Re: Well Ordering the Reals

David Kastrup said:
> 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.
Oh come on. That set has one element, not an infinite number. Do you know what
"number of" means? If there is a box with n chickens in it, then the NUMBER OF
chickens in the box is n. That doesn't mean one of the chickens is NAMED "n".
Sheesh. And, no, I am not talking about "cardinality", but Bigulosity, or
actual set size, equal to the integral quantity of atomic elements present in
the set. This definition is not circular. Infinite quantities are defined in
terms of finite quantities, and infinite sets in terms of infinite quantities.
There is nothing circular about it, except when you bring your standard
concepts and mix them with mine. They don't play well together. So tell your
system to keep its axioms and definitions to itself, or my system will kick its
ass, and take its milk money. Cardinality, indeed! :D
>
> >> 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.
Not as moronic as an infinite set with a finite number of elements in it. In
your parlance, you call such finite sets infinite, which I refuse to do, since
they are not truly infinite.
>
> >> >> 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).
>
>
Oh Jimminy! I already did that many times. Okay, this is a new thread, so here
goes. For any real quantity x:

1) 0<x<=1 -> finite(x)
2) finite(x) <-> finite(0-x)
3) finite(x) <-> finite(1/x)
4) not(finite(x) or zero(x)) -> infinite(x)

No mention of sets. Of course, the operations need definition, but for the
purposes of distinguishing finite and infinite, they are assumed.

--
Smiles,

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

Relevant Pages

  • Re: Galileos Paradox
    ... Tony Orlow wrote: ... infinite quantities. ... measure with count in the infinite is the task here, ... subsets of the reals, and their quantities need not correlate. ...
    (sci.math)
  • Re: Galileos Paradox
    ... Eckard Blumschein wrote: ... No. Infinite quantities include e.g. an infinite amount of points. ... are NEVER mathematicians? ...
    (sci.math)
  • Re: Hardy-Weinberg law
    ... >> You cannot simply talk about ratios between two infinite quantities. ... *unless* you specify a sampling strategy. ...
    (sci.bio.evolution)
  • Re: infinity
    ... >>> about arithmetic with infinite quantities. ... >>> Being infinite and being finite are qualities that sets can have. ... > makes no more sense than conflating metres and grams. ... >> time singularity AT noon, ...
    (sci.math)
  • Re: Cantors circular "proof" that evens = integers
    ... you said that the proof was circular because a "bijection was ... Perhaps if you had thought more about the underlying mathematics, and less about dialectic "victories," you would have noticed that yourself. ... Doesn't EVERY website, book, and article state, categorically, that one of the differences between finite and infinite sets is that an infinite set is equinumerous with a proper subset of itself? ... Now, am I actually supposed to believe that the authors are claiming this NOT because the sets are infinite, but because of some set of axioms that is used along with the infinite sets? ...
    (sci.logic)