Re: Inside or Outside ?



Rupert wrote:

Nam Nguyen wrote:

Rupert wrote:


Nam Nguyen wrote:


P(x) df= Ayz[((x /= 0) /\ (x /= S0) /\ (S0 < y) /\ (y < x)) -> ~(x=y*z)]


How about

P(x) df= ((x /= 0) /\ (x /= SO)) /\ Ayz[((S0 < y) /\ (y < x)) ->
~(x=y*z)]

Sure. I don't see anything wrong with your def. (Did you see anything
mine?)



Yes, by your definition 0 and 1 would be prime.

I see: x was free in mine. Thanks.

"2 is a prime" then is just P(SS0). The key question is then can we
prove P(SS0), without depending on a) the assumed PA's consistency

Yes.



and b) proof by contradiction?


Define when a proof does not use proof by contradiction.

Assuming T is consistent, a proof that "does not use proof by
contradiction" is a proof that is not a proof-by-contradiction.



What's a proof-by-contradiction?

One in which you assume ~(conclusion) and prove (F /\ ~F). The
proof of (A /\ ~A) cited before is not such a proof.

--
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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

  • Re: Inside or Outside ?
    ... Rupert wrote: ... without depending on a) the assumed PA's consistency ... Define when a proof does not use proof by contradiction. ... when we inhale and exhale. ...
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  • Re: Inside or Outside ?
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    (sci.logic)
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