On vector spaces over specific fields without choice. (English) Zbl 1278.03082

In what follows, \(\mathbf{ZF}\) is Zermelo-Fraenkel set theory, \(\mathbf{AC}\) is the axiom of choice and \(\mathbf{ZFC} = \mathbf{ZF} + \mathbf{AC}\). \(\mathbf{ZFA}\) is the set theory obtained by weakening the axiom of extensionality in order to allow the existence of atoms. \(\mathbf{MC}\) is the axiom of multiple choice, which is the statement “For every family \(\{X_i: i \in I\}\) of non-empty sets there exists a function \(F\) with domain \(I\) such that, for every \(i \in I\), \(F(i)\) is a non-empty, finite subset of \(X_i\).” \(\mathbf{AC}_{\text{fin}}\) is the axiom of choice restricted to families of non-empty finite sets. \(\mathbf{VnD}\) denotes van Douwen’s choice principle, which states: “Every family \(\mathcal{A} = \{ (A_i, \leqslant_i): i \in I\}\) of linearly ordered sets isomorphic with \(\mathbb{Z}\) (considered with the usual ordering) has a choice function.” \(\mathbf{BPI}\) is the Boolean prime ideal theorem, which states: “Every Boolean algebra has a prime ideal.”
In the paper under review, the authors introduce a number of notations for statements about vector spaces; such notations are parametrized by \(F\), which is supposed to represent a field. Let us present some of these notations: \(\mathbf{B}(F)\) is the statement “Every vector space over \(F\) has a basis.” \(\mathbf{BE}(F)\) (basis extension) is: “Every independent subset of a vector space \(V\) over \(F\) can be extended to a basis for \(V\).” \(\mathbf{BEB}(F)\) (basis extension from a basis) is: “For every vector space \(V\) over \(F\) and every basis \(\mathcal{B}\) for \(V\), if \(X\) is any independent subset of \(V\) then there is a subset \(S \subseteq \mathcal{B}\) such that \(X \cup S\) is a basis for \(V\).” \(\mathbf{BEBW}(F)\) (weak basis extension from a basis) is: “For every vector space \(V\) over \(F\), if \(V\) has a basis then any independent subset of \(V\) can be extended to a basis for \(V\).” \(\mathbf{S}(F)\) (direct summand) is: “For every vector space \(V\) over \(F\) and every subspace \(W\) of \(V\) there is a subspace \(W'\) of \(V\) such that \(V = W \oplus W'\,\).” \(\mathbf{D}(F)\) (see [M. Morillon, Commentat. Math. Univ. Carol. 50, No. 3, 421–431 (2009; Zbl 1212.03034)]) is: “For every vector space \(V\) over \(F\), there is a non-zero linear functional \(f:V \rightarrow F\).” \(\mathbf{SLin}(F)\) is: “For every system \(S\) of linear equations over \(F\), \(S\) has a solution (in \(F\)) if, and only if, every finite subsystem of \(S\) has a solution (in \(F\)).” It is worthwhile remarking that for all the statements \(\psi(F)\) listed above one has “\(\forall F (\psi(F))\)” stablished as a theorem of \(\mathbf{ZFC}\).
The main goal of the paper is to investigate the consistency strength of statements as the ones presented in the preceding paragraph, relating them to various choice principles. Before presenting their own results, the authors summarize a number of known results on the subject. Such previous results, generously compiled by the authors, include (among many others) the following ones: A. Blass showed in [Contemp. Math. 31, 31–33 (1984; Zbl 0557.03030)] that the statement “For every field \(F\), every vector space over \(F\) has a basis” implies \(\mathbf{MC}\) in \(\mathbf{ZF}\). Since \(\mathbf{AC}\) and \(\mathbf{MC}\) are equivalent in \(\mathbf{ZF}\) (cf. [U. Felgner and T. Jech, Fundam. Math. 79, 79–85 (1973; Zbl 0259.02052); T. Jech, The axiom of choice. Amsterdam-London: North-Holland Publishing Company; New York: American Elsevier Publishing Company (1973; Zbl 0259.02051)]), Blass’s result gives us the equivalence within \(\mathbf{ZF}\) of the statements \(\mathbf{AC}\) and “\(\forall F(\mathbf{B}(F))\)” . The authors present a short proof that, for any field \(F\), \(\mathbf{BE}(F)\) implies all of \(\mathbf{B}(F)\), \(\mathbf{S}(F)\) and \(\mathbf{BEBW}(F)\). M. N. Bleicher has shown in [Fundam. Math. 54, 95–107 (1964; Zbl 0118.25503)] several results such as the following: (i) In \(\mathbf{ZFA}\), \(\mathbf{AC}\) is true if, and only if, there is a field \(F\) such that \(\mathbf{B}(F)\) and \(\mathbf{BEB}(F)\); (ii) If for some field \(F\), \(\mathbf{BEB}(F)\) is true, then \(\mathbf{AC}_{\text{fin}}\) is true – and therefore “\(\exists F (\mathbf{BEB}(F))\)” is not provable in \(\mathbf{ZF}\); and (iii) For any field \(F\) of characteristic \(0\), \(\mathbf{MC}\) is equivalent to \(\mathbf{S}(F)\). Morillon [loc. cit.] has proved that, for any field \(F\), \(\mathbf{D}(F)\) is equivalent to each of \(\mathbf{DE}(F)\) and \(\mathbf{DS}(F)\), where \(\mathbf{DE}(F)\) is “For every non-trivial vector space \(V\) over \(F\), for every subspace \(W\) of \(V\) and for every linear functional \(f:W \rightarrow F\) there is a linear functional \(g: V \rightarrow F\) such that \(f \subseteq g\)” , and \(\mathbf{DS}(F)\) is: “For every non-trivial vector space \(V\) over \(F\) and for every \(a \in V \setminus \{0\}\), there exists a linear functional \(f:V \rightarrow F\) such that \(f(a) = 1\).”
Bleicher [loc. cit.] addressed the question of whether the use of \(\mathbf{AC}\) is essential in proving the statement “\(\exists F (\mathbf{B}(F))\)” , and, if the answer is positive, he asked if its full strength is essential, that is, if “\(\exists F (\mathbf{B}(F))\)” is equivalent to \(\mathbf{AC}\) or to some weak form of it. The research done by the authors and materialized in the paper under review was motivated by such questions posed by Bleicher, besides the per se interest of studying the consistency strenghts of statements concerning vector spaces over specific fields and comparing them with several choice principles. Typical results of the paper are as follows; among many other results, it is shown in the paper that:
In the Dawson-Howard permutation model of \(\mathbf{ZFA}\) (which is referred to as \(\mathcal{N}29\) in [P. Howard and J. Rubin, Consequences of the axiom of choice. Providence, RI: American Mathematical Society (1998; Zbl 0947.03001)]), for every field \(F\) there exists a vector space \(V\) over \(F\) which has no basis. This gives a positive answer to Bleicher’s question of whether the use of \(\mathbf{AC}\) is essential in the proof of “\(\exists F (\mathbf{B}(F))\)”.
The statement “\(\exists F (\mathbf{BEBW}(F))"\) implies \(\mathbf{VnD}\), and therefore one has that “\(\exists F (\mathbf{BEB}(F))\)” also implies \(\mathbf{VnD}\).
For any field \(F\), \(\mathbf{D}(F)\) is equivalent to \(\mathbf{SLin}(F)\); in addition, it is shown that \(\mathbf{BPI}\) implies \(\mathbf{D}(F)\) for every finite field \(F\) and, for every field \(F\), \(\mathbf{S}(F)\) implies \(\mathbf{D}(F)\).
“\(\forall F (\mathbf{B}(F))\)” is independent of the theory \(\mathbf{ZFA} + (\forall F(\mathbf{D}(F))\).

The paper ends with several open questions, such as: does “\(\forall F(\mathbf{B}(F))\)” imply \(\mathbf{AC}\) within \(\mathbf{ZFA}\)? Does “\(\forall F (\mathbf{BE}(F))\)” imply \(\mathbf{AC}\) in \(\mathbf{ZF}\)? Is there a field \(F\) for which \(\mathbf{B}(F)\) implies \(\mathbf{AC}\)? Is there a field \(F\) for which \(\mathbf{B}(F)\) does not imply \(\mathbf{AC}\)? Does \(\mathbf{BPI}\) imply “\(\forall F(\mathbf{D}(F))\)”? Does \(\mathbf{B}(\mathbb{R})\) imply that \(\mathbb{R}\) can be well ordered?


03E25 Axiom of choice and related propositions
03E35 Consistency and independence results
15A03 Vector spaces, linear dependence, rank, lineability
Full Text: DOI


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