Re: Calculus XOR Probability

Virgil said:
In article <MPG.1edcf295ebad989d98ad0c@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

stephen@xxxxxxxxxx said:
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
stephen@xxxxxxxxxx said:

Oh I am quite sure he will change his definition of "actually
infinite" to exclude this example. Of course, his definition of
"actually infinite", like all his definitions of "infinite" is
somewhat circular, so it is not really a definition at all.

Hand waving.

??? Do you know what hand waving means? Your use of the word here
suggest that you do not.

It means you make statements like "his definitions are circular"
without basis. It's a false hand-waving claim regarding what I've
claimed.

Since TO's definitions are almost totally without mathematical basis,
and frequently circular, they are themselves a virulent form of hand
waving.

TO seems quite incapable of giving any unambiguous mathemaitally sound
definitions.

Furthermore, TO continues to make claim after claim for which he cannot
provide any mathematically valid justifications, heaping Pelion on Ossa.


Anyway, your definition of "actually infinite" contains the word
"infinite", which is why I said your definition was somewhat
circular. Your definition of "actually infinite" is meaningless
until you define what "an infinite number of elements" between two
elements.

See my previous post to Virgil. Between 0 and 1 are provably an
infinite number of reals

But what does TO mean by "an infinite number of reals"? TO has
misdefined "infinite" so many times that one can never be certain of
what he means by that word unless he spells it out.


, given the Archimedean principle E x,z e R:
x<z -> E y e R: x<y ^ y<z. When you can map your set to the reals in
the unit interval, then it's actually infinite.

The Archimedean principle only guarantees countably many reals between
any two given reals. And mere countably infinite TO has repeatedly
called finite.

So, there are countably many reals in [0,1]. You might want to read up on that.


So that It now appears that either the rationals are as infinite as the
reals or the reals are as countable as the rationals, at least according
to what TO says here.

Countability is a dumb criterion. The reals in [0,1] number Big'un, and over
the real line, Big'un^2. The rationals form a Big'un x Big'un grid, with many
quantitative values being repeated in this matrix. In fact, most to the values
are repeats, as every single one has a multitude equal to it. The number of
these repeat values is precisely the number of reals which are NOT rationals.
The number of repeat hyperrationals is precisely equal to the number of
hyperirrationals.



This is meaningless because it does not define what he means by
"an infinite number" of elements between two elements. Using the
standard definition of "infinite", there are an infinite number
of elements between 1 and w. But Tony is using some different
definition of "infinite" which he is unable to articulate and
only he can comprehend.


Is w a member of the set? No. So, that statement doesn't apply to
the elements of the set, does it?

w is a member of the set {1, 2, 3, ..., w }, which is the set I am
talking about. You need to pay attention to what people actually
write.

Oh. So now omega is a natural?

Not at all. To seems bound and determined to jump to unwarranted
conclusions.

If w is the set of naturals, then w union {w} is a perfectly reasonable
set, but is NOT a set of only natural numbers.


The it's kind of irrelevant to the question of whether the set of naturals is
infinite or not.

--
Smiles,

Tony
.

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