That recursive definitions are valid-meaning that a recursive definition identifies a unique function-is a theorem of set theory, the proof of which is non-trivial.
32.
This comes in contrast with the direct meaning of the notion of semantic consequence, that quantifies over all structures in a particular language, which is clearly not a recursive definition.
33.
In a function defined by a recursive definition, each value is defined by a fixed first-order formula of other, previously defined values of the same function or other functions, which might be simply constants.
34.
Existence and uniqueness of the general transfinite recursive definition of sets was demonstrated in 1928 by von Neumann for both Zermelo-Fraenkel set theory and Neumann's own set theory ( which later developed into NBG set theory ).
35.
Therefore any recursive coding of a nonstandard model onto the natural numbers, along with a recursive definition ( acting on codes ) for the model's addition and multiplication operations will give a recursive separator, which cannot exist.
36.
A "'recursive definition "'( or "'inductive definition "') in mathematical logic and computer science is used to define the elements in a set in terms of other elements in the set ( Aczel 1978 : 740ff ).
37.
The topological recursion is then a recursive definition of infinite sequences of symmetric meromorphic n-forms \ omega _ { g, n } on \ Sigma ^ n, with poles at ramification points only, for integers ge " 0 such that 2g-2 + n > 0.
38.
One such function, which is provable total but not primitive recursive, is Ackermann function : since it is recursively defined, it is indeed easy to prove its computability ( However, a similar diagonalization argument can also be built for all functions defined by recursive definition; thus, there are provable total functions that cannot be defined recursively ).
39.
In fact, the model of " any " theory containing PA obtained by the systematic construction of the arithmetical model existence theorem, is " always " non-standard with a non-equivalent provability predicate and a non-equivalent way to interpret its own construction, so that this construction is non-recursive ( as recursive definitions would be unambiguous ).
40.
Normalisability is a rare feature of most non-trivial type theories, which is a big departure from the logical world . ( Recall that almost every logical derivation has an equivalent normal derivation . ) To sketch the reason : in type theories that admit recursive definitions, it is possible to write programs that never reduce to a value; such looping programs can generally be given any type.
