# Article

Full entry | PDF   (0.4 MB)
Keywords:
choice principle; vector space; base for vector space; nonzero linear functional; norm on vector space; Fraenkel--Mostowski permutation models of ${\rm ZFA}+\neg{\rm AC}$; Jech--Sochor first embedding theorem
Summary:
In set theory without the axiom of choice (${\rm AC}$), we study certain non-constructive properties of infinite-dimensional vector spaces. Among several results, we establish the following: (i) None of the principles AC$^{\rm LO}$ (AC for linearly ordered families of nonempty sets)---and hence AC$^{\rm WO}$ (AC for well-ordered families of nonempty sets)--- ${\rm DC}({<}\kappa)$ (where $\kappa$ is an uncountable regular cardinal), and "for every infinite set $X$, there is a bijection $f\colon X\rightarrow\{0,1\}\times X$", implies the statement "there exists a field $F$ such that every vector space over $F$ has a basis" in ZFA set theory. The above results settle the corresponding open problems from Howard and Rubin "Consequences of the axiom of choice:", and also shed light on the question of Bleicher in "Some theorems on vector spaces and the axiom of choice" about the set-theoretic strength of the above algebraic statement. (ii) "For every field $F$, for every family $\mathcal{V}=\{V_{i}\colon i\in I\}$ of nontrivial vector spaces over $F$, there is a family $\mathcal{F}=\{f_{i}\colon i\in I\}$ such that $f_{i}\in F^{V_{i}}$ for all $i\in I$, and $f_{i}$ is a nonzero linear functional" is equivalent to the full AC in ZFA set theory. (iii) "Every infinite-dimensional vector space over $\mathbb{R}$ has a norm" is not provable in ZF set theory.
References:
[1] Blass A.: Ramsey's theorem in the hierarchy of choice principles. J. Symbolic Logic 42 (1977), no. 3, 387–390. DOI 10.2307/2272866 | MR 0465865 | Zbl 0374.02037
[2] Blass A.: Existence of bases implies the axiom of choice. Axiomatic Set Theory, Contemp. Math., 31, Amer. Math. Soc., Providence, 1984, pages 31–33. DOI 10.1090/conm/031/763890 | MR 0763890 | Zbl 0557.03030
[3] Bleicher M. N.: Some theorems on vector spaces and the axiom of choice. Fund. Math. 54 (1964), 95–107. DOI 10.4064/fm-54-1-95-107 | MR 0164899 | Zbl 0118.25503
[4] Halpern J. D., Howard P. E.: The law of infinite cardinal addition is weaker than the axiom of choice. Trans. Amer. Math. Soc. 220 (1976), 195–204. DOI 10.1090/S0002-9947-1976-0409183-1 | MR 0409183
[5] Howard P., Rubin J. E.: Consequences of the Axiom of Choice. Mathematical Surveys and Monographs, 59, American Mathematical Society, Providence, 1998. DOI 10.1090/surv/059 | MR 1637107 | Zbl 0947.03001
[6] Howard P., Tachtsis E.: On vector spaces over specific fields without choice. MLQ Math. Log. Q. 59 (2013), no. 3, 128–146. DOI 10.1002/malq.201200049 | MR 3066735 | Zbl 1278.03082
[7] Howard P., Tachtsis E.: No decreasing sequence of cardinals. Arch. Math. Logic 55 (2016), no. 3–4, 415–429. DOI 10.1007/s00153-015-0472-5 | MR 3490912
[8] Howard P., Tachtsis E.: On infinite-dimensional Banach spaces and weak forms of the axiom of choice. MLQ Math. Log. Q. 63 (2017), no. 6, 509–535. DOI 10.1002/malq.201600027 | MR 3755261
[9] Jech T. J.: The Axiom of Choice. Studies in Logic and the Foundations of Mathematics, 75, North-Holland Publishing, Amsterdam, American Elsevier Publishing, New York, 1973. MR 0396271 | Zbl 0259.02052
[10] Läuchli H.: Auswahlaxiom in der Algebra. Comment. Math. Helv. 37 (1962/1963), 1–18 (German). DOI 10.1007/BF02566957 | MR 0143705
[11] Lévy A.: Basic Set Theory. Springer, Berlin, 1979. MR 0533962
[12] Morillon M.: Linear forms and axioms of choice. Comment. Math. Univ. Carolin. 50 (2009), no. 3, 421–431. MR 2573415 | Zbl 1212.03034
[13] Rubin H., Rubin J. E.: Equivalents of the Axiom of Choice, II. Studies in Logic and the Foundations of Mathematics, 116, North-Holland Publishing, Amsterdam, 1985. MR 0798475

Partner of