Enumerable

A recursively enumerable set is recursive if its complement is recursively enumerable.

For the latter, see citation An important property of the primitive recursive functions is that they are a recursively enumerable subset of the set of all total recursive functions (which is not itself recursively enumerable).

Gödel's incompleteness theorem shows that no consistent, recursively enumerable theory (that is, one whose theorems form a recursively enumerable set) in which the concept of natural numbers can be expressed, can include all true statements about them.

In particular, the fuzzy set of logically true formulas is recursively enumerable in spite of the fact that the crisp set of valid formulas is not recursively enumerable, in general.

Also, since all functions in these languages are total, algorithms for recursively enumerable sets cannot be written in these languages, in contrast with Turing machines.

An analogue of Rice's theorem for recursive sets One can regard Rice's theorem as asserting the impossibility of effectively deciding for any recursively enumerable set whether it has a certain nontrivial property.

Boolos's proof proceeds by constructing, for any computably enumerable set S of true sentences of arithmetic, another sentence which is true but not contained in S. This gives the first incompleteness theorem as a corollary.

Effective axiomatization A formal system is said to be effectively axiomatized (also called effectively generated) if its set of theorems is a recursively enumerable set (Franzén 2004, p. 112).

However it does not have a recursively enumerable set of axioms, and thus does not satisfy the hypotheses of the incompleteness theorems.

If one takes all statements in the language of Peano arithmetic as axioms, then this theory is complete, has a recursively enumerable set of axioms, and can describe addition and multiplication.

Indeed one can run the Turing Machine and accept if the machine halts, hence it is recursively enumerable.

In order of increasing expressive power it includes regular (or Type-3) grammars, context-free (or Type-2) grammars, context-sensitive (or Type-1) grammars, and recursively enumerable (or Type-0) grammars.

In other words, the enumerable first n bits of Omega are highly compressible in the sense that they are limit-computable by a very short algorithm; they are not random with respect to the set of enumerating algorithms.

In particular, this means that, given a computably enumerable set of axioms, there are Diophantine equations for which there is no proof, starting from the axioms, of whether the set of equations has or does not have integer solutions.

Presburger arithmetic is complete, consistent, and recursively enumerable and can encode addition but not multiplication of natural numbers, showing that for Gödel's theorems one needs the theory to encode not just addition but also multiplication.

Recursively enumerable languages are not closed under set difference or complementation.

The set difference L - P may or may not be recursively enumerable.

The theory of first-order Peano arithmetic is consistent, has an infinite but recursively enumerable set of axioms, and can encode enough arithmetic for the hypotheses of the incompleteness theorem.

We say that s is decidable if both s and its complement –s are recursively enumerable.