Re: infinity


Tony Orlow (aeo6) wrote:
> Randy Poe said:
> >
> > aeo6 Tony Orlow wrote:
> > > > But the set of all k can be Cantor-infinite even though each k in that
> > > > set is finite.
> > > Yes, even though you have a finite number of finite terms, the sum is Cantor-
> > > infinite. Doesn't that indicate a problem with Cantor's definitions/
> >
> > No. What's the problem? As you have repeatedly admitted,
> > there is no end to a list of the values in such a set.
> > Most of us have no problem calling a thing which is
> > unending "infinite", and have a great deal of trouble
> > understanding how anybody could apply the adjectives
> > "finite" and "unending" to the same thing.
> Even though there is no end to the possible values for a natural number, and
> yet you apply the label "finite" to natural numbers?

There is no end to the possible values. I apply the name
"infinite" to the possible values.

There is an end to every particular natural number. I
apply the label "finite" to each number, as the list of
its digits has an end in every base.

Why this confusion between an individual number and the
set of all numbers? When you say I apply the label "finite"
to "natural numbers", is your choice of language deliberately
ambiguous (do you mean "any particular natural number" or
"the collection of natural numbers"?) or do you really
not see the difference between each element and the
entire collection?

> This is the root of the
> problem here. I am only saying the apparently unending set is finite because
> you are claiming the apparently unending set of values is finite.

No, the unending set of values is not finite.

Each individual number doesn't have a set of values, it
has a single, finite value. When I say that a given
number is finite, I have not made a statement about
the set of values.

> You allow infinite naturals and this problem goes away.

No it doesn't. It just avoids discussing the properties of
the collection of finite naturals. It's a distraction.

> > Despite your faith-based assertions that you can't, the
> > actual proofs that you can are elementary.
> Faith-based?

Yes.

> I have offered nothing but numerical arguments

Each of which eventually ends up with a faith-based
statement, a leap beyond what is justified by the logic.

> and caclulations

<snort>

> > > If adding one natural to the set at a time can produce an infinite set by
> > > incrementing the set size repeatedly,
> >
> > It can't.
> Then the recursive definition of the natural numbers defines a finite set.

No, the definition of natural numbers does not require
that the collection be built one at a time. There is
nothing in the definition that makes such a requirement
on the set.

It DOES require that each ELEMENT of the set be reachable
by adding one to one a finite number of times. Again you
have gotten confused between the elements and the set.
The recursive definition defines the elements. Once the
elements are defined, the set is immediately defined as
the collection of everything that meets that definition.

This isn't your usual quantor dyslexia, it's some other
sort of element-set dyslexia.

> Be serious.

I'm quite serious. Which part did you think was a joke?

> > > then incrementing the values in it can
> > > just as easily produce an infinite value.
> >
> > It can't.
>
> Look up infinite series, for god's sake.

Be more specific. Tell me what part of "infinite series
for god's sake" implies that one can reach an produce
value by a process of incrementing.

Perhaps you should "look up infinite series for god's
sake" and familiarize yourself with what it means for
a series to diverge. One thing it doesn't mean is that
anything ever reaches an infinite value.

- Randy

.

Relevant Pages

  • Re: Galileos Paradox and the Project of the Reals
    ... the positive integers which I defined the other day. ... A finite real, then, may be defined as any finite natural, or any number between any two finite naturals on the real line, by subdivion of the unit interval. ... We can also construct a linear enumeration of the reals using powers as I suggested with the H-riffic numbers. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... If a quantitative set is mapped in ascending order from the naturals, with each increment in the domain, the range increases by some amount. ... you had said that the existence ... Like it's the number of unit intervals, and the number of reals in the unit interval. ... You are using a form of infinite induction, making a claim for an infinite set based on all finite initial segments of it. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... If a quantitative set is mapped in ascending order from the naturals, with each increment in the domain, the range increases by some amount. ... Like it's the number of unit intervals, and the number of reals in the unit interval. ... You are using a form of infinite induction, making a claim for an infinite set based on all finite initial segments of it. ... don't have a definition for an arbitrary set of its "standard ordering" ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... If a quantitative set is mapped in ascending order from the naturals, ... number of reals on the line. ... to the subsequent logic that claims such a set cannot have infinite values. ... standard orderings, since sets in general don't come with little tags ...
    (sci.math)
  • Re: infinity
    ... >>> surjection into a proper subset is a generally good definition. ... >> about the surjection definition of infinite, ... All finite naturals are finite, ... Constant relationships such as this can be taken to infinity, ...
    (sci.math)