow. C is definable, ergo, its consistency is coded
>>by a sentence saying that there is no derivation of 0=1 from axioms
>>that meet C's definition.

>Sorry, I miswrote; a few lines down I gave a codeable sentence expressing
>Con(C). The intended point above was that any sentence taking the
>definition of C into account will not be a standard Goedel sentence, but
>something with several layers more quantifier alternation. (Which is
>also what prevents a standard Goedelian argument from being carried out.)

--
-Matthew P Wiener (weemba@sagi.wistar.upenn.edu)

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