Re: Well Ordering the Reals

Daryl McCullough said:
> Tony Orlow says...
>
> >Daryl McCullough said:
>
> >> I've already given you my objection. Your ordering
> >> has infinite descending chains.
> >
> >That's because it's an infinite set, dingbat.
>
> You think every infinite set has an infinite descending
> chain? So you think that it is impossible to well-order
> any infinite set? Yet, you claim to have well-ordered the
> reals.
If one can, as you did, select an infinite element, can form a set by
considering elements finitely less than that element, then any "uncountable"
set has subsets which are "infinite" descending chains. If this countable
subset is considered an infinite descending chain, then how can one possibly
create a well ordering on ANY uncountable set?
>
> On the one hand, "well-ordering" implies that there
> are no infinite descending chains. On the other hand,
> your ordering has infinite descending chains. Together,
> these two pieces of information tell you that your
> ordering is *not* a well-ordering. Yet you claim it
> is a well-ordering.
I thought the set as a whole needed a first element. When you define
"infinite" descending chains using the finite naturals, well, that makes that
task on Hilbert's list simply impossible.
>
> What do you mean by that? You say it's a well-ordering.
> You admit that it has infinite descending chains. But
> the first statement contradicts the second. Why are you
> content with having self-contradictory beliefs?
I am not. Obviously, the concept of a well ordering on any "uncountable" set is
self-contradictory.
>
> >If that were a valid objection, then why would Hilbert and
> >Goedel even consider the idea?
>
> Because there is no (known) proof that the reals can be
> well-ordered, and there is no (known) proof that they cannot be
> well-ordered.
If one can take any infinite element, and count backwards as the finite
naturals, and call that an infinite descending chain, then there's your proof
that it can't be done, according to these definitions. Perhaps it's not a
"well-ordering" after all, but it is an important enumeration of the real
numbers. I suppose Ross's infinitesimal-based well ordering is not a well-
ordering either, since we can start at 1 and count backwards by iotas, and have
an infinite descending chain as well. Given your definitions, how can no one
have proven that it can't be done? It seems obvious.
>
> --
> Daryl McCullough
> Ithaca, NY
>
>

--
Smiles,

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

Relevant Pages

  • Re: Well Ordering the Reals
    ... >>> That none of my bits are in infinite positions, and yet the set covers all the ... >> You seem to be confused between bijection and well-ordering. ... > ordering on an uncountably infinite set without countably infinite descending ... It is not trivial to create infinite descending chains in an order ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... It's an infinite descending chain. ... >> proving that your ordering is not a well-ordering. ... >> Daryl McCullough ...
    (sci.math)
  • Re: Review of Mueckenheims book.
    ... infinite series of transpositions leave the order type unchanged is ... suppose there is a well-ordering of the continuum. ... shows the uncountability of a set by constructing another number of a ...
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  • Re: An uncountable countable set
    ... > finitely many or infinitely many transpositions of elements, ... transformations are only transformations that change order ... the infinite set becomes empty. ... That it is not possible to do so without well-ordering all elements. ...
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  • Re: Well Ordering the Reals
    ... >>mathematical notation, and therefore is not available to you in proving my ... > to use your notation to prove that it's not a well-ordering? ... I never used infinite bit strings. ... >>Can you provide some infinite descending chain using ...
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