[FOM] On Godel's Enigmatic Footnote 48a
Alasdair Urquhart
urquhart at cs.toronto.edu
Wed Sep 1 14:14:30 EDT 2004
Jeffrey Ketland's conclusion that Goedel already knew in 1931 a lot
of the content of Tarski's famous truth paper of 1936 is,
I think, basically correct. One source for this is his
recollections as given to Hao Wang in 1976 ("Some Facts about Kurt
Goedel", JSL Vol. 46, 653- 659). The crucial passage here is on page
654. Goedel says that he discovered the incompleteness theorem as
follows. He set himself the task of proving analysis consistent, assuming
all of elementary number theory.
"He represented real numbers by formulas (or sentences) of number
theory and found he had to use the concept of truth for sentences
in number theory in order to verify the comprehension axiom for analysis.
He quickly ran into the paradoxes (in particular, the Liar and Richard's)
connected with truth and definability. He realized that truth in number
theory cannot be defined in number theory and therefore his plan
of proving the relative consistency of analysis did not work. He went on to
draw the conclusion that in suitably strong systems ... there are undecidable
propositions." (654)
It is interesting to notice here that Goedel started from the idea that the
notion of arithmetical truth is solid and reliable, but when he came to
publish the results, he replaced the notion of truth as far as possible by
finitary notions (hence the rather artificial notion of "omega consistency"
in the published paper).
It's often overlooked that there are really two papers on truth by Tarski.
The first was delivered in Warsaw, 21 March 1931. The second is that
published in German with a Postscript reflecting Tarski's reaction to
Goedel's incompleteness paper. The first paper says:
1. There is no exact definition of the notion of truth in ordinary
(semantically closed) languages, because of the Liar paradox;
2. In a simple type hierarchy truth for order n is definable in order n+1;
3. There is NO definition of truth for the whole typed language (because
transfinite types are not allowed).
The Postscript, reacting to Goedel, says very different things -- and this is
what most people remember as Tarski's results on truth. One of the things
that Tarski says is that he was too much under the spell of the theory of
semantical categories, and hence only considered languages of finite order.
It's only in the Postscript that he considers languages of transfinite order;
he explicitly refers to Goedel's famous footnote 48a in this connection.
Goedel was always very generous in giving credit to Tarski. As the recently
published volumes of his correspondence show, there was an extremely warm
friendship between the two logicians.
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