Re: G.H.Hardy gave an invalid proof of Infinitude of Primes in "AMathematician's Apology"

tomlewton1_at_hotmail.com
Date: 03/29/05 

Date: 29 Mar 2005 13:59:02 -0800

> You mistake method with specific example.
>
> In the *indirect method* yes of course 510511 is necessarily prime
because
> your total universe space of primes is your assumption that
2,3,5,7,11,13,17
> were all the primes that existed. In that restricted space 510511 is
> necessarily prime and larger than 17 and hence the proof.
>
> In the *indirect method* let us assume 3,5 was the total universe of
primes
> and thus P!+1 yields 16. It is necessarily prime because the
assumption was
> that 3,5 were the only primes in existence.
>
> Now, shifting to the *direct method* which Euclid's ancient proof was
a
> direct method of increasing set cardinality. Given the set
{2,3,5,7,11,13,17}
> then in the proof 510511 is no longer necessarily prime and a *prime
factor*
> search must be added into the proof yielding two new primes of 19 and
> possibly 26869??. The set {3,5} also requires a prime-factor search
to yield
> the new prime not in the list which is "2".
>
> So in the Indirect Method there is never a search for a prime-factor
because
> P!+1 is necessarily prime. In the Direct Method where every set of
primes is
> increased with a new prime not in the set itself requires a *prime
factor
> search*.
>
> You cannot mix the two.
>
> The two mistakes Hardy made was that he thought Euclid's proof was
Indirect
> when it was Direct all along and secondly, Hardy does a prime-factor
search
> in a Indirect method.
>
> If Infinitude of Primes proof were written out in solely the language
of
> Symbolic Logic then it would be evident that a "Prime Factor Search"
arises
> only in the Direct set theory method.

there are two possibilities.
1. you have no clue what you are talking about
2. you are a troll
i'm erring on the side of 2.

Relevant Pages

  • Re: G.H.Hardy gave an invalid proof of Infinitude of Primes in "AMathematicians Apology"
    ... > were all the primes that existed. ... > that 3,5 were the only primes in existence. ... > direct method of increasing set cardinality. ...
    (sci.logic)
  • Re: IPP - proof and question
    ... the primes can be listed p1, ... Then because of the unique factorization theorem, ... existence of the prime N not one of the p1,...,pk. ... Obviously the only reason for this reasoning by AP ...
    (sci.logic)
  • Re: Infinitude of Primes - Proof and (more important) question
    ... the primes can be listed p1, ... Then because of the unique factorization theorem, ... means it divides 1, which is absurd. ... existence of the prime N not one of the p1,...,pk. ...
    (sci.math)
  • Re: induction vs Cantor
    ... >> Cantor's proof is similar to Euclid's proof of the existence ... >> In Euclid's case, you assume that you have a FINITE set of primes, ... >> of reals, and then show that there is a real that is not in that set. ... >> This shows that the set of all reals is not COUNTABLY INFINITE. ...
    (sci.math)
  • Re: induction vs Cantor
    ... >> Cantor's proof is similar to Euclid's proof of the existence ... >> In Euclid's case, you assume that you have a FINITE set of primes, ... >> of reals, and then show that there is a real that is not in that set. ... >> This shows that the set of all reals is not COUNTABLY INFINITE. ...
    (sci.logic)