Re: A new Arithmetic Principle?
Jim Spriggs wrote:
Nam Nguyen wrote:
Consider the following (proposed) Arithmetic Principle,
say, "Anti-Induction" [or just "AI"]:
(1) There _exists_ an arithmetic number that we don't know
if it's even. [I.e. no formalization can assert that it's even]
or a more generalized version, named "gAI" which is:
(2) If P is an arithmetic property that:
a) if P is true for an arithmetic number n0
b) if P(nk) is true for an nk >= n0, then there
exist an nk' >= n0 and nk'' >= n0 such that
P(nk') is true and P(nk'') is false.
Basically gAI would state that certain arithmetic knowledge is not
"inductive".
Could/should we adopt gAI?
In the presence of what other axioms?
Not "axioms", imho. Given that Arithemtic is used for the
foundation/framework of, say FOL, then the question would
be about what kind of impact to the overal FOL based mathematics.
If we do, what would be the possible
consequences?
---Nam Nguyen
.
