Fodor's lemma: Difference between revisions
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==Importance to apierology== |
==Importance to apierology== |
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Since \(\omega_1\) is regular, setting \(S=\{<\omega_1\}\), Fodor's lemma implies there does not exist a fundamental sequence system that both assigns a sequence to all countable ordinals, and has the Bachmann property. Fodor's lemma may fail without choice |
Since \(\omega_1\) is regular, setting \(S=\{<\omega_1\}\), Fodor's lemma implies there does not exist a fundamental sequence system that both assigns a sequence to all countable ordinals, and has the Bachmann property. Fodor's lemma may fail without choice<ref>A. Karagila, [https://arxiv.org/abs/1610.03985 Fodor's Lemma can Fail Everywhere]</ref>, but instead we can use a weakened version known as '''Neumer's theorem''' to prove this result, in which the set \(S\) is removed and replaced with \(\{<\kappa\}\) in all cases. |
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==References== |
==References== |
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* E. Tachtsis, [https://www.ams.org/journals/proc/2020-148-03/S0002-9939-2019-14794-8/S0002-9939-2019-14794-8.pdf Juhász's topological generalization of Neumer's theorem may fail in ZF] (2019). Corollary 2.7. |
* E. Tachtsis, [https://www.ams.org/journals/proc/2020-148-03/S0002-9939-2019-14794-8/S0002-9939-2019-14794-8.pdf Juhász's topological generalization of Neumer's theorem may fail in ZF] (2019). Corollary 2.7. |
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* Karel Hrbacek & Thomas Jech, Introduction to Set Theory, 3rd edition, Chapter 11, Section 3. |
* Karel Hrbacek & Thomas Jech, Introduction to Set Theory, 3rd edition, Chapter 11, Section 3. |
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Latest revision as of 21:26, 29 August 2023
Fodor's lemma (or the pressing-down lemma) is a lemma proven by Géza Fodor in 1956. The lemma states that when \(\kappa\) is an uncountable regular cardinal and \(S\) is a stationary set of ordinals \(<\kappa\), any regressive function \(f:S\to\{<\kappa\}\) must be constant on a stationary set of ordinals \(<\kappa\).
Importance to apierology[edit | edit source]
Since \(\omega_1\) is regular, setting \(S=\{<\omega_1\}\), Fodor's lemma implies there does not exist a fundamental sequence system that both assigns a sequence to all countable ordinals, and has the Bachmann property. Fodor's lemma may fail without choice[1], but instead we can use a weakened version known as Neumer's theorem to prove this result, in which the set \(S\) is removed and replaced with \(\{<\kappa\}\) in all cases.
References[edit | edit source]
- E. Tachtsis, Juhász's topological generalization of Neumer's theorem may fail in ZF (2019). Corollary 2.7.
- Karel Hrbacek & Thomas Jech, Introduction to Set Theory, 3rd edition, Chapter 11, Section 3.
- ↑ A. Karagila, Fodor's Lemma can Fail Everywhere