Re: A new Arithmetic Principle?


Nam Nguyen 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]

In what language are you supposed to be asserting this?
You can't have a setence in traditional FOL that means
this. I suppose you could approximate "know" in some
modal logic. But in that case you would need to say
which modal logic you are advocating.

> >
> > 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.

A whole lot of properties satisfy this.
In particular, the properties "being even",
"being odd", and "being divisible by n", for each n,
satisfy this. This has nothing to do with induction.
If you want to be anti-inductive then you can just take
some denials of instances of the usual induction axiom
as new axioms.

> If a) and b) are satisfied, then there exists an arithmetic
> number m that we don't/can't know if P(m) is true.

Again, you can't say that, not without a modal logic.
You can say that P(m) is not provable, but even there,
having your provability predicate encode REAL ACTUAL
provability as opposed to some formal approximation thereof
requires some meta-theoretical presumptions.

> > Basically gAI would state that certain arithmetic
> > knowledge is not "inductive".

In the context of axioms about arithmetic, there is
more than one meaning of "inductive". In this room,
all knowledge is deductive anyhow.

> >
> > Could/should we adopt gAI?

No, we can't adopt it because it is incoherent

> > If we do, what would be the possible
> > consequences?
Until you state it IN the context of some LOGIC,
it CAN'T have any "consequences".

.

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