Re: An uncountable countable set

On Mon, 23 Oct 2006 15:00:58 -0400, Tony Orlow <tony@xxxxxxxxxxxxx>
wrote:

Lester Zick wrote:

[. . .]

Unfortunately, transfinitology exists, despite the fact that it makes no
sense underneath the hood. When it comes to arithmetic on them, it's one
big kludge. But, there are forms of infinite numbers upon which one can
define arithmetic. They just have nothing whatsoever to do with omega or
the alephs.

Not sure what you're talking about here, Tony. Lots of things exist in
the sense of having been defined. That doesn't make them true and
doesn't mean they form any basis for the truth of other things defined
on them. There's no shortage of things other than infinity on which to
define arithmetic.

What is log2(0)?

-00? Not sure what this is in aid of, Tony. What is log0(0)? For that
matter what is log0(00) or log-00(00) or x!=0? There are all kinds of
restrictions around 0 and 00 precisely because 0 is not a natural
number.

Yet I've also been considering what it looks like you're trying to do
with trans finite arithmetic.In particular it occurs to me that if one
takes +00 to be larger than any positive finite -00 correspondingly
must be smaller than any negative finite such that your concept of
circularity among arithmetic numbers might be combined in the
following way: [-00, . . . 3, 2, 1, 0, 1, 2, 3 . . . +00]. The only
difference would be that whereas +00 represents the number of
infinitesimals, -00 would represent the size of infinitesimals. Thus
we'd have a positive axis with the number of infinitesimals and a
negative axis with the size of infinitesimals. At least that's the
best I can make of the situation.

Well, I rather think of 1/oo as the size of infinitesimals, or more
precisely, for any specific infinite n, 1/n is a specific infinitesimal
value. When it comes to the number circle, in some ways oo and -oo can
be considered the same so the number line forms an infinite circle, but
in others, such as lim(n->oo) as opposed to lim(n->-oo), there is a very
clear difference between the two. I think it's a bit like the
wave-particle dualism for physical objects, and may actually be directly
connected.

Well now you're just back to the idea of arithmetic as some kind of a
TOE, Tony. It's very simple. The only mechanical definition for 00 is
any finite/0. And if that product can't be defined than neither can
00. There is no specific size to infinitesimals because they're an
process not a static thing. Any series of infinitesimals varies in
size continuously. There's a reciprocity between number and size for
any infinite series but no "circle" between them.

Technically, the number of reals in the unit interval (0,1] is Big'un.

But the point is that they're within the interval. There is no
infinite set 1, 2, 3 . . . 00 outside of some interval.

That's also the infinite length of the real number line, in unit
intervals. The unit infinitesimal is Lil'un, or 1/Big'un. Now they're
all specific and related to spatial measure and quantity. :)


On the other hand if you want to do transfinite arithmetic you might
ask yourself what the results of 00-00 or 00/00 are. The latter can be
addressed through application of L'Hospital's rule but I don't know
any way to address the former.

~v~~
The formulas that lend themselves to L'Hospital's Rule usually cannot be
simplified any further to resolve that problem. Subtracting one simple
formula from another is just a matter of combining like terms and
finding the most significant to see if you get a finite result through
mutual cancellations.

But L'Hospital's rule applies to ratios, Tony. It only gives the
finite ratio between infinities. If you subtract 1/0 from 2/0 what do
you get? They both already have common denominators so the answer
would seem to be 1/0 which still remains infinite. Kluge is the right
word for transfinite arithmetic.

~v~~
.

Relevant Pages

  • Re: Galileos Paradox
    ... conventional algebra dealing with finites such as r+y=z and so on. ... infinitesimals such as dr doesn't alter the size of finites such as r. ... Not if you're talking set "theory" Tony. ... definitely finitely integrated and thereby transformed into cardinals. ...
    (sci.math)
  • Re: Galileos Paradox
    ... conventional algebra dealing with finites such as r+y=z and so on. ... infinitesimals such as dr doesn't alter the size of finites such as r. ... Not if you're talking set "theory" Tony. ... definitely finitely integrated and thereby transformed into cardinals. ...
    (sci.math)
  • Re: Galileos Paradox
    ... conventional algebra dealing with finites such as r+y=z and so on. ... Robinson's Nonstandard Analysis is THE first rigorous treatment of infinitesimals, apparently, and Internal Set Theory is a set-theoretic axiomatization of it, apparently. ... Not if the derivative is 0dr, Tony. ... and using the assumption of a unit infinity equal to the number of reals per unit interval. ...
    (sci.math)
  • Re: infinity
    ... David R Tribble said: ... Yes, all infinitesimals have a vlaue of zero, on the finite ...
    (sci.math)
  • Re: infinity
    ... Tony: ... Yes, all infinitesimals have a vlaue of zero, on the finite ... Which would seem to be saying that these: ...
    (sci.math)