Re: infinity

>> I'm talking about ways to make Leibniz' infinitesimal
>> analysis, with its correct results, rigorous.

Yes, that can be done using Non-Standard Analysis, as pioneered by
Abraham Robinson. When I say 18th century uses of infinitesimals were
non-rigorous, I am referring to many of its inconsistent uses and
improper assumptions (some of which were noted at the time, such as by
Bishop Berkeley).

>> (What is mathematical rigor?)

Rigorous approaches in mathematics essentially got its start in the
19th century. Prior to that, the uses of infinity and the
infinitesimal were inconsistently, where seemingly correct answers were
embraced while logical paradoxes ignored. Without a framework of
logical progression behind it, steps were made without justification,
and there was no sense of certainty as to why a particular process
worked.

Bernhard Bolzano and Karl Weierstrass were two of the earliest
mathematicians to have a sense of nailing down the steps of Calculus
and Analysis. Removing the unnecessary appeals to infinitesimals, they
helped give us what we know of today as delta-epsilon proofs. Georg
Cantor helped clean up the fuzziness in our thinking with Set Theory,
marvelously polished off by Zermelo & Fraenkel with an axiomatic
system. Peano arithmetic is a logic axiomatic basis for early
mathematics. With Kurt Godel, limitations to our assumptions was put
into place by the incorporation of incompleteness and relative
consistency proofs. Today, mathematics is a very specific, very
rigorous world, in which circular arguments and fuzzy definitions have
no place. (This is primarily why cranks feel that they are being
conspired against, since they generally lack rigor.) Despite their
many posts, cranks typically are unwilling to be disciplined in their
thinking, relying on faulty intuition rather than proof.

>> 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.

It certainly does not imply that. Well-ordering the reals will use a
different <= operation than the normal ordering. What is required by a
well-ordering is any comparison operation (which we denote <= but is
not the usual less than or equal to) such that for any a,b in R either
a<=b or b<=a and that any non-empty subset of R has a "first" element
(in the sense of our new <=).

>> 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.

Of course you can have any number of such systems. If you believe you
have an alternative one, please list the axioms in your framework (and
any rules of inference if different from the usual). If your logical
system is something other than ZFC, please list this too. Anything
short of that is just mental fuzziness, and probably the source of your
incorrect thinking.

>> 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.

No that's just plain silly. It's like saying "if it's between any two
integers, it's an integer" and promote all the reals to integers? How
about redefining "rationals" to include the irrationals? The whole
point of definitions is to be able to distinguish between separate
concepts. That's why we call the integers something different than
reals or complexes. If you want only integers, you say integers. If
you can to include 1/2, pi, etc., you talk of the reals. If you wish
to extend to infinitesimals, then you speak of the hyper-reals (or
whatever your new axiomatic system produces).

>> Consider infinitesimals as point particles. The universe
>> is infinite: infinite sets are equivalent.

This is precisely what I am referring to when I speak of non-rigorous
thinking. Mixing crank cosmology with crank mathematics is quite
common for cranks, since they care more about some philosophical agenda
than they do in the rigorous pursuit of the truth. They hand wave away
uncomfortable ideas (or drown those who discover them, as the
Pythogoreans did), and embrace a naive simple-mided world view, without
the annoyances of multiple infinities or general relativity.

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

Oh physician, heal thine own mathematics first.

>> 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.

Ross, your passion for the study of these things is laudable.
Unfortunately, your lack of mathematical training leaves you at a
disadvantage. I often feel like I am trying to explain the difference
between red and green to a color-bind individual. Fortunately, this is
easily repaired. You might consider taking some evening courses in
mathematics and work your way up to these concepts. Your first heavy
analysis course would probably be the watershed point.

>> I tend to trust my mathematical intuition.

Unfortunately, this is your greatest weakness (the trust part, not the
intuition). Mathematicians require proof. Intuition is a great source
of initial investigation, but in the end, every step must be justified,
and every assumption either previously proven or used as an axiom in
the system.

>> So: well-order the reals.

Sigh. Didn't I do this one already? Here it is again <grin>:

Take any arbitrary 1-1 mapping F from the reals R to the power set of
natural numbers P(N). By the Axiom of Choice, we know we can
well-order P(N), so take any such well-ordering, <=. Define a <=
operation (obviously different from the standard one) on R such that
for a,b in R, a <= b whenever F(a) <= F(b). You have now well-ordered
R. Creating F and well ordering P(N) are left as exercises. :-)

Hope that helps,

Jonathan Hoyle

.

Relevant Pages

  • Re: Why? [was Re: Cantor`s powerset theorem is false?]
    ... there is no bijection between the reals and the naturals. ... a measure of size is not very useful in constructive mathematics. ... the infinitesimals and infinities were immensely easier to work with. ... Frege set theory, which was inconsistent. ...
    (sci.logic)
  • Re: Uncountable sets in CZF?
    ... naturals bijectively to the reals. ... relatively modern response to the vagaries of infinitesimals of Newton ... from the naturals to the reals was the definition of the Natural/Unit ... properties similar to EF in monotonically mapping. ...
    (sci.math)
  • Re: how to list all of the real numbers
    ... of an infinite set by the axiom of infinity as generally stated (that ... maps the first integer after zero to the first real number after zero ... in the total linear and well-ordering there of the reals, ... With regards to the notions of infinitesimals and where they exist the ...
    (sci.math)
  • Re: INFINITY Revisited
    ... > interval) additonal balls always remain to be added and to be removed. ... >> Any digit string that contains a decimal digit at digit position n for ... >> infinitesimals will surely not make the set any smaller. ... In order to "throw out infinitesimals to make the reals", ...
    (sci.math)
  • Re: An online source for a hyperreal primer?
    ... he's saying in the hyperreals. ... countably saturated extension of the reals.) ... Notice that these links focus more on the infinitesimals ... The Transfer Principle looks like voodoo mathematics to me. ...
    (sci.math)