[FOM] Recursion Theory and Goedel's theorems

[Peter Michael Gerdes]

On Aug 2, 2007, at 12:55 PM, Charles Silver wrote:

> 	I want to make a small point, which from my point of view is
> significant.  Take this statement of Gödel's theorem: IF PA is
> consistent, then there is a sentence G that is not provable, nor is  
> ~G.
> 	To me, all allegations that G can be "seen" to be true (without
> qualification) neglect the fact that PA must first be "seen" to be
> consistent.  The theorem itself doesn't refer to truth or falsity at
> all (by Gödel's intention).  So, one has to "see" or "interpret" G to
> be true.   As mentioned, though, this is impossible without first
> "seeing" or "interpreting" PA to be consistent.
> 	(This small point may be unimportant in the present context, but
> unalloyed statements that Gödel's G is true allow people like Lucas
> and Penrose to make all sorts of [what seem to me to be] erroneous
> claims. [See Putnam and Feferman])
>
> Charlie Silver

I don't know about Lucas but what Penrose seems to need is that we  
can *unerringly* determine truths like "PA is consistent." (or does  
he even need a non \delta^0_2 sentence? I don't remember)

So long as we only 'see' that PA is  consistent because of  
potentially fallible intuition and the empirical failure to find a  
proof of inconsistency so far this doesn't cause any problems.   
Sometimes we may guess that an inconsistent theory is really  
consistent (like Frege did with naive set theory) but the longer we  
go without a proof of contradiction and the stronger our intuition  
that such a system must be consistent the less likely we are mistaken.

In short we make statements like "'apples fall' is true" all the time  
about physical facts that we don't know for certain but are just very  
very sure about.  I see no problem doing the same for mathematical  
statements like 'PA is consistent.'

Peter

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