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From: minsky@media.mit.edu (Marvin Minsky)
Subject: Re: On the possibility of enumerating all formal systems.
Message-ID: <1995Aug8.213111.25163@media.mit.edu>
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Date: Tue, 8 Aug 1995 21:31:11 GMT
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Xref: glinda.oz.cs.cmu.edu sci.logic:13770 comp.ai.philosophy:31452

In article <408dkt$khk@nntp5.u.washington.edu> forbis@cac.washington.edu (Gary Forbis ) writes:

>I have seen the suggestion in several discussions that there is a
formal way to enumerate all possible formal systems.  Some seem to claim the
formalization is trivial.  I'd like to explore this claim.

It is trivial, because the term "enunerate" means a countably infinite
listing.  One way would be to enumerate all finite sets of Post
Productions.  Diagonalization is not a problem, because mathematicians
agree that a formal system is defined to have only a finite number of axioms.

There are formalisms that allow the use of "axiom schemas" each of
which generates an infinite number of axioms.  I suppose such a system
isn't considered to be a genuine formal system unless this is just a
matter of notational convenience, of the form of abbreviations of
expressions from a higher order
logic that has only a finite number of axioms.

I presume that it is impossible to effectively enumerate all
*consistent* formal systems. Can't think of an easy proof of this,
though.

[...]

>Is there a reason to believe there are at most a finite number of possible
>axioms and thereby a finite number of formal systems?

 Of course there are an infinite number of possible axioms,
such as "2 is even", "4 is even", etc.


