[FOM] Did G?del's result come as a surprise to Bertrand Russell?

Alasdair Urquhart urquhart at cs.toronto.edu
Sat Apr 3 10:07:37 EDT 2010

On Wed, 31 Mar 2010, hdeutsch at ilstu.edu wrote:

> Fascinating! Did Goedel actually regard his results as paradoxical.
> What is the evidence of this?  As I understand it, Carnap didn't
> understand the result at first, but was the first to produce an
> explicit proof of the diagonal lemma. Is that true?

I am sorry that I expressed myself unclearly here.  What I meant to say
was: "Goedel thought that Russell regarded the incompleteness
theorem as some kind of new paradox."  I believe that this is what
Goedel said in coversation with Hao Wang.

Of course Goedel did not consider his results to be paradoxical.
This is because he distinguished sharply between truth and
formal provability -- this distinction of course is the
heart of the incompleteness theorems.  But I would argue
that Russell never properly understood this distinction.
Look at how Principia Mathematica expresses the rule of
modus ponens, for example:

 	*1.1  Anything implied by a true elementary proposition is true.  Pp.

Whitehead and Russell use the notion of truth freely in expressing
their primitive propositions, and do not clearly demarcate it from
formal provability (in fact, PM has no clear concept of formal
provability at all).  So, if we assume that Russell continued in old age 
not to distinguish the two concepts, Goedel's results WOULD appear 
paradoxical to him.

I think we tend to underestimate nowadays how difficult it was for
Goedel's contemporaries to understand the nature of the advances
he had made.  Look at the correspondence between Goedel
and Zermelo -- you see many misunderstandings on Zermelo's part.

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