Re: infinity

David R Tribble said:
> David R Tribble said:
> >> But you were talking about a binary number with an infinite number
> >> of digits. Which is an ill-defined concept, since N contains only
> >> finite natural numbers; there is no infinite natural "number".
> >>
>
> Tony Orlow (aeo6) wrote:
> >> That is your opinion, based on a flawed proof.
>
> >> 2. You further assert that there exists an infinite natural number,
> >> which we'll call W.
> >
> > I assume no such thing. One can add 1 and get W+1. One can have W-1 and W/2.
> > So, you lost me there. You claim all (countable) infinities are the same. I
> > reject that notion.
>
> >> Did I break any of your rules?
> >
> > Yes. You declared all infinities equal, assumed you had some smallest
> > infinity,w hich is impossible, and then claimed W-W=0, which is hardly
> > agreed upon, and generally considered to be undefined, because infinities
> > actually DO come in different forms.
>
> Well, I agree that my math was bogus, but I thought I was following
> your rules when I wrote it. Apparently I wasn't.
That's okay. I DO consider 000...001:000...000 to be a different number than
000...000:111...111, although their ratio is essentially 1. It's like the
difference between 0.111... and 1.000...
>
>
> It only took about 50 posts, but now I think I understand where you're
> coming from. You deny that any countably infinite set A can be put
> into a one-to-one correspondence with any another countably infinite
> set B. You don't believe that there is only one countably infinite
> set cardinality (aleph_0), but that there are different sizes of
> (countable?) infinities.
Well, I do believe one can form bijections between countably infinite sets, but
I don't agree that a bijection between two infinite sets makes them exactly
equal. The mapping function describes the relationship between the infinite
sets. As a very simple example, we map the naturals to the evens using a
mapping function f(x)=2x. Taking the inverse of the mapping function, we get
that the set of evens is N/2, or 1/2 the size of the set of naturals. So, the
second part of your understanding is correct. There are, in my opinion, a full
spectrum of countable infinities.
>
> In other words, we cantorians believe we can prove that each ball
> that is removed from the vase can be matched one-for-one with every
> ball that was added to the vase, even though it *seems* that there
> are nine times as many balls added than removed. But you believe
> that the sets are both infinite but have a different number of
> elements.
Exactly.
>
> Since you are making a claim that flies in the face of last 120
> years of mathematical thought, it seems only reasonable to expect
> you to provide more convincing arguments for your claim.
>
> -drt
>
>

I am not sure what would convince you, since it is accepted in this area that
results are bound to be "unintuitive". But, common intuition says that a proper
subset is always smaller than its superset, as is true with finite sets. So,
given this very basic rule of thumb, it seems ridiculous intuitively that there
are as many evens as naturals, since every even is a natural, but only half the
naturals are even. Similarly, it seems wrong to say that the number of reals in
[0,1) is the same as the number of reals in [0,2), since the second includes
all the reals in the first, plus an additional infinity of them. While the
Cantorian bijection-implies-equality approach is simple and neat in ways, it is
in contradiction to basic intuitions about proper subsets, so it seems to me
that an alternative is needed that satisfies intuitions better and avoids the
many absurdities that are derived from Cantor's methods. It has seemed to me
that the major objection to this effort is that it is simply not possible, and
yet, given a few basic rules from areas such as algebraic formulas for mapping
functions on quantities, properties of symbolic systems, trees, Turing
machines, and other recursive and therefore infinite systems, it is possible to
derive methods that provide more exact comparisons between infinities. Of
course, the properties of these different constructs differ somewhat, so a
single rule will not cover them all. I believe the desire to have a single
simple rule for dealing with all infinities is rather unreasonable, and a
better goal is to have a small set of rules that deal with each of the infinite
constructs, and ways of combining them for hybrid situations. Indeed, I started
this effort several months ago here based largely on intuition, having fallen
into it after discussing what seemed like everything under the sun, and having
formed my opinion so 25 years ago and decided to explore other approaches to
infinity. In discussing this here, and in being pressed for exact
justifications, my position is not only more refined, but I have developed at
least the beginnings of these alternative approaches. Certainly, it is not
easy, but I am afraid to say that I have always found this area of math to be
mistaken, and consider the study of zeroes and infinities to be not only a
mathematical one, but one that has philosophical implications at the highest
levels, and one that needs to be explored in detail to attain a real
undertanding of the universe as a whole. After all, the wellspring of Zero,
India, soon came to the concept of Infinity, and it is no coindidence, in my
opinion, that India is also the source of the first modern concepts of God.
Now, that may not be a proper justification for you, depending on your
spiritual stance, but hopefully the satisfaction of our intuitions
mathematically, and an end to this ongoing battle over absurd results, would
free up a lot of time on both sides, and hopefully open the door to more
productive exploration of the limits (or lack thereof) of numbers.

Sorry for the long-winded response, but I guess I should make my personal
thoughts known on the subject, so you know where I am coming from, and not just
think I am being a troll or a purposely annoying crackpot. Whew! Did that help?


--
Smiles,

Tony
.

Relevant Pages

  • Re: Why does Cantor a target for cranks?
    ... relativity and quantum ... intuitions may be offended in people, ... integrated approach toward the infinite, ... characterizes the naturals, is violated by the ...
    (sci.math)
  • Re: Why does Cantor a target for cranks?
    ... relativity and quantum ... intuitions may be offended in people, ... more cohesive and integrated approach toward the infinite, ... count=max, which characterizes the naturals, is violated by the von ...
    (sci.math)
  • Re: Why does Cantor a target for cranks?
    ... certainly runs counter to intuitions which some feel ... integrated approach toward the infinite, ... half the naturals ... integration between the various areas of mathematics. ...
    (sci.math)
  • Re: Why does Cantor a target for cranks?
    ... certainly runs counter to intuitions which some ... integrated approach toward the infinite, ... you can draw a bijection between, say, the reals ... characterizes the naturals, is violated by the ...
    (sci.math)
  • Re: Why does Cantor a target for cranks?
    ... yet it certainly runs counter to intuitions which some feel ... more cohesive and integrated approach toward the infinite, ... half the naturals are even, and all the evens are natural.<< ... seek greater integration between the various areas of mathematics. ...
    (sci.math)