Re: Relative Cardinality

In article <1121513224.807178.207430@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

> 207. Virgil Jul 13, 3:08 pm
> Then that numeral exists, but the numeral may well exist otherwise, and
> the number may well exist even if no numeral can be found for it in
> some
> system of numerals. The existence of a number is not dependent on the
> existence of a particular form of numeral to represent it.
>
> For example, does WM wish to say that a "number" which has an exactly
> known continued fraction representation cannot be a number unless it
> also has an entirely known decimal, or other n-ary representation?
>
> WM
> Continued fraction representations, as, for instance, given by Lord
> Brouncker or Euler for pi, are as useful as infinite series as given by
> Vieta, Gregory-Leibniz, Euler, but those representations do not, in
> general, allow for comparison of magnitude.

But does such a "number" exist in WM's peculiar worldview?
What about the continued fraction represented by
(1:2,2,2,2,...)
>
>
> 208. Virgil Jul 13, 2:43 pm
> By what supernatural powers does WM assert that he can know what I can
> or
> cannot imagine?
>
> WM
> By some general knowledge of psychology and of brain functions.

There are more things in my imagination that are dreamed of in your
philosophy.

> Every honest man will concede that he is not capable of obtaining
> ratios to better than 10 % uncertainty from imagined geometric
> figures like circles, squares or triangles.

Where does this come from? No honest person would come up with any such
declaration ofnumerical morality without rock solid evidence to back it
up. And I see no such evidence of honesty.


>
> The set of place values required by naturals, counting from the lowest
> place value upward, has no end.
>
> > Or do you mean the set of digits is infinite?
>
> The the set of place values required has no end.
>
> > Of course, if there are
> > infinitely many numbers, then the set of digits is infinite.

>
> Virgil
> "Has no end".
>
> WM
> = is not finite = infinite (Latin: in = English no, Latin: finis =
> English: end)
>
> > Or easier: Use the unal system, where 7 is represented by IIIIIII.
> > There are infinitely many natural numbers, but thee are only finitely
> > many strokes?
>
> Virgil
> In any one number, the "number of" strokes ends, but in the set of all
> naturals, the "number of strokes" does not end.
>
> WM
> In the set of naturals the number of strokes does not end and the
> number of number does not end either, but the number of numbers is
> always just as finite as the number of strokes.
>
> Virgil
> That is to say that the set of them is not finite.
> WM
> Fine, we are getting closer together. Please introduce "potentially" in
> order to eliminate any ambiguity, and we are ready.

Why should I introduce any such irrelevancy.

The "reality" of every number is at best potential, as none of them are
actual in any physical sense.


But in the imagination, which is the only place that any numbers can be
actual, infinite cardinality is as actual as finite cardinality as soon
as anyone imagines it.

> 210. Virgil Jul 13, 3:27 pm
>
> If mathematicians have one idea of what constitutes a number and
> anti-mathematicians like WM have a different idea, which idea does WM
> suppose that mathematicians will use?
>
> WM
> Mathematics has become an indefinite feeling of ideas meanwhile?

Not so indefinite, at least in the mathematical world of ideas. But all
(pure) mathematics is equally indefinite in the physical world, as that
is not where it lives.
>
> Virgil
> It is a matter of definition what are to be called and what are not to
> be called numbers, and mathematicians are of necessity the ones doing
> the defining.
>
> WM
> Mathematicians? or such who believe that they are mathematicians?

The community of mathematicians, which can be reasonably clear about who
belongs, and WM does not.
>
>
> 212. Virgil Jul 13, 2:37 pm
>
> > > WRONG! WM conflates infinitely many with infinitely large. There are
> > > infinitely many rationals in any non-degenerate real interval, but in
> > > bounded intervals, none of them are infinitely large, so that nfinitely
> > > many does not imply infinitely large.
>
> > I spoke of natural numbers.
>
> Every natural "IS" a rational, in an obvious sense so that unless WM
> insists on having infinitely large rationals (and reals and complexes
> and quaternions, etc.) he is being clearly inconsistent.
>
> WM
> But not every rational is a natural! You argued that a set of rationals
> can contain infinitely many finite elements. I did not argue against
> that, but said that an actually infinite set of naturals cannot exist
> without an actually infinite element.

WRONG! That is like saying that every sequence converges.
>
> > If you represent the natural numbers in the unal system, i.e., 3 = III
> > etc, then there are as many strokes as there are numbers. Neither gets
> > infinite without the other.
>
> Virgil
> It depends on what one means by infinite in this context. My meaning is
> "without end".
>
> The sequence of naturals is without end. But there is no single natural
> that is without end.
>
> WM
> And just that is to be taken as potential infinity or we get a
> contradiction.

"WE" meaning WM, since nobody else gets any contradictions.

> Every natural is actually finite. Therefore we cannot
> have a set of actually infinitely many elements. It is very simple once
> you got it.
Every rational and every real and every complex is actually finite and
they all contain all of the naturals, so that WM now is requiring that
there be infintie rationals and infintie reals and infintie complexes
and God she knows what else.
>
>
> 213. Virgil Jul 13, 2:31 pm
>
> With the exception of countably many rationals with dual
> representations
> in decimal notation, one does not worry about two numbers having
> differing digits 'only' at one place value. The issue is only whether
> the numerals differ at SOME place value.
>
> WM
> And in order to detect that difference, you must compare all digits up
> to the first one where a difference occurs.
>
> Virgil
> Also, there is nothing in the axioms of the reals that requires any
> real
> to even have a representation as an n-ary numeral. In terms of the
> axioms, no such representations are required.
> 214. Virgil Jul 13, 3:48 pm
> Who gives WM the right to determine what mathematicians will choose
> call
> a number?
>
> Since mathematicians are the ones using the word, they get to decide,
> at
> least in mathematical contexts.
.

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