A note on Łoś theorem without the Axiom of Choice

• Autorzy: Usuba, Toshimichi
• Języki publikacji: EN

Abstrakty

EN

We study some topics around Łoś’s theorem without assuming the Axiom of Choice. We prove that Łoś’s fundamental theorem on ultraproducts is equivalent to a weak form that every ultrapower is elementarily equivalent to its source structure. On the other hand, it is consistent that there is a structure M and an ultrafilter U such that the ultrapower of M by U is elementarily equivalent to M, but the fundamental theorem for the ultrapower of M by U fails. We also show that weak fragments of the Axiom of Choice, such as the Countable Choice, do not follow from Łoś’s theorem, even assuming the existence of non-principal ultrafilters.

Słowa kluczowe

EN

Axiom of Choice; Łoś’s theorem; symmetric extension; ultrapower; ultraproduct

• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2024
• Tom: Vol. 72, no 1
• Strony: 17--44
• Opis fizyczny: Bibliogr. 13 poz.

Twórcy

autor

Usuba, Toshimichi

• Faculty of Science and Engineering, Waseda University, Tokyo, 169-8555 Japan,

Bibliografia

• [1] A. Blass, A model without ultrafilters, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), 329–331.
• [2] H.-D. Donder, Regularity of ultrafilters and the core model, Israel J. Math. 63 (1988), 289–322.
• [3] Y. Hayut and A. Karagila, Spectra of uniformity, Comment. Math. Univ. Carolin. 60 (2019), 285–298.
• [4] H. Herrlich, Axiom of Choice, Lecture Notes in Math. 1876, Springer, 2006.
• [5] P. E. Howard, Łoś’ theorem and the Boolean prime ideal theorem imply the axiom of choice, Proc. Amer. Math. Soc. 49 (1975), 426–428.
• [6] P. E. Howard and J. E. Rubin, Consequences of the Axiom of Choice, Math. Surveys Monogr. 59, Amer. Math. Soc., 1998.
• [7] T. Jech, The Axiom of Choice, Stud. Logic Found. Math. 75, North-Holland, 1973.
• [8] A. Kanamori, The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings, 2nd ed., Springer, 2009.
• [9] A. Karagila, Iterating symmetric extensions, J. Symbolic Logic 84 (2019), 123–159.
• [10] J. Łoś, Quelques remarques, théorèmes et problèmes sur les classes définissables d’algèbres, in: Mathematical Interpretation of Formal Systems, North-Holland, 1955, 98–113.
• [11] D. Pincus and R. M. Solovay, Definability of measures and ultrafilters, J. Symbolic Logic 42 (1977), 179–190.
• [12] M. Spector, Ultrapowers without the axiom of choice, J. Symbolic Logic 53 (1988), 1208–1219.
• [13] E. Tachtsis, Łoś’s theorem and the axiom of choice, Math. Logic Quart. 65 (2019), 280–292.

Identyfikatory

DOI

10.4064/ba240406-10-8