Re: infinity

Jonathan Hoyle wrote:
> >> You describe a misuse, but the results of infinitesimal
> >> analysis exactly match those of the integral calculus.
>
> Getting the correct answer does not justify the logic of the approach.
> 18th century infinitessimal use was never done rigorously, as they
> could be done today (although delta-epsilon proofs pretty much remain
> the standard).
>
> >> Countably infinite sequence of iota sum to one.
>
> Actually, this was one of the problems. Countable Additivity of 0's
> gives you 0, not 1. It's one of the reasons that the rationals on
> [0,1] are of measure 0, but the irrationals on [0,1] are of measure 1.
>
> >> When you well-order the reals, then via an extension of Cantor's
> >> first (proof of the "uncountability" of the real), nested intervals, a
> >> well-ordered set of nested intervals is generated with a cardinality
> >> of whatever well-ordered set you map to the reals.
> <snip>
>
> I am not following your line of reasoning, as you appear to be adding
> unnamed assumptions to jump from point to point. First of all, there
> are only a countable number of non-overlapping open intervals, since
> (as you point out) each open interval contains a distinct rational.
> Thus it cannot be of the same cardinality as the reals. Your lack of
> rigor and fuzziness of argument makes it difficult to understand what
> you are saying here, but I think your error is assuming that these
> nested intervals are non-overlapping. Since every open interval
> (regardless of size) is of positive length, it is impossible to have
> one interval per real without overlapping.
>
> >> I think countable vis-vis uncountable additivity is
> >> basically equivocation about non-empty measure
> >> "zero". They're infinitesimals, not zero, in the limit
> >> zero, but infinitesimals.
>
> Not in Standard Analysis they're not. If you wish to use Non-Standard
> Analysis, then you can shrink the open intervals to infinitessimal size
> to have non-overlapping coverage over the standard reals, but you still
> are left with the problem on the hyper-reals, as you cannot cover them
> without overlapping, as every infinitessimal open interval contains a
> hyper-rational.
>
> >> Measure theory basically says "from zero to one, the measure is
> >> one", but cardinality is not shown to matter because the cardinality
> >> of [0,1], [0,2] is the same, but their measure is different.
>
> Cardinality does matter, as your example demonstrates my point. As you
> correctly point out, [0,1] and [0,2] both have the same cardinality
> (uncountable), yet their measures are different. Therefore,
> Uncountable Additivity does not hold in Measure Theory. No one is
> claiming that Uncountable Additivity holds here. However, Countable
> Additivity does hold, and nothing you have stated here contradicts
> that. The fact that Countable Additivity holds while Uncountable
> Addivity does not, is an important distinction between infinite
> cardinalities.
>
> >> What I hoped you might mention would be any "demonstrably
> >> correct result" about measure zero sets dense in the reals.
>
> The demonstrable examples from Measure Theory (that I can think of) are
> those from Probability Theory, since these can be sampled and measured
> experimentally. (I'm sure there are wiser people out there who can
> come up with other examples.) Behavior of measure zero subsets will
> have no effect on the probabilistic outcome, so a "demonstration" of
> this would be one you cannot see. I realize that this doesn't help
> much. Certainly you can do thought experiments, but these are not
> demonstrable: Ask God to pick a random number from U(0,1); what is the
> probability that this number is algebraic? (Answer: 0)
>
> My use of demonstrable examples is to show that Countable Addivity does
> hold, whereas Uncountable Additivity does not. Failing to see this
> distinction is in part why many of the ancients were so wrapped up in
> paradoxes of infinity that they just gave up. Like you, they were not
> open-minded enough to accept that there are different sizes of
> infinity. (They had an excuse though, as they lived prior to Georg
> Cantor's discoveries.) Countable Additivity is such an integral part
> of modern mathematics, that many of the advances of this century would
> not have taken place without it. Knowing when you can add an infinite
> number of addends and when you cannot, may be the key to helping your
> resolve your internal paradoxes.

I'm talking about ways to make Leibniz' infinitesimal analysis, with
its correct results, rigorous. (What is mathematical rigor?)

Iota values are not zeroes, they're iota values.

About the extension of Cantor's first, another way of considering it is
that it illustrates a flaw in the reasoning of there being
uncountability of the reals.

Well-order the reals. That implies adjacent points in the normal
ordering of the reals, and Cantor's first can be seen to not hold.

You mention "Standard" and "Nonstandard" analysis, plainly I am saying
that there are non-"Standard" forms of analysis that are not
"Nonstandard", because "Nonstandard" does not include everything that
is not "Standard", nonstandard.

Andrej Bauer on FOM:
http://www.cs.nyu.edu/pipermail/fom/2005-October/009125.html

Hmm... "nilpotent" infinitesimals.

The reals are complete, there are everywhere and only reals between
zero and one. If it's between zero and one, it's a real number.
That's saying that infinitesimals are real numbers too.

Finding an actual experiment that would illustrate the difference would
be useful. Consider infinitesimals as point particles. The universe
is infinite: infinite sets are equivalent.

Will someone go tell the physicists there can't be a theory of
everything unless it's consistent and complete.

I'm glad you're willing to read and consider what I have to say instead
of dismissing it out of hand. I've carefully investigated theories of
the infinite and criticize them, and I say several of the exact same
things as I did before five or six years of research, and vigorous
debate, in the problem areas. I tend to trust my mathematical
intuition.

So: well-order the reals.

Ross

.

Relevant Pages

  • 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: 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, 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: Dial 999 for the real number line
    ... length n as initial strings followed by the expansion of pi after the nth ... long decimal expansion followed by an infinite expansion. ... part can be as long as we like, we have an infinite sequence of initial ... dense in the reals. ...
    (sci.math)
  • Re: Dial 999 for the real number line
    ... Keep in mind that whatever system of axioms you come up with will ... proofs of uncountablility of the reals, but that is not going to happen. ... I have a hang-up, it's about infinite sequences. ... My point is that mathematical existence has nothing to do with physical ...
    (sci.math)