Re: Well Ordering the Reals

David R Tribble said:
> Tony Orlow wrote:
> > I think if you claim to have an infinite number of finite elements, you need to
> > address what this means. Apparently it doesn't mean that there is any infinite
> > number of elements in the set, or it would have elements with an infinite
> > number of predecesors, and those would have infinite values. In any set of
> > successive naturals starting at 1, any largest element always equals the set
> > size, and any set size always equals the largest element. This is a constant
> > equality for every case proven inductively, and there is no reason to imagine
> > this breaking down at oo.
>
> So oo just acts like a really, really, really big finite number, then?
>
> If oo and all the other "infinite" numbers behave just like the finite
> numbers, then what is the difference between them? Why bother having
> two kinds of numbers at all if they act the same way under all of the
> same rules? Why not just say all numbers are naturals (or reals),
> and dispense with "finite" and "infinite"?
Because when you deal with the real expanse, or discrete successions that don't
end, you cannot measure them in the normal way where there are two identifiable
ends and a distance between them. In these cases we should be trying to extend
the concept to finite numbers in as consistent a manner as possible. There's no
sense in making things more complicated than they have to be. If some things
behave differently, that's to be expected, but if some things actually work in
the infinite case, that's a plus.
>
>
> > So, why do you have a set of successive naturals
> > starting at 1, with only finite values and no specific largest one, but with a
> > supposedly specific set size, which is infinite? This was the very first thing
> > about this theory that turned me from it years and years ago, and I still
> > believe this is the core issue with the theory.
>
> Yes, it is obviously the core issue of your unbelief of all of set
> theory.
Well, it sounds rather inconsistent, doesn't it, an unending set with a
particular size which is infinite, but no infinite values or positions in the
set? It simply doesn't make any more sense to me that the square circle.
>
>

--
Smiles,

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

Relevant Pages

  • Re: Well Ordering the Reals
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  • Re: Well Ordering the Reals
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  • Re: Two results of set geometry
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  • Re: Well Ordering the Reals
    ... > Tony Orlow wrote: ... >> where there IS a most significant digit. ... > But x and y have an infinite number of digits; ...
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  • Re: An uncountable countable set
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