Primitive Recursive Ordered Fields and Some Applications

Effective Presentations of Mathematical Structures

“…In particular, we prove effective versions of universality for the space C[0, 1] of continuous functions on the unit interval, the Urysohn space, and Cantor space (among Stone spaces). We also continue the systematic development of primitive recursive (punctual) analysis which was initiated in [DMN21] and, in the context of ordered fields, proposed in [SS21]. We prove computable versions of the universality results, and then we also establish their primitive recursive analogs.…”

“…The main goal is to give a systematic primitive recursive analysis of primitive recursive closure theorems in algebra [DM23]. The next result of the thesis is the following theorem which, in particular, answers the aforementioned question raised in [SS21] by Selivanov and Selivanova. Theorem 6.…”

“…For instance, every computable algebraic field (over Q) can be com-putably embedded into a computable presentation of the algebraic closure acl(Q) of Q. The study of the primitive recursive content of such results has been initiated by Selivanov and Selivanova [SS21] who gave a primitive recursive way of embedding a primitive recursive Archimedean ordered field into its primitive recursive real closure (under a mild extra effectiveness condition on the field). The general case was left open.…”

“…Beginning with the mid-eighties pretty much all computable analysis has been done using general Turing computability; see, e.g., [PER89], [Wei00]. Recently in [DMN21] and [SS21], it has been proposed to revive this program of applying primitive recursiveness in analysis using modern methods. For instance, very recently Selivanov and Selivanova [SS21] have established a primitive recur-sive ('punctual') version of the Ershov-Madison theorem [Ers68,Mad70] establishing that every primitive recursive Archimedean field can be primitively recursively embedded into its primitive recursive real closure.…”

“…Recently in [DMN21] and [SS21], it has been proposed to revive this program of applying primitive recursiveness in analysis using modern methods. For instance, very recently Selivanov and Selivanova [SS21] have established a primitive recur-sive ('punctual') version of the Ershov-Madison theorem [Ers68,Mad70] establishing that every primitive recursive Archimedean field can be primitively recursively embedded into its primitive recursive real closure. Here we want to test our methods to establish primitive recursive versions of several universality results in the theory of separable spaces.…”