Re: Well Ordering the Reals

David Kastrup said:
> Tony Orlow <aeo6@xxxxxxxxxxx> writes:
>
> > David Kastrup said:
> >> Tony Orlow <aeo6@xxxxxxxxxxx> writes:
> >>
> >> > Daryl McCullough said:
> >> >> Tony Orlow says...
> >> >>
> >> >> >Oh come on. That set has one element, not an infinite number. Do
> >> >> >you know what "number of" means?
> >> >>
> >> >> Nobody knows what *you* mean by "number of elements". They know
> >> >> what people *normally* mean, but you have explicitly rejected that
> >> >> definition.
> >> >
> >> > You HAVE no definition for number of elements for infinite sets. You
> >> > have cardinality instead, which is not a particular number of
> >> > elements, but an equivalence class that ignores actual numbers.
> >>
> >> It is an equivalence class, period. Numbers are not relative for
> >> establishing order among sets. It turns out that they are a
> >> convenient help with finite sets, though.
> > Right, so cardinality <> number of elements = set size.
>
> You are being disingenuous: you complain that you find cardinality not
> meeting your "number of elements" moniker without even defining what
> _that_ is supposed to mean.
Set size, the way I and many see it, should change when elements are added or
removed. When you add elements, the set size should increase, and when you
remove elements it should decrease. In absence of such behavior, I don't
consider transfinite cardinality to be a good measure of set size at all.
>
> >> > I don't reject bijections. I just consider them insufficient in
> >> > themselves to declare equal set size or number of elements.
> >>
> >> You came up with no sensible objection, though.
> >
> > I certainly have. It violates basic properties of sets, such as
> > removing elements implying smaller set size.
>
> How is that a "basic property"? Adding 1 to each element of a set
> that has numbers in a ring is a reversible operation, so it can hardly
> "imply" smaller set size. And yet adding 1 to each element of N is
> _exactly_ equivalent to removing 0 from the set, without adding any
> element to it.
Yes, because you do not consider what happens to the set at the other end. This
is another generalization from finite sets which should be preserved: when you
multiply all elements by a value, you should also be considered to be
multiplying the entire range by that value. I'd like to see that behavior
preserved.
>
> So it seems that removing elements does _not_ imply smaller set size,
> unless there is no 1:1 mapping back from the set with the removed
> element to the original set.
You know my position is that bijection alone does not mean same size for
infinite sets.
>
> > That's a pretty basic flaw, the way I see it. I mean, you have a
> > set of rationals dense in the reals, and a set of naturals sparse in
> > the reals, and you equate them.
>
> You are babbling incoherent nonsense again. Nobody is talking about
> the rationals or reals here.
I am, right now. You have one natural for every unit of value on the number
line, and an infinite number of rationals for every unit. Your theory, using a
convoluted bijection, declares these two sets equal, when there are obviously
infinitely times as many rationals as naturals. That makes absolutely no sense
to me. I don't agree with it.
>
> > I object to that.
>
> To an incoherent strawmen invented by yourself.
So, the set of naturals does not have the same cardinality as the set of
rationals?
>
> >> Different from what? You consider surjectability not a good
> >> measure of relative set size. But what you have come up as a
> >> substitute is mere handwaving and babbling. And however
> >> insufficient or dissatisfactory you may consider surjectability as
> >> a measure of set size, your handwaving and babbling is worth
> >> nothing at all.
>
> > Bull. I have offered the inverse function rule and N+S^L to deal
> > with quantitative and symbolic sets, respectively.
>
> Babbling and handwaving again. What does this have to do with N? And
> what framework for sizes would that be.
The inverse function rule is used for comparing any two formulaically defined
quantitative sets. N is the unit discrete infinity and all such sets are
compared with N. What do you mean by "framework for sizes"?
>
> >> There are no "actual numbers" describing the set size of the naturals.
> >
> > That is true. There is no exact number.
>
> This is nonsense. There is no "number", where "number" is an entity
> obeying the usual arithmetic laws. Period. But whatever moniker you
> want to assign to the set size, it is exact, since it is the size of a
> completely defined set.
Um, no, I disagree, and so does internal set theory.
>
> >There is only the identity function between value and count which
> >defines it. That is why it is the unit discrete infinity.
>
> More babbling and handwaving.
It's mutual.
>
> >> You have to invent new numbers for that, and if you do that, you
> >> have to define the exact meaning of those inventions with regard to
> >> sets.
> >
> > I already gave the above definition for N,
>
> You gave nothing whatsoever.
I did, despite your inability to understand.
>
> > and said it's not a number. When speaking of it as a number, it
> > actually serves as a variable which can assume finite or infinite
> > values.
>
> Babbling and handwaving. The set N is defined as a fixed and static
> entity. Its set size does not "assume" values. It is a fixed and
> static entity, too.
Wrong. It is defined recursively as a process of element generation. You cannot
name where it ends. Without an end, it is unmeasurable. You can declare aleph_0
to be the size of the set, but you will always have folks like me telling you
the system is incorrect. Sorry.
>
>

--
Smiles,

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

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