On the solvability of systems of linear equations over the ring $\\mathbb{Z}$ of integers

[摘要] We investigate the question whether \\emph{a system $(E_i)_{i\\in I}$ of homogeneous linear equations over $\\mathbb{Z}$ is non-trivially solvable in $\\mathbb{Z}$ provided that each subsystem $(E_j)_{j\\in J}$ with $|J|\\le c$ is non-trivially solvable in $\\mathbb{Z}$} where $c$ is a fixed cardinal number such that $c< |I|$. Among other results, we establish the following. (a) The answer is `No\' in the finite case (i.e., $I$ being finite). (b) The answer is `No\' in the denumerable case (i.e., $|I|=\\aleph_{0}$ and $c$ a natural number). (c) The answer in case that $I$ is uncountable and $c\\le\\aleph_{0}$ is `No relatively consistent with $\\mathsf{ZF}$\', but is unknown in $\\mathsf{ZFC}$. For the above case, we show that ``\\emph{every uncountable system of linear homogeneous equations over $\\mathbb{Z}$, each of its countable subsystems having a non-trivial solution in $\\mathbb{Z}$, has a non-trivial solution in $\\mathbb{Z}$}\'\' \\textbf{implies} (1) the Axiom of Countable Choice (2) the Axiom of Choice for families of non-empty finite sets (3) the Kinna--Wagner selection principle for families of sets each order isomorphic to $\\mathbb{Z}$ with the usual ordering, and is \\textbf{not implied by} (4) the Boolean Prime Ideal Theorem ($\\mathsf{BPI}$) in $\\mathsf{ZF}$ (5) the Axiom of Multiple Choice ($\\mathsf{MC}$) in $\\mathsf{ZFA}$ (6) $\\mathsf{DC}_{<\\kappa}$ in $\\mathsf{ZF}$, for every regular well-ordered cardinal number~$\\kappa$. We also show that the related statement ``\\emph{every uncountable system of linear homogeneous equations over $\\mathbb{Z}$, each of its countable subsystems having a non-trivial solution in $\\mathbb{Z}$, has an uncountable subsystem with a non-trivial solution in $\\mathbb{Z}$}\'\' (1) is provable in $\\mathsf{ZFC}$ (2) is not provable in $\\mathsf{ZF}$ (3) does not imply ``every uncountable system of linear homogeneous equations over $\\mathbb{Z}$, each of its countable subsystems having a non-trivial solution in $\\mathbb{Z}$, has a non-trivial solution in $\\mathbb{Z}$\'\' in~$\\mathsf{ZFA}$.

[学科分类] 物理化学和理论化学

[关键词] Axiom of Choice;weak axioms of choice;linear equations with coefficients in $\\mathbb{Z}$;infinite systems of linear equations over $\\mathbb{Z}$;non-trivial solution of a system in $\\mathbb{Z}$;permutation models of $\\mathsf{ZFA}$;symmetric models of $\\mathsf{ZF}$DOI: DOI 10.14712/1213-7243.2015.207AMS Subject Classification: 03E25 03E35 PDF