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Many years ago I ran into the following proof of Godel's first incompleteness theorem (here $T$ is an "appropriate" theory of arithmetic.):

First, we observe that Tarski's undefinability theorem can be slightly tweaked to say that for all $M\models T$, the theory of $M$ can't be the standard part of an $M$-(parameter-freely-)definable set. Next, by the usual representability arguments we have that every computable set is invariantly definable: if $X\subseteq\mathbb{N}$ is computable and $M\models T$ then $X$ is the standard part of a definable set in $M$. Now if $S$ were a computable satisfiable completion of $T$ we would get a contradiction by looking at $M\models S$. If we want, we can then replace "satisfiable" with "consistent" via the completeness theorem.

My question is:

  • Where did this argument appear?

I'm not interested in the question of whether this is a "genuinely different" argument; rather, I'm just interested in its presentation in terms of invariant definability. This was a notion first introduced by Kreisel following Godel/Herbrand and subsequently studied by others (see e.g. 1, 2, 3). My vivid recollection is that the argument above appeared as a footnote in the first paper mentioned above, but it turns out it's not there after all.

(A related question is when the variant of Tarski's theorem mentioned above first appeared explicitly. Of course the proof really is a trivial variation of the usual argument, but I'm still interested in when it was observed.)

Noah Schweber
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  • FWIW my personal take is that it is meaningfully different from the other arguments I'm familiar with, but for a very silly reason - see here. That said, it does also have a minor convenience advantage when we generalize it to higher logics appropriately (e.g. here by using its generalization I avoid needing to talk about the proof theory of infinitary logic). – Noah Schweber Mar 12 '20 at 15:10

1 Answers1

1

Based on the lack of reference provided here and at MO, as well as my own lack of success in finding a reference, I'm going to tentatively say that this particular approach has not appeared as such in print. I'll delete this answer of course if someone supplies a reference in which it appears, but I think at present it's appropriate to move this off the unanswered queue (I've made my answer CW since the line between answer and non-answer here is a bit blurry, so this doesn't feel quite right as a reputation-producing answer).

Noah Schweber
  • 245,398