Observe that the predicate <, as restriction of the order on Z to N, is already definable in (N; +}.

A relation is D-definable or simply definable when the structure is understood, if it can be defined by a first-order formula on D.

1, it can be interpreted as saying that the first-order definable sets in (Z; +, 0,1, <} are exactly the rational subsets of [Z.

Our decidability result is based on the equivalence between definable subsets in the structure [Z.

Conversely, if for some subset Y of X the set [tau] (Y) is definable by some Z- (resp.

We recall that the problem of deciding whether or not a relation definable in Buchi arithmetic can be actually defined in Presburger arithmetic has been positively answered in [18].

The definable criterion for definability in Presburger arithmetic and its applications.

If, however, x is definable, then we do have that lon(x)[element of]DOn; and this suggests a small modification of Russell's conditions, as follows: For given properties [phi] and [psi], and (possibly partial) function [delta]: 1) w = {x: [phi](x)} exists and [psi](w) 2) if x is a subset of w such that [psi](x): a) [delta](x) [not an element of]x and b) [delta](x)[element of]w If these conditions are satisfied we still have a contradiction.

DOn is clearly definable, and as we have already observed, for definable x, Transcendence and Closure hold.

phi](x) is "x is a natural number (= finite ordinal) definable in less than 19 words", so w={n; n is a natural number definable in less than 19 words}, [DN.

In this, [phi](x) is "x is a definable real number between 0 and 1", so that w is the set of such reals, DR; [psi](x) is "x is definable" and [delta](x) is diag(x), a real, defined by diagonalisation on x, in such a way as to ensure that [delta](x)[not an element of

delta] is a function, [sigma](a), defined by some suitable technique of diagonalisation so that if a is any definable set [sigma](a)=[alpha] where [alpha]=<[alpha][not an element of

Now, if a is definable and a[subset or equal to]Tr: [sigma](a)[element of]a [implies] <[alpha][not an element of]a>[element of]a [implies] <[alpha][not an element of]a>[element of]Tr [implies] [alpha][not an element of]a (by the T-schema) [implies] [sigma](a)[not an element of]a Hence [sigma](a)[not an element of