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nomadreid
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- TL;DR
- Does Sergei Artemov's version of consistency (references given) which allows PA to prove its own consistency lead anywhere?
I have come across several preprints by Prof. Sergei Artemov, for example
https://arxiv.org/abs/2508.20346
https://arxiv.org/abs/2403.12272v1
(If it has appeared in a peer-reviewed journal yet, I have not found it.)
I have questions about the general approach.
First, my main question: whether my general understanding of what he wishes to do is correct.
Roughly, it appears that he says that the usual definition of consistency is too strong; he substitutes a weaker definition of the consistency (of PA) every finite subset of axioms (of Peano Arithmetic, PA) is consistent. He then proves (via "selectors") that PA fulfills this weaker version, so that PA can prove its own consistency. (Unlike Gentzen's proof of the consistency of PA.) He says that the Gödel proof is not violated because that proof required the stronger version of consistency. He concludes that therefore Hilbert's program is not dead after all, since his definition of consistency is the more appropriate definition. Corrections to my understanding would be highly welcomed.
Secondly, and this is a set of more general questions; whether this version of consistency can be of any use to the rest of logic or is just philosophical quibbling, whether there are any major weaknesses in his approach (rigorous enough? questions of infinity appropriately handled? ), whether his main goal (reviving the Hilbert Program) is indeed fulfilled by his procedure, and whether this procedure for PA would be extendible to other systems, e.g. ZFC. Any other relevant remarks would also be appreciated.
Many thanks in advance.
https://arxiv.org/abs/2508.20346
https://arxiv.org/abs/2403.12272v1
(If it has appeared in a peer-reviewed journal yet, I have not found it.)
I have questions about the general approach.
First, my main question: whether my general understanding of what he wishes to do is correct.
Roughly, it appears that he says that the usual definition of consistency is too strong; he substitutes a weaker definition of the consistency (of PA) every finite subset of axioms (of Peano Arithmetic, PA) is consistent. He then proves (via "selectors") that PA fulfills this weaker version, so that PA can prove its own consistency. (Unlike Gentzen's proof of the consistency of PA.) He says that the Gödel proof is not violated because that proof required the stronger version of consistency. He concludes that therefore Hilbert's program is not dead after all, since his definition of consistency is the more appropriate definition. Corrections to my understanding would be highly welcomed.
Secondly, and this is a set of more general questions; whether this version of consistency can be of any use to the rest of logic or is just philosophical quibbling, whether there are any major weaknesses in his approach (rigorous enough? questions of infinity appropriately handled? ), whether his main goal (reviving the Hilbert Program) is indeed fulfilled by his procedure, and whether this procedure for PA would be extendible to other systems, e.g. ZFC. Any other relevant remarks would also be appreciated.
Many thanks in advance.