Re: Review of Mueckenheims book.

On 1 Mrz., 03:30, "*** T. Winter" <***.Win...@xxxxxx> wrote:

> You apparently do not read hat other write, for instance
> http://zimmer.csufresno.edu/~larryc/proofs/proofs.contradict.html
> One of the first proofs by contradiction is the following gem
> attributed to Euclid

Read what I write, for once. The above page attributes to Euclid the
theorem "there are infinitely many primes".

Of course. That is the modern version to say: No (finite) set of
primes contains all primes.

>http://mathworld.wolfram.com/ProofbyContradiction.html
> For example, the second of Euclid's theorems starts with the
> assumption that there is a finite number of primes.

And when you follow the link you will find: "Euclid's second theorem states
that the number of primes is infinite".

Of course. That is the modern version to say: No (finite) set of
primes contains all primes.

Neither is true, which you would have known when you had written what I did
write and followed the link. The actual proof was the theorem that
"given a collection of primes, there is a prime not in the collection".
But see the reference I gave. (Did you read it, or the summary I wrote?)

> > Because the proof does not show that there is a contradiction with a
> > largest prime.
>
> It does.

That is a corollary that dates from beyond Euclid's days. And it is in
that corollary that contradiction is used.

That is no corollary but it is a modern version to state Euclid's
proof.

> > Proofs by infinite descent are proofs by contradiction.
>
> That does not mean that alls proofs by contradiction are proof by
> infinite descent. But Euclid's proof falls just into this category,
> namely a proof by infinite increase (instead of decrease).

Eh, where is the infinite increase?

The prime constructed from set S1 can be included and the proof can be
repeated. Do you really think that Euclid did not see this?

> > Note that there is an initial assumption that is contradicted. Euclid's
> > original proof is a direct proof that given a collection of primes that
> > there is a prime not in the collection.
>
> In your version I would only read that some collection does not
> contain every prime.

That is not *my* version, it is Euclid's original version.

Do you think Euclid stopped 10 cm in front of the finishing line?


Did you follow
the link I provided? I think you did not even bother to consider it. And
you state that I do not follow *your* links, which I did.

In your http://www.syllogismos.it/history/Icme10-TSG17.pdf
I found a distinction between Euclid's and Euler's proof, not by
Euclid's and a modern corollary of this proof. Futher the different
concepts of infinity by Euclid, Euler, Erdoes and Fuerstenberg. But
even there is stated that *all* these proofs including that original
proof by Euclid are done by reductio ad absurdum, what is also called
proof by contradiction.

Regards, WM

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