Re: infinity

>> 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.

.

Relevant Pages

  • Re: infinity
    ... >>> Countably infinite sequence of iota sum to one. ... > Thus it cannot be of the same cardinality as the reals. ... then you can shrink the open intervals to infinitessimal size ... > Uncountable Additivity does not hold in Measure Theory. ...
    (sci.math)
  • Re: Rational numbers, irrational numbers: each dense in real numbers
    ... generated in a particular way thus that the open intervals (each ... If you're claiming an uncountably-infinite set of disjoint open ... of items in the *observable* universe almost surely is finite. ... Universe is finite or infinite, the number of subsets is two to ...
    (sci.math)
  • Re: Distinct linear orderings on Z
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    (sci.math)
  • Re: Cantors diagonal proof wrong?
    ... >> So you claim that all infinite sets have the same cardinality? ... >What I don't understand is what the notion of cardinality really is. ... on about, it's not mathematics, it's a confession of faith (and ... or on whether "axioms" in one system or another ...
    (sci.math)
  • Re: An uncountable countable set
    ... You agreed that it is not a cardinality ordering. ... 'infinite') in one ordering, which is not a cardinality ordering, is ... problem is that you spout about non-standard analysis while you haven't ... and mathematics that that book uses as that book is written ...
    (sci.math)