Re: Review of Mueckenheims book.

In article <1172673465.908565.212680@xxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
On 26 Feb., 04:01, "*** T. Winter" <***.Win...@xxxxxx> wrote:
In article <1172406666.259855.247...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueck...@xxxxxxxxxxxxxxxxx writes:
On 25 Feb., 05:24, "*** T. Winter" <***.Win...@xxxxxx> wrote:
> > Where is the assumption that p1, p2, ..., pr are all the prime
> > numbers? It is simply stated that giving a number of prime
> > numbers we can find another one.
>
> Didn't you see:
> http://zimmer.csufresno.edu/~larryc/proofs/proofs.contradict.html

Do you *read* what I write? I even made a comment about it. It is just
below what you quoted above.

The comment is wrong.

Oh.


> Infinitely Many Primes
> One of the first proofs by contradiction is the following gem
> attributed to Euclid.
> Theorem. There are infinitely many prime numbers.

Not the original theorem. You apparently do *not* read what I write.

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".

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".

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.

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?

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. 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.

> > What does that mean?
>
> You cannot express sqrt(2) as precisely as desired by numbers.

Makes no sense at all. sqrt(2) in base sqrt(2) is 10.

You cannot express the base sqrt(2) by the unit 1.

But the expression I use is precise enough to do calculations with it.
You are leaving the realm of reality.

Please try to think a bit deeper than usual: If It has been proven
that one cannot prove a definable well ordering of the reals then such
a well ordering cannot be done because doing so would prove the
possibility of doing so, which cannot be proved, as has been proven.

There is a difference between "the possibility" and "the provability".
But apparently you fail to see the distinction.

> > > > Page 90 of your book:
> > > > (2) Komprehensionsaxiom
> > > > For each set A there exists a set B, that contains those
> > > > elements of A that in addition obeys some well-defined
> > > > property E.
> > > > This axioms states what subsets do exist. And if some property
> > > > E is not valid in a model, there is no subset.
>
> And what are "each" and "exists" and "those elements"?

Do you not know the meanings of the words you use in your book?

I think I know them as far as one can know them. And I know too what a
subset of N is.

From outside the model. When you are arguing about Skolem's countable
model, you should stay within that model.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.