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

Herrlich, Horst ; Tachtsis, Eleftherios

Commentationes Mathematicae Universitatis Carolinae, Tome 58 (2017) no. 2, pp. 241-260.

Résumé

We investigate the question whether 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 ``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 $\mathbb{Z}$'' 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 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 ``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 $\mathsf{ZFA}$.

Détail

DOI : 10.14712/1213-7243.2015.207

Classification : 03E25, 03E35
Keywords: 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 $\mathsf{ZF}$