A sequential Martin-Lof test (briefly, M-L test) is a recursively enumerable set
U [subset of equal to] N x [Q.
A problem is considered to be partially decidable, semidecidable, solvable, or provable if A is a recursively enumerable set
He argues that the intuitively provable arithmetic sentences constitute a recursively enumerable set
, which has a Godel sentence which is itself intuitively provable.
is less than]z]-programs do not have effective syntax in the sense that there is no recursively enumerable set
S of safe [Datalog.
Let an infinite, enumerable set
of point particles [p.
n] (the phase) is defined on some enumerable set
l; the sequence of phases (In) take values on state space I.
Since we may be reluctant (as are Nagel and Newman) to claim that 'the' human mind is capable of recognizing all arithmetic truths, this theorem does not in itself support Nagel and Newman However, it does lend some support because it can be proven as a consequence of the representability of any recursively enumerable set
of strings S within any reasonable arithmetic axiomatization T.
bar] [summation over (term)] * is a computably enumerable set
, then L[([X.
Solovay, "Recursively enumerable sets
modulo iterated jumps and extensions of Arslanov's completeness criterion," The Journal of Symbolic Logic, vol.
Other chapters describe the properties of recursively enumerable sets
, the link between computability theory and Godel's incompleteness theorem, relative computability and degrees of unsolvability, and polynomial time computability.
With only a few exceptions--for example, the material about recursively enumerable sets
, which the reader skims or skips at his/her later peril--these guidelines work well.
Attempts to loosen the constraints by going beyond recursively enumerable sets
of theorems fare no better because of other more specialised reasons: the inability to deal with infinite sets in one case and the failure to incorporate higher order logics in another.