Re: Well Ordering the Reals

David R Tribble said:
> David R Tribble said:
> >> At any rate, we could try a similar (but different) addition using
> >> your notation (where c and d are not the same as a and b above):
> >> c = 0:101010...101010
> >> d = 0:110110...110110
> >> + ------------------
> >> s = 0:1000000...000000
> >>
> >> Hmm. It looks like s is one bit longer than c and d, even though
> >> they are all (supposedly) infinitely long numbers. Or perhaps
> >> we've simply just run out of bits, even though each number has an
> >> infinite supply of them?
> >
>
> Tony Orlow wrote:
> > So what? There are larger infinities and smaller infinities. Haven't I been
> > consistent in saying even one bit makes a difference to an infinity, and even
> > one additional element? When you add two digital numbers of a given length,
> > what are the chances that the sum will be one bit longer? about 50/50. What's
> > wrong with that? Does it work TOO much like finite math for you?
>
> The "in-" of "in-finite" doesn't mean anything to you, does it?
> It's pretty clear, at least to most of us, that "infinite" means
> "not finite", and thus that "rules of finites" are not the same as
> "rules of infinites". But you don't get that.

Infinite numbers are numbers, like finite numbers, and any rules that apply to
numbers in general apply to both. So, the differences between finite and
infinite are balanced by the similarities between all quantities. If we can
establish rules that apply to all numbers, then that overrides the distinction
of finiteness, barring any conflict. So, the goal should be to maximize the
common properties, and minimize the distinctions necessary between the two, to
have the most consistent system possible.


>
> When you say a number like x = 1:1010...1010 has N digits,
> where N is supposedly infinite, this means that 2x has N+1
> digits and x^2 has 2N digits, etc. What good is N for, then?

For comparing sets formulaically, when they cannot be directly measured due to
infinity.

> It's completely arbitrary and essentially meaningless, since it
> doesn't really tell us anything useful about infinite numbers.
> N might just as well be finite, for all the good it does you.
> And indeed, that's how you treat it.

Indeed, as I've said, it is a type of variable which can take of finite or
infinite values, given the strict identity function between position and value
for each element in the set, which affords it inductive existence beyond the
finite. It allows us to say that, for S mapped from N given f(x), and g(f(x))=f
(g(x))=x, the size of S is g(N), and more specifically, within range [x,y], S
contains floor(g(y)-g(x)) elements. This allows us to compare finite and
infinite sets ina consistent manner. I can't say it any simpler than this.

>
> But that simply reflects your whole problem with standard math.

My problem is a set of axioms which lead to erroneous conclusions.

> You don't really seem capable of comprehending what "infinite"
> really means. Oh sure, you say things like "never ending", and
> "continues forever", and "unending supply", which implies that
> you might just get it. But then everything else you say makes it
> obvious that infinity is just a really big finite number to you.
> I can only conclude that, as bright as you seem to be, you
> cannot concieve of anything being truly infinite.

And it doesn't even remotely occur to you that there might be more than one
perspective, besides your arbitrarily chosen set of axioms and all its
variations? It doesn't make sense that a set with a range of values might
require that range for proper measurement, even if that range is infinite? I
mean, you are saying my infinities are not really infinite, while I am saying
your countable infinities are not infinite. I say your general limit ordinal
approach is flawed, and you say my value range approach is flawed. they are
both attempts to put some kind of label at infinity with a concept that we can
build on. Aleph_0 is not infinite in my book, but 1:000...000 is, since it has
a 1 bit in an actually infinite position. The time has come for math to get
back to the study of quantity.

>
>

--
Smiles,

Tony
http://www.people.cornell.edu/pages/aeo6/WellOrder/
.

Relevant Pages
Loading