Re: Well Ordering the Reals

Tony Orlow says...
>
>Daryl McCullough said:

>> What you are calling an "actually infinite set" is what everyone
>> else would call an "uncountably infinite set". With that interpretation,
>> your claim is correct: It is impossible to enumerate all the elements
>> of an uncountable set. That's why it's called "uncountable".
>>
>> On the other hand, what you call a "finite but unbounded set" is what
>> everyone else would call a "countably infinite set".

>As I pointed out in the post you responded to here (which part was
>snipped) the Wolfram definition of "enumerate" mentions nothing
>about finiteness.

What you are calling "finite naturals" is what everyone else
calls "naturals". To enumerate a set means to set up a correspondence
between that set and the naturals. An enumeration means the same
thing as a sequence, which is a function whose domain is the
naturals (what you call the "finite naturals").

>Got a reference which does? The reals are countable.

In standard terminology, a set S is countable if there is
a surjection f from the naturals (what you'd call the "finite
naturals") to S. A surjection f from N to S is a function such
that every element of S is in the image of f.

What that means is that a set is countable if it is possible
to order the elements of the set such that every element has
finitely many predecessors (elements that come "earlier" in
the ordering). As I said, standard terminology uses the term
"countably infinite set" to mean approximately what you mean by
"finite unbounded set".

The reals are not countable, by this definition.

>Was Hilbert a crackpot for suggesting that this may be the case?

You are getting confused. Hilbert was not talking about enumerations,
he was talking about well-orderings. A well-ordering on a set S is a
binary ordering relation r that is total, transitive, and well-founded.
A relation r is said to be well-founded if every subset S' of S has
a smallest element, according to ordering r. For every well-ordering
there is a corresponding ordinal, which means that there is a bijection
m from S to a set of ordinals such that

r(x,y) -> m(x) < m(y)

Hilbert wondered whether the reals could be well-ordered. He definitely
didn't wonder if the reals could be countable. He certainly knew that
they weren't countable (according to the standard definition of "countable").

--
Daryl McCullough
Ithaca, NY

.

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