Re: Inside or Outside ?



Aatu Koskensilta wrote:

Nam Nguyen wrote:

Rupert wrote:

Okay, well let's clarify what your definition of a proof is. On the
usual definition of a proof from a given set of axioms in the predicate
calculus, you never assume anything in proofs at all. Each step of the
proof is either an axiom or a logical axiom or a formula that follows
from previous formulas in the proof by means of one of the rules of
inference.


"My" definition of a (general) proof? You asked for a def. of a
*specific* kind of proof. Where did you or I even wonder about
a general definition of a proof? Technicality errors is one thing,
unless you claim all proofs *must be* proofs-by-contradiction,
could you stop this nonsensical keyboard-"fiddling" and answer
the rather direct question: can there be a proof of P(SS0) that's
not a proof by contradiction?


Rupert's point is that there are different deductive systems for first order logic, and in some the notion of a proof by contradiction is relatively clear - such as sequent calculi and natural deduction formulations - and in some not so clear, such as Hilbert-style systems. In any case, P(n) for a prime n will be provable with a direct proof embodying the calculation verifying that no m > 1 and < n is a factor of n in any sensible deductive system in which the notion of direct proof makes sense.

Thanks, Aatu, for the input. IIRC, FOL used in Shoenfield's book (that
I've used as a reference on and off) is a Hilbert system. But in any
rate I begin to see it now: after all S (the successor function) would
exhaust all the prime numbers, so the direct proof embodying the
verification is possible. As stated before, that wasn't obvious to
me.

So this is a dash to my hope of well-defining the prime-theorems
that are different from the 2-theorems. Perhaps the assumption PA is
consistent might help. But I don't know and back to the drawing board
I guess.


--
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What we call 'I' is just a swinging door which moves
when we inhale and exhale.
Shunryu Suzuki
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Relevant Pages

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