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From: minsky@media.mit.edu (Marvin Minsky)
Subject: Re: On the discovery of formal systems (Was: On the possibility of enumerating all
Message-ID: <1995Aug11.020854.8099@media.mit.edu>
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Cc: minsky
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References: <40b48s$1f0@nntp5.u.washington.edu> <1995Aug9.232702.1506@media.mit.edu> <40dnrb$6pb@nntp5.u.washington.edu>
Date: Fri, 11 Aug 1995 02:08:54 GMT
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Xref: glinda.oz.cs.cmu.edu sci.logic:13890 comp.ai.philosophy:31552

In article <40dnrb$6pb@nntp5.u.washington.edu> forbis@cac.washington.edu (Gary Forbis ) writes:
>In article <1995Aug9.232702.1506@media.mit.edu>, minsky@media.mit.edu (Marvin Minsky) writes:
>|> No need to cheat.  All you need to do is enumerate the integers in
>|> binary notation.  Then, for example, the 10th axiom set will be
>|> represented by 1010 -- and you interpret this as 
>|>   
>|>   Axiom 4 = Yes
>|>   Axiom 3 = No
>|>   Axiom 2 = Yes
>|>   Axiom 1 = No
>|> 
>|> This will enumerate all the finite sets of axioms in a nice order, and
>|> no set will be covered twice (which would not matter anyway).
>
>This certainly enumerates all the finite sets of axioms but unless which 
>axioms are which is known, I'm willing to bet that any particular formal
>system is not one of the first n formal systems listed (n being some finite
>integer.)

A family of formal systems must operate on subsets of some set master
set of well-formed formulas.  It should be easy to enumerate these in
some lexicographic form, as simple strings of symbols.  an easy way to
do all this at once is to enumerate all strings in base 5 notation
where any substring of 1s and 0s designates a symbol, the digit 2
means end of symbol, the digit 3 means end of axiom, and the digit 4
means end of set of axioms.  This enumerates all finite sets of finite
axioms, and there's no problem at all of seeing which axioms are
which.

Perhaps you should read a good text on recursive function theory.
It's grand fun.  Much better than Westerns or Detective Stories, in my
view.  The plots are deeper, and the characters more interesting.  If
you can find a copy of my out-of-print book, Computation, there's a
lot about this sort of thing in the chapter on Post Normal Systems.



