That is very similar to the proof to which I refer. The claim is that in any consistent first-order theory T, of any kind, there can be no definable subset of the plane having every definable unary set as a section, even consistently. I gave a presentation of this argument in my seminar at Oxford in 2021, and a version of the argument appears in section 7 of my recent paper on Fregean abstraction (https://arxiv.org/pdf/2209.07845.pdf). But I also have a short paper on the three views of Tarski, where this is theorem 3. I’ll send you the paper.

But I see now that you are focussed on the sentential version, and so your argument is indeed different than the argument of mine to which I refer. I’ll take a look at your paper as soon as I can.

]]>Tarski’s theorem that Albert and I use in our paper is the sentential one, and does not refer to models at all, it is the spartan version going back to Tarski that says that if T is a theory in a language L that has the property that the function that maps each unary L-formula phi(x) to phi(#(phi)), where #(phi) is a code for phi, then there is no unary L-formula V(x) such that T proves equivalences of the form psi V(#(psi)) for all L-sentences psi.

As pointed out in the second paragraph of Remark 2 of the paper with Albert, a very special case of Theorem A+ of our paper shows that by using Rosser’s trick and Tarski’s theorem (in the above format) one can prove the first incompleteness theorem in the general form that says for all c.e. consistent extensions of the (very) weak arithmetical theory R (and in particular for all consistent c.e. extensions of Robinson’s Q) are incomplete.

In contrast to the usual proofs of the first incompleteness theorem, due to the bare bones machinery used in the proof, our proof of incompleteness is nonconstructive and does not yield an algorithm such that, given a description of the theory T (that extends R) as input, the algorithm outputs a sentence that is undecidable in T.

Perhaps this simple way of proving the first incompleteness theorem was noticed earlier; given the paramount stature of the incompleteness theorem and the large number of people who have written about it since the 1930s, one would expect for it to have been noticed before; but so far Albert and I don’t know of any. However, after completing our paper we were informed that our proof technique was anticipated by Emil Jeřábek to prove a related result in his MO answer in 2016, (URL: https://mathover

ow.net/q/257044). The latest version on arXiv does include the reference to Emil’s priority in formulating the proof method.

Thanks for this link—I’ll take a look. I also have been advocating this proof method, and have described it from a few years ago as my favorite proof of incompleteness (https://twitter.com/JDHamkins/status/1468677783505313795, unrolled here: https://threadreaderapp.com/thread/1468677783505313795.html). This proof is a chapter in my forthcoming book on incompleteness, which is the basis of this course. The point I emphasize is that one can prove Tarski’s theorem in an essentially Russellian manner, without any Gödelian self-reference, and this shows how truly very close to Gödel incompleteness Russell was.

Mel Fitting also has an account of this proof method in his article for the Smullyan volume, and he describes Smullyan as emphasizing that much of the fascination people express for Gödel’s theorem is better directed at Tarski’s theorem, and I think he is totally right. (But meanwhile, Mel goes for the sentential version of Tarski, instead of the satisfaction version, and for this reason he has to to internalize the syntax to the object theory. But a simpler proof is possible, avoiding all the internal syntax, if one goes for satisfaction without requiring substitution, as I explain in my book chapter.)

]]>That makes my story more plausible. Banach-Tarski showed that some of our naive intuitions had to go—because it was after both general relativity and quantum theory, either continuity of matter or isotropy of space were candidates and mathematicians chose to give up on continuity. But if they had worked with RVM more, perhaps if Ulam’s results had been found earlier, by the time the crisis came they would have had enough intuition that it was harmless and consistent that they would have kept it because it was so powerful (proving Con(ZF) and not-CH and much more besides).

]]>Dropping continuity seems like a rather large change.

]]>I’m not sure, but I shall find out and report back here about that.

]]>If general relativity had been developed before quantum mechanics, when “continuous matter” was still a possibility, people would have found the hypothesis of a real-valued measure on all subsets of the continuum plausible, and after working with it for decades, would have a strong intuition for it being consistent, so that the Banach-Tarski theorem would have led them to drop the requirement of rotation invariance rather than that of continuous space (general relativity is fine with anisotropy). Godel’s results on L would merely be considered to indicate that ZFC was too weak, and the power of RVM would have been even more appreciated because it implies Con(ZFC) by taking L(kappa) for weakly inaccessible kappa. All large cardinals below measurables, after Solovay’s work, would be regarded as unproblematic.

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