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Article 5984 of comp.ai.philosophy:
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>From: ramsay@unixg.ubc.ca (Keith Ramsay)
Subject: A reflection principle (was Re: penrose)
Message-ID: <1992May30.032735.22750@unixg.ubc.ca>
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References: <atten.706786286@groucho.phil.ruu.nl> <1992May29.053625.6202@sics.se> <1992May29.112749.14160@CSD-NewsHost.Stanford.EDU>
Date: Sat, 30 May 1992 03:27:35 GMT
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Torkel Franzen mentioned the reflection principle, by which one takes
a theory T and adds as a new axiom schema

           BewT(x) --> x

where BewT is provability in T. (To be more precise, in `BewT(x)' one
should have in place of `x' the number encoding x. I omit this
notation here and in what follows.)

It is true that there are consistent theories which become
inconsistent upon adding this axiom schema. For example, consider the
theory S obtained from Peano arithmetic by adding the (false) axiom
"Peano arithmetic is inconsistent" added. BewS(1=0) is a (false)
theorem in this system, and BewS(1=0) --> 1=0 yields an inconsistency.

Sound theories T, however, are a different story.

In article <1992May29.112749.14160@CSD-NewsHost.Stanford.EDU> 
costello@CS.Stanford.EDU (T Costello) writes:
|I am afraid I did not make myself quite clear.  Given a first order
|language, take Robinson arithmetic, or a system at least as strong.
|Take the set of axioms and the inference rules.  For the language
|construct a mapping from its symbols to numbers, and hence from
|numbers to equations.  Now consider the elementary predicate defined
|as follows.  It is true of the encoding of each axiom, if it is true
|of an implication and it first part it is true of its second.
|
|We add that this predicate is true of the encoding of the statement
|that any number it applies to is true.  This predicate exists, call it
|B. It of course already has a aritmetical name. A model of this can be
|demonstrated.

If I understand you correctly, you're adding axioms

      B( B(x) --> x),
      B( axiom ) for each axiom of Q,
      B(x) & B(x-->y) --> B(y),

to a theory Q like Robinson arithmetic. (I omit notation for encoding
statements as numbers- at little risk of confusion.) This does give a
sound theory, which I'll call T, if B is interpreted correctly.

|Now consider the reflection principle you suggested, that is the
|connecting of the predicate that states that the arithmetic coding of
|a proof of a statement implies a statement.  Our notion of proof is a
|proof from the axioms given above.  We use the same encoding scheme.
|We can write this as

|Bew x \supset x

Or in my notation, BewT(x) --> x.

|But we see from our definition of B above that we can derive that Bx
|\supset Bew x.

I don't see how.

|Thus by Montague's result we have an inconsistent system.  

I don't think this is true.

If we interpret B as "is provable in Q' + the axiom schema `x <-->
B(x)'", for a stronger system Q' than Q, then all the axioms appear to
hold true, including the reflection schema. But B(x) --> BewT(x) does
not, if we take x to be an arithmetic statement which is provable in
S', but not in T.

Followup directed to sci.logic.
-- 
Keith Ramsay
ramsay@raven.math.ubc.ca


