Re: Well Ordering the Reals

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?

>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?

>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?

>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"?

You talk about a "recursive structure" but what does that mean?

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?

>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.

--
Daryl McCullough
Ithaca, NY

.

Relevant Pages

  • Re: Well Ordering the Reals
    ... >>> to have an infinite quantity of elements. ... that the Dedekind definition of an infinite set as one ... If you say that the collection of finite naturals ... an infinite set is defined by a recursive structure ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... If the Dedekind definition of "infinite ... If there are some circumstances in which the Dedekind definition is ... I asked you what basis you have for believing that the Peano axioms ... Why do you think that a recursive structure yields an infinite ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... stevendaryl3016@xxxxxxxxx (Daryl McCullough) writes: ... > That's a recursive definition of the set S. Is S infinite? ... > Why do you think that a recursive structure yields an infinite ... you are falling into Tony math by confusing ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... then the Dedekind definition is not valid. ... >> define an infinite set (according whatever non-Dedekind definition ... >> of infinite you are using). ... Circular definition! ...
    (sci.math)
  • Re: Cantors proof
    ... They define ordinals and cardinals, ... > then use the Dedekind definition of infinite in the axiom of infinity ...
    (sci.math)