Re: infinity

David Kastrup said:
> "Ross A. Finlayson" <raf@xxxxxxxxxxxxxxx> writes:
>
> > David R Tribble wrote:
> >> Ross A. Finlayson said:
> >> >> I think there are much better ways to gauge the
> >> >> relative sizes or densities of infinite sets than cardinality, ...
> >> >
> >>
> >> Tony Orlow wrote:
> >> > Right on! See? I am not entirely alone.
> >>
> >> See what? Of course you're not the only one who doesn't accept the
> >> rules of standard mathematics. Is that supposed to count for
> >> something?
> >
> > No, asymptotic density from number theory is quite standard. Also it
> > gives some very exact and intuitive results, eg, half of the integers
> > are even.
>
> Uh, no, it doesn't. Take the sequence
>
> 1,2,4,3,6,8,5,10,12,7,14,16 ...
>
> Then you get the very exact and intuitive result that two thirds of
> the integers (all of which are contained in the sequence) are even.
You just love misarranging things don't you, Kastrup? This is the same as your
idiotic explanation for the vase that you have +10-1-1-1-1-1-1-1-1-1-1+10-1-1-
1-1-1-1-1-1-1-1..... That's not what you have. Maintain proper order and get
proper results.
>
> > That a proper subset of any set is smaller, in terms of a definite
> > size relation, seems rather obvious and is sometimes on sci.math
> > attributed in validation to Fred Katz.
>
> It does not seem "rather obvious" to me that the set of naturals
> becomes smaller if I replace every of its element by its unique
> successor, and yet this operation results in a proper subset of the
> naturals.
That's only because you have no concept of a vlaue range in your set
description. You might want to pack one of those in your bag the next time you
go hiking on Infinity Mountain.
>
> > Basically within a number system, there is ample reasoning to suggest
> > half of the non-zero integers are positive, half of the real numbers
> > between zero and one are less than one half, etcetera.
>
> But it does not make sense that if I square all of the real numbers
> between zero and one, a reversible operation, that the unique and
> reversible images of half of the reals suddenly occupy only a quarter
> of the reals. Where did the rest go, and how does it reemerge when
> doing a square root?
When you square all numbers from 0.000...001 through 0.500...000, you now have
a set of numbers from 0.000...000:000...001 through 0.250...000:000...000. Your
missing half of the squares are buried within that set of second-level
infinitesimals. Actually, that is a simplification. Most of them are in that
range from 0.000...000:000...001 and 0.000...001, since the numbers between
0.000...001 and 0.250...000 are squares of numbers from 0.000...001 and
0.500...000, where the number of digits is now 1/2 of what it was before, and
the number of elements is the square root of what it was before. You have,
within the range 0.000...001 and 0.500...000, the square root of that range
being squares of numbers in the range. The other N-sqrt(N) numbers have been
compressed into the level of the sub-infinitesimal.
>
> > It's generally accepted that the set is smaller than its powerset,
> > among the reasons being that the powerset of the set with n elements
> > has 2^n elements.
>
> Uh, no. That is just an argument valid for finite sets. The main
> reason is that any presumed mapping from set to powerset can be shown
> to be incomplete, without any reference to an actual count of members.
That argument was used by Cantor in consideration of the Continuum Hypothesis,
was it not, by suggesting 2^omega as a possible value for c? That rule holds
for all sets, finite and infinite, by laws of combinatorics.
>
>

--
Smiles,

Tony
.

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