Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Path: utzoo!mnetor!uunet!husc6!bloom-beacon!gatech!udel!burdvax!bpa!sjuvax!jserembu
From: jserembu@sjuvax.UUCP (J. Serembus)
Newsgroups: sci.math,sci.philosophy.tech
Subject: Re: G"odel made easy (Part I)
Message-ID: <891@sjuvax.UUCP>
Date: Fri, 2-Oct-87 09:57:19 EDT
Article-I.D.: sjuvax.891
Posted: Fri Oct  2 09:57:19 1987
Date-Received: Wed, 7-Oct-87 01:33:27 EDT
References: <362@prlb2.UUCP>
Reply-To: jserembu@sjuvax.UUCP (J. Serembus)
Organization: St. Joseph's University, Phila. PA.
Lines: 24
Keywords: G"odel's incompleteness theorem
Xref: mnetor sci.math:2196 sci.philosophy.tech:507

In article <362@prlb2.UUCP> ronse@prlb2.UUCP (Christian Ronse) writes:
>
So, given these axioms
>and rules of logical inference, one can derive statements called theorems.
>These theorems are of course true statements within that axiomatic system, but
>they do not necessarily coincide with all true statements, even not with all
>`provable' statements.


	This seems to me to be a fairly ambiguous and inaccurate way of
describing what takes place when one has an axiom system along with a rule
or rules of inference.  Theorems are those statements which are derivable
or provable from the axioms or other theorems by means of the rules of 
inference.  At this level they should not be referred to as true.
It is only when one is concerned with semantics that the notion of truth 
comes into play.  One of the major tasks of anyone working with a logic of
the kind in question is to try to determine if the set of all statements
flagged as `derivable` or `provable` by the deductive apparatus (syntax)
is coextensive with the set of all statements flagged as `true` or `valid`
by the semantics.  In essence, Goedel showed that they are not coextensive.
There are statements that fall into the latter set but not the former.  Or,
to put it naively,  there are "more" statements in the latter set than in 
the former one.  Or yet again, The former set is a proper subset of the 
latter one.