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AnotherKevin comments on Gödel and Bayes: quick question - Less Wrong
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Comments (36)
The key is not just addition and multiplication, the key is addition and multiplication of whole numbers. The first-order theory of the real numbers, by contrast - with addition and multiplication - is decidable. And probability deals with real numbers, not integers.
On top of that you have the issue that the "axioms" of probability are not axioms in the sense of an axiomatic system, they're just "axioms" in the sense of "requirements".
In short this whole post seems badly misinformed.
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
(\exists%20x((x%20\in%20\mathbb%20R%20)%20\vee%20P(x))%20\iff%20\exists%20y%20((\text{y%20is%20algebraic})%20\vee%20P(y))%0A)
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!