Two notes on vector spaces with recursive operations.
J. C. E. Dekker · Notre Dame Journal of Formal Logic · 1971
In [1] the author studied an tf 0 -dimensional vector space U F over a countable field F; it consists of an infinite recursive set ε F of numbers (i.e., non-negative integers), an operation + from ε F x ε F into ε F and an operation • from F x ε F into ε F .If the field F is identified with a recursive set, both + and are partial recursive functions.Let β be a subset of ε F .We call β a, repere, if it is linearly independent; β is an a^repere, if it is included in a r.e.repere.A subspace V of U F is an a-space, if it has at least one abasis, i.e., at least one basis which is also an α-repere.We write c for the cardinality of the continuum.It can be shown [l,pp.367, 385, 386 and 2, §2] that among the c subspaces of ΊΪ F there are c which are α-spaces and c which are not.The present paper* contains improvements of two results obtained in [1].Henceforth the notations and terminology of [1] will be used.