Re: Well Ordering the Reals

Daryl McCullough said:
> Tony Orlow says...
> >
> >Daryl McCullough said:
> >> Tony Orlow says...
> >>
> >> >I already defined finite, infinite and infinitesimal quantities, Daryl.
> >>
> >> For the record, give a definition of what it means for a set
> >> to have an infinite quantity of elements. The closest you have
> >> come is saying that 1/0 is infinite, but how does that apply
> >> to sets? What does it mean for a set to have 1/0 elements?
>
> >I said if it's not zero and not finite, which I explicitly defined, then it's
> >infinite, as a quantity.
>
> The question is what does "number of elements" mean
> (to you, that is).
>
> >I said this before, that the Dedekind definition of an infinite set as one
> >which injects into a proper subset is valid, so long as you do not limit the
> >elements in some way that causes them to only have finite indices in the set.
>
> That doesn't make a bit of sense. If the Dedekind definition of "infinite
> set" is valid, then it follows that the collection of finite naturals
> is an infinite set. If you say that the collection of finite naturals
> is *not* an infinite set, then you are rejecting the Dedekind definition.
> So what definition are you using?
I said, "so long as you do not limit the elements in some way that causes them
to only have finite indices in the set." By limiting the naturals to finite
values, which is unecessary, you have limited the indices, as you admit, to
finite values. No element has an infinite number of predecessors. Therefore, in
my mind, there is not an infinite number of elements.
>
> >The Peano axioms define an infinite set
>
> They define a Dedekind infinite set. But what basis do you have
> for believing that it isn't one of those funny "finite but unbounded"
> sets?
It is only finite if you restrict the value range to a finite value. You say
there is no infinite difference between any two naturals, so there can be no
infinite set ebtween any two naturals, so there is no infinite set. Like i
said, if you restrict the digit positions in the reals in [0,1] to finite
values, you also have not truly enuemrated the entire infinite set.
>
> >but when you restrict them to finite
> >values, you make it impossible for the set to actually include an infinite
> >number of numbers.
>
> What does it mean to have an "infinite number of numbers", if you
> don't mean Dedekind infinite?
It means elements with infinite indexes in the sequence, when linearly ordered.
It means there are elements with an infinite number of predecessors in the
order. It means there are pairs of elements separated by an infinite number of
other elements.
>
> >In general, an infinite set is defined by a recursive structure
> >wherein every element has at least one successor in the set, and
> >where the number of successor operations is actually an infinite
> >number
>
> What is an infinite number? Can you define infinite set without
> using the word "infinite"?
I think that is a risky proposition. What makes it truly infinite without some
infinite quantitative aspect?
>
> You talk about a "recursive structure" but what does that mean?
A structure defined using recursion, such that each element, except perhaps one
or more root elements, are generated by some set of rules from a previous
element or elements.
>
> A recursive *definition* is one in which the thing being defined
> is used in the definition. For example, you are defining some
> object X and your definition is in terms of a function f:
>
> X = f(X)
>
> I don't know what you mean by a "recursive structure". Do
> you mean anything that satisfies a recursive definition?
Yes, I think, where each element has at least one sucessor. It may have more,
as in the binary tree, but that can always be collapsed into a linear order.
>
> >and not restricted to finite values. If we allow all finite
> >iterations, but no infinite ones, then we have a boundless
> >set which is nonetheless finite in scope. I hope that
> >clarifies my position on the subject.
>
> No, it doesn't. What is an infinite number? What is a
> recursive structure? If you are going to redo all of
> mathematics, you have to figure out what your primitive
> concepts are, and define everything in terms of those.
That is true if I am really using the terms in some new sense. In these cases,
I don't think I am. Is there something that really strikes you as wrong in what
I am suggesting? Do you see a particular problem with this approach, besides,
of course, that I am an idiot?
>
> --
> Daryl McCullough
> Ithaca, NY
>
>

--
Smiles,

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

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