If you’re attempting to define N as a first order predicate, that doesn’t work; you’ve defined it in terms of itself. You can’t directly define predicates recursively; predicates must be finite. If you want to do get a “recursive” predicate you have to do quite a bit more work than that, and in particular you need tools not available in the first order theory of the reals (with addition and multiplication, as usual).
Your definition of Z has additional minor problems; you mean and, not implies. (X>=0 ⇒ N(X)) is automatically satisfied for any X<0.
Your last statement is correct (if a bit less general than it could be :) ), though your notation is a bit strange. (Again, assuming + and * as usual.)
Z is defined correctly. When X >= 0 the formula becomes N(X) AND TRUE when X < 0 the formula becomes TRUE AND N(0-X).
Otherwise I was confused. I was trying to define N implicitly which I should have recognized as invalid. Explaining what I was trying to say at the end would be pointless given that I didn’t say it and it’s also wrong =P. Mea culpa
Either I am confused or this discussion is confused.
N(X) iff (X=0) || ((X > 0) && N(X-1)) iff X is natural or 0
Z(X) iff ( (X >= 0) → N(X) ) && ( (X < 0) → N(0 - X) ) iff X is an integer
equivalently
I’m also under the impression that the algebraic numbers are countable, dense in R, and that
Edit: note to all, mixing latex and plain text on a line looks messy. Further edited for formatting due to lack of preview.
If you’re attempting to define N as a first order predicate, that doesn’t work; you’ve defined it in terms of itself. You can’t directly define predicates recursively; predicates must be finite. If you want to do get a “recursive” predicate you have to do quite a bit more work than that, and in particular you need tools not available in the first order theory of the reals (with addition and multiplication, as usual).
Your definition of Z has additional minor problems; you mean and, not implies. (X>=0 ⇒ N(X)) is automatically satisfied for any X<0.
Your last statement is correct (if a bit less general than it could be :) ), though your notation is a bit strange. (Again, assuming + and * as usual.)
Might I ask what the relevance of all this is?
Z is defined correctly. When X >= 0 the formula becomes N(X) AND TRUE when X < 0 the formula becomes TRUE AND N(0-X).
Otherwise I was confused. I was trying to define N implicitly which I should have recognized as invalid. Explaining what I was trying to say at the end would be pointless given that I didn’t say it and it’s also wrong =P. Mea culpa
Oh, you said “and”; my apologies, I implicitly read an “or” there!