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From: jlg@lanl.ARPA
Newsgroups: net.physics
Subject: Re: predicting the universe with computers
Message-ID: <19974@lanl.ARPA>
Date: Mon, 21-Jan-85 16:19:30 EST
Article-I.D.: lanl.19974
Posted: Mon Jan 21 16:19:30 1985
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Organization: Los Alamos National Laboratory
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>      The halting problem states that no Turing machine can be built
> that decides whether some other arbitrary Turing machine will halt.
> But there is no rule against predicting what a Turing machine will do
> for the next little while.  The halting problem is more of a comment on
> the inadequacy of our logic systems along the lines of Russell's
> Paradox, than it is a statement about the universe.
> 
>      Turing machines also have the property that they have an infinite
> tape.  This is the practical problem that one encounters when trying to
> build a halt detector.  But it has not been shown that the universe is
> infinite.  In fact I believe that Einstein proved that the universe is
> finite.


The halting problem is less like Russell's Paradox than it is like Godel's
Theroem.  It doesn't require the infinite tape of the turing machine, but
only that some state variable have unbounded range (I learned a proof of
it with only one such variable in an otherwise finite machine with finite
input).  However, a corollary of the halting problem does imply that no
machine can predict its own behaviour, much less the behaviour of anything
more complex than itself (even without the unbounded variable).

Einstein not only DIDN'T prove the universe finite, he didn't believe it.
For a long time Einstein supported the steady-state theory (more because he
was a relativist and didn't want to admit that there were special reference
times than because he was attached to an infinite duration universe).  He
even introduced a so-called cosmological constant into his equations for
general relativity in order to prevent it from predicting that the universe
would expand or contract.  When the evidence came that the universe was
expanding, Einstein retracted the cosmological constant - calling it the
worst error he had ever made (to introduce an idea merely to support his
own preconceived beliefs).  Still, the universe could be finite or infinite
since general relativity doesn't exclude either possibility.