Re: Well Ordering the Reals

Tony Orlow <aeo6@xxxxxxxxxxx> writes:

> David Kastrup said:
>> Tony Orlow <aeo6@xxxxxxxxxxx> writes:
>>
>> > David Kastrup said:
>> >> Tony Orlow <aeo6@xxxxxxxxxxx> writes:
>> >>
>> >> > So, countable means finite but unbounded, as I've said for months.
>> >>
>> >> No. It does not mean finite either which way. It does not mean
>> >> finite for the member values (in fact, it means nothing whatsoever
>> >> for the member values except that they are distinguishable), and it
>> >> certainly does not mean finite for the set.
>> >
>> > Essentially it does. A countable set consists only of elements
>> > with a finite number of predecessors.
>>
>> An infinite number of them, yes. According to which definition
>> does that make the set finite?
>
> The definition that says the set is infinite iff it contains an
> infinity NUMBER of elements. This is not the Dedekind definition,
> but the new, improved Orlow definition.

So {infinity} is supposed to be an infinite set? Or do you mean any
set X with an infinite cardinality(X) is supposed to be infinite?
What does it mean to have an infinite cardinality, then? Your
definition seems circular.

>> Fine. So you agree that the set of (finite) naturals as defined by
>> the Peano axioms is infinite as defined by Dedekind.
> Sure, as defined by Dedekind, but I think that definition has
> issues, because the set SIZE is not actually infinite, but
> unboundedly finite.

Unboundedly finite is an oxymoron.

>> >> And that means that _any_ unbounded (and any non-empty open)
>> >> ordered set is infinite.
>> >
>> > In the Dedekind sense, yes, I know. I don't need this repeated.
>>
>> But that's what mathematicians are talking about when talking about
>> "infinite sets", and you have not come up with any different
>> definition. So you have no business complaining.

> I most certainly have said that I believe that definitions can start
> with sets or with quantities, and have offered a set with an
> infinite QUANTITY of elements as the difnition of an infinite set.

Then you need to define "infinity QUANTITY". You are being circular
(and self-inconsistent, but that's another matter).

--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
.

Relevant Pages

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