On certain non-constructive properties of infinite-dimensional vector spaces

摘要

In set theory without the axiom of choice (${m AC}$), we study certain non-constructive properties of infinite-dimensional vector spaces. Among several results, we establish the following (i) None of the principles AC$^{m LO}$ (AC for linearly ordered families of nonempty sets)---and hence AC$^{m WO}$ (AC for well-ordered families of nonempty sets)---reak${m DC}({