[FOM] Re: On Gödel's Enigmatic Footnote 48a

[Jeffrey Ketland]

Dear Panu,

>> In the Postscript, Tarski wrote,
>>
>>          The definition of truth allows the consistency of a deductive
>>          science to be proved on the basis of a metatheory which is
>>          of higher order than the theory itself.
>
>Which is, as I always like to remind, false. ACA_0 can give an adequate
>truth definition for PA, but can't prove Cons(PA)...

Agreed. ACA_0 is materially adequate (i.e., there is an L_2 formula True(x)
such that ACA_0 proves all instances of True("A")<->A, with A arithmetic)
but still conservative over PA. However, note that ACA_0 should in any case
count as a defective theory of arithmetic truth in that it does not prove
the inductive clause for negations,

      forall x in Sent(L_1), neg(x) is true iff x is not true

(informally, "truth commutes with negation")

And similarly ACA_0 does not prove the other Tarski inductive clauses.

Moving from ACA_0 to ACA---i.e., induction scheme is now extended to all L_2
formulas---permits the proof of these clauses. Then ACA is equivalent to the
truth theory Tr(PA) which does prove Con(PA) and the reflection principles
for PA. Relative to the aim of having a good theory of arithmetic truth, ACA
is good and ACA_0 is bad. The situation is similar with NBG over ZF.

Overall, all this appears to strengthen the point that I keep going on
about, that the property of material adequacy alone is insufficient for a
good theory of truth. A good theory of truth should explain why accepting a
theory S commits you also to "S is true" and the resulting reflection
principles. By definition, a conservative (i.e., deflationary) theory cannot
do this. This is why disquotational truth theories are defective.

I agree that Tarski was somewhat unclear about this, because he seems to
suggest in the Postscript to the 1935/6 paper that material adequacy alone
is enough for "the consistency of a deductive science to be proved".
However, this is a minor slip, and the point about restricting the induction
scheme, so that it does not apply to higher-order concepts, is a more recent
observation.

Best wishes --- Jeff

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Jeffrey Ketland
School of Philosophy, Psychology and Language Sciences
University of Edinburgh, David Hume Tower
George Square, Edinburgh  EH8 9JX, United Kingdom
jeffrey.ketland at ed.ac.uk
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~