Re: infinitely many nn's = infinite nn's?

Phil wrote:
Virgil wrote:
In article <45F466BD.9060809@xxxxxxxxxxxxx>,
Phil <toob-headman@xxxxxxxxxxxxx> wrote:

However, that can only be true if there can be several sequential
irrationals, with no rational in sight.

Not so, however much it seems so. There are never two or more reals "in
a row" without others between them, regardless of their rationality or
lack therof.

Sorry, I meant that there must be a sequence of irrationals -- without
worrying about their being "next to" each other -- with no rationals
between them.

Easily shown to be false.

If a rational number exists between ANY two irrationals,
meaning we cannot under ANY circumstances have several irrationals "in a
row," with no rationals between ANY of the irrationals in question,

True.

then
there cannot even be more irrationals than irrationals, let alone
infinitely more.

You haven't seemed to grasp the point that in mathematics we don't say
things unless we can prove them. You keep making assertions without
offering any proof. What you wrote doesn't logically follow.

Well, I'm not saying that proofs don't exist that rationals exist
between ANY two irrationals, I'm just pointing out that this REALLY IS
incompatible with the proofs that there are infinitely many more
irrationals.

You may believe this, but you haven't shown it.

If you have 10 of x for every 1 of y, there simply is no
way to arrange these x's and y's such that for ANY two x's, there is a
y.

Irrelevant, since no one is saying there are ten irrationals for every
rational.

Now, if we want to say that we are limited by potential infinity when
mentally examining a structure with actually infinitely many elements, I
would certainly agree that we cannot mentally examine the elements
individually, but in the limit, where actual infinity either CAN be
used, or at least predicted, there cannot be a y between ANY two x's.

"Potential infinity" and "actual infinity" are not meaningful terms in
modern mathematics. They were used long ago. If you wish to use them in
a proof, you must define them.

No flaw. One can actually prove the density of positive rational quite
easily:

Note that a/b < c/d, where a,b,c and d are positive integers, if and
only if a*d < b*c.

Let a, b, c and d be naturals such that a/b < c/d as rationals, then
with a little algebra, one can prove a/b < (a+c)/(b+d) < c/d.
Similarly for negative rationals.
And between any rational and zero is half that rational.

I don't have a problem with that, any more than I have a problem with
the Achilles saying that we can never leave the room, but I want to
point out that this proof uses the assumptions of potential infinity,
not actual infinity.

No idea what the "assumption of potential infinity" is. And, no idea
where in the proof you think it is used. I don't see any such assumption
being used.

Let x and y be irrationals with x < y, so that y-x > 0.
By the Archimedean property of the reals there in a natural n such that
n*(y-x) > 2.

Okay, without reading the rest of your proof yet, and not saying this is
bad, but you do know that the Archimedean postulate is a way of (1)
eliminating infinitesimals, by (2) again placing us under the
limitations of potential infinity.

Nonsense. The Archimedian property of the reals follows from the
properties of a complete ordered field. It also can be proved from any
of the constructions of the real numbers.

Since y - x > 2/n, there must be some integer k with n*y > k > n*x,
or, equivalently, with y > k/n > x.

So that k/n is then a rational between x and y.

Without question, under potential infinity, two irrationals are a FINITE
distance apart -- since infinitesimals don't exist under either
potential infinity or the Archimedean postulate (the Archimedean
postulate basically DEFINES a "finite region," in which any two reals
have a ratio that is "finite but unbounded," i.e., limited by potential
infinity) -- and we can ALWAYS find a rational between two real numbers
that have a finite difference between them. Just don't be surprised if
the proofs that the irrationals are infinitely more numerous than the
rationals use actual, rather than potential, infinity.

The phrases "potential infinity" and "actual infinity" are not
meaningful.

You seem to be confused as to what the real numbers are. The real
numbers are the unique complete ordered field. All their properties must
be logically deduced from the axioms for a complete ordered field.

Cantor's diagonal
proof, for example, uses aleph-0 rationals, infinitely many rationals,
to produce infinitely many more irrationals.

It does nothing of the sort.

By the way, did you ever ask yourself how INFINITELY many reals can
exist on the line segment [0,1] (probably proved using actual infinity)
with a FINITE distance between each and every one of them (probably
proved using potential infinity)? And here I thought that the sum of
infinitely many finite distances always equaled an infinite distance;
shows you how dumb I am!

Indeed it does. You really should try to think logically.

Or for that matter, why modern mathematicians
believe that they can take the results of proofs that ASSUME potential
infinity, and combine them with results from proofs that ASSUME actual
infinity, and NOT create a contradictory mess? Just thought I would ask ...

Because we haven't a clue what you mean by "potential infinity" and
"actual infinity". You might wish to learn some mathematics rather than
assuming you know it.

Look, if you want to be a crank, that is your choice. But, if you want
to learn something, you need to go through an argument step by step. You
are very confused as to what reasoning is valid and what reasoning is
not valid. You keep making assertions that can't be justified. The fact
that you think you can justify them just shows that you are confused.
Have you taken any math classes in school?

--
David Marcus
.

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