Independence from ZFC: Difference between revisions
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For any computable and arithmetically sound axiomatic theory T, there exists an integer <math>N_T</math> such that T cannot prove the values of BB(n) for any <math>n \ge N_T</math>. For [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] (ZF), this value is known to be in the range: | For any computable and arithmetically sound axiomatic theory T, there exists an integer <math>N_T</math> such that T cannot prove the values of BB(n) for any <math>n \ge N_T</math>. For [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] (ZF), this value is known to be in the range: | ||
<math display="block">6 \le N_{ZF} \le | <math display="block">6 \le N_{ZF} \le 385</math> | ||
This lower bound comes from the fact that [[BB(5)]] has been proven in Rocq. The upper bound comes from an explicit TM which enumerates all possible proofs in ZF and halts if it finds a proof 0 = 1. Assuming ZF is consistent, then it cannot prove its own consistency, hence it cannot prove whether this specific TM halts. | This lower bound comes from the fact that [[BB(5)]] has been proven in Rocq. The upper bound comes from an explicit TM which enumerates all possible proofs in ZF and halts if it finds a proof 0 = 1. Assuming ZF is consistent, then it cannot prove its own consistency, hence it cannot prove whether this specific TM halts. | ||
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|[https://github.com/andrew-j-wade/metamath-turing-machines Github NQL] ([https://github.com/andrew-j-wade/metamath-turing-machines/commit/30d2e3194866615f68dd2f5101cb300fb039adca commit]) | |[https://github.com/andrew-j-wade/metamath-turing-machines Github NQL] ([https://github.com/andrew-j-wade/metamath-turing-machines/commit/30d2e3194866615f68dd2f5101cb300fb039adca commit]) | ||
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|16 July 2025 | |16 July 2025 | ||
| | |Andrew J. Wade | ||
|[https://github.com/andrew-j-wade/metamath-turing-machines Github NQL] ([https://github.com/andrew-j-wade/metamath-turing-machines/commit/5d676aec074a94f598959cb3b7733a8f7781762f commit]) | |[https://github.com/andrew-j-wade/metamath-turing-machines Github NQL] ([https://github.com/andrew-j-wade/metamath-turing-machines/commit/5d676aec074a94f598959cb3b7733a8f7781762f commit]) | ||
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|[https://hachyderm.io/@mscibing/114985452862976915 Mastodon Post] | |||
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Revision as of 16:36, 3 September 2025
For any computable and arithmetically sound axiomatic theory T, there exists an integer such that T cannot prove the values of BB(n) for any . For Zermelo–Fraenkel set theory (ZF), this value is known to be in the range:
This lower bound comes from the fact that BB(5) has been proven in Rocq. The upper bound comes from an explicit TM which enumerates all possible proofs in ZF and halts if it finds a proof 0 = 1. Assuming ZF is consistent, then it cannot prove its own consistency, hence it cannot prove whether this specific TM halts.
Scott Aaronson conjectured in his Busy Beaver Frontier survey[1] that .
Axiom of Choice
Due to Shoenfield's absoluteness theorem, it is known that any TM proven non-halting in ZFC can also be proven non-halting in ZF (and the converse is trivially true), therefore
Therefore we refer to ZF and throughout this article since adding the Axiom of Choice does not have any effect on Turing machine decidability.
History
There is no one authoritative source on the history of TMs independent of ZF, this is our best understanding of the history of TMs found. Mostly these are taken from Scott Aaronson's blog announcements and Busy Beaver Frontier or self-reported by the individuals who discovered them.
| States | Date | Discoverer | Source | Verification |
|---|---|---|---|---|
| 7910 | May 2016 | Adam Yedidia and Scott Aaronson | Yedida and Aaronson 2016[2] | |
| 748 | May 2016 | Stefan O’Rear | Github NQL file, Busy Beaver Frontier[1] | |
| 745 | July 2023 | Johannes Riebel | Riebel 2023 Bachelor Thesis[3] | |
| 643 | July 2024 | Rohan Ridenour | Github NQL, Aaronson Announcement | |
| 636 | 31 August 2024 | Rohan Ridenour | Github NQL (commit) | |
| 588 | 12 July 2025 | Andrew J. Wade | Github NQL (commit) | |
| 549 | 16 July 2025 | Andrew J. Wade | Github NQL (commit) | |
| 385 | 6 Aug 2025 | Andrew J. Wade | Mastodon Post |
References
- ↑ 1.0 1.1 Scott Aaronson. 2020. The Busy Beaver Frontier. SIGACT News 51, 3 (August 2020), 32–54. https://doi.org/10.1145/3427361.3427369
- ↑ A. Yedidia and S. Aaronson. A relatively small Turing machine whose behavior is independent of set theory. Complex Systems, (25):4, 2016. https://arxiv.org/abs/1605.04343
- ↑ Riebel, Johannes (March 2023). The Undecidability of BB(748): Understanding Gödel's Incompleteness Theorems (PDF) (Bachelor's thesis). University of Augsburg.