Most general first order theorems are not recursively enumerable

Most General First Order Theorems are not Recursively Enumerable

The first order formulas preserved under ultrafilter extensions are not recursively enumerable

Proof: By assumption, φinf is preserved under ultrafilter extensions. The second conjunct of θ, i.e., ∀xy.(x = y → Rxy) is also preserved under ultrafilter extension, since it is modally definable using global modality. Finally, consider the third conjunct of θ. From the fact that M |= ∀xy.(x = y → Rxy), we can derive that the denotation of R in ueM includes all pairs of ultrafilters (u, v) suc...

The Primitive Recursive Functions are Recursively Enumerable

Meta-operations on primitive recursive functions sit at the brink of what is computationally possible: the semantic equality of primitive recursive programs is undecidable, and yet this paper shows that the whole class of p.r. functions can be enumerated without semantic duplicates. More generally, the construction shows that for any equivalence relation ≈ on natural numbers, N/ ≈ is r.e. if ≈ ...

The Recursively Enumerable Degrees

Decision problems were the motivating force in the search for a formal definition of algorithm that constituted the beginnings of recursion (computability) theory. In the abstract, given a set A the decision problem for A consist of finding an algorithm which, given input n, decides whether or not n is in A. The classic decision problem for logic is whether a particular sentence is a theorem of...

Recursive and Recursively Enumerable

I. Basic concepts 1. Recursive enumerability and recursiveness. We consider infinite sequences of non-negative integers, free from repetitions. A familiar equivalence relation between such sequences is that based on sets: two sequences are equivalent when they enumerate the same set. The equivalence classes under this relation, with the necessary operations introduced, form a system isomorphic ...