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Koepke 1984 consistency strength free subset property omega
theorem_2_2: Koepke's lower bound: if Fr_ω(ω_ω, ω) holds, then some inner model has a measurable cardinal at most ω_ω.
theorem_4_4: Koepke's upper bound: if κ is a measurable cardinal, then Fr_ω(ω_ω, ω) holds in some two-stage generic extension of V.
Peter Koepke, The Consistency Strength of the Free-Subset Property for omega_omega. The Journal of Symbolic Logic 49 (1984), 1198-1204, DOI 10.2307/2274272. The copy read for this card, from the author's preprints page (math.uni-bonn.de/people/koepke), prints "©1984, Association for Symbolic Logic" at the foot of its second page (article p. 1198; the text layer renders the symbol as "0"), and its first page is the JSTOR cover sheet, whose "you may use content in the JSTOR archive only for your personal, non-commercial use" is the platform's notice, every other right reserved.
Koepke settles a question of Devlin by proving that Fr_omega(aleph_omega, omega) - every structure whose universe includes omega_omega and which has at most omega functions and relations has an infinite free subset of omega_omega - is equiconsistent with a measurable cardinal. Theorem 2.2 gives the lower bound: under Fr_omega(omega_omega, omega) some inner model has a measurable cardinal at most omega_omega, proved via the Dodd-Jensen covering theorem for the core model K after a warm-up argument through L yielding 0-sharp (Theorem 2.1). Theorem 4.4 gives the converse: from a measurable cardinal a two-stage generic extension forces Fr_omega(omega_omega, omega), adapting Shelah's forcing but needing only a coherent sequence of Ramsey cardinals, obtained in section 3 as an end segment of a Prikry sequence. Section 1 supplies the combinatorics, including Lemma 1.1 on strongly free sets and Lemma 1.2, by which the least kappa with Fr_omega(kappa, omega) is weakly inaccessible or of cofinality omega. Koepke notes that omega_omega can be replaced by cardinals such as omega_{omega+omega} and omega_{omega*omega}, and announces without proof that "higher" core models make Fr_omega(omega_{omega_1}, omega_1) equiconsistent with omega_1 measurable cardinals, adding that these results form part of his doctoral thesis. The paper does not mention set mappings; the corpus's pages on Problem 623 cite it for Theorems 2.2 and 4.4, which fix the consistency strength of Fr_omega(omega_omega, omega).
Bears on.
- #623: Fr_omega(aleph_omega, omega) implies the problem's positive answer, by coding a set mapping as countably many functions (an observation recorded on the Theorem 4.4 page), so by Theorem 4.4 a positive answer holds in a forcing extension of a model with a measurable cardinal. Theorem 2.2 bears on the problem only through the converse direction, from a positive answer to Fr_omega(aleph_omega, omega), which Lee's unrefereed 2026 note claims and which this card does not check.
Results.
- Theorem 2.2 (p. 1201): Fr_omega(omega_omega, omega) implies that some inner model has a measurable cardinal at most omega_omega.
- Theorem 4.4 (p. 1204): if kappa is a measurable cardinal, then Fr_omega(omega_omega, omega) holds in some two-stage generic extension of V.
Further results, without pages of their own:
- Lemma 1.1 (pp. 1198-1200): for an infinite cardinal lambda, Fr_mu(kappa, lambda) gives every structure S with kappa a subset of S and at most mu functions and relations a free subset of kappa with monotone enumeration (x_i : i < lambda) such that, for each i < lambda, no ordinal in [x_i, x_i^+) (x_i^+ the smallest cardinal above x_i) lies in the substructure generated by the ordinals below x_i and the x_j with i < j < lambda; in particular x_i does not.
- Lemma 1.2 (p. 1200): for an infinite cardinal lambda and kappa the least cardinal with Fr_omega(kappa, lambda), kappa is a limit cardinal, satisfies Fr_mu(kappa, lambda) for every mu < kappa, is weakly inaccessible or has cof(kappa) = cof(lambda), and is at least omega_lambda.
- Theorem 2.1 (p. 1200): Fr_omega(omega_omega, omega) implies that 0-sharp exists.
No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.