Fodor's lemma

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

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, 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

• 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.