Re: Does the Calculus rest on Euclid?

David C. Ullrich wrote:

On Fri, 09 Jun 2006 11:27:19 -0400, Hatto von Aquitanien
<abbot@xxxxxxxxxxxxxx> wrote:
A real number is a set S of rationals with the following properties:

(i) S is non-empty
(ii) S does not contain every rational
(iii) If s, t are rational, s is in S and t > s, then t is in S
(iv) S has no smallest element.

Pi is not a member of the set of rationals.

No, it's not. The definition above does not require that
every real number be rational. A real number _is_ a certain
_set_ of rationals.

I do not accept your definition
of the real numbers.

Guffaw. It's not my definition, it's a version of one of
the standard definitions. Saying you do not "accept"
a _definition_ is silly.

Yes, by the definition above there _is_ a real number
commonly called pi. And it's irrational.

If a real number is a "set of rationals", and there is only one real number
designated by the concept indicated by the phrase "a real number", then, in
the case of pi either the set called pi is identical to the object called
pi, or we are avoiding (or more correctly, evading) the original topic.

No assumptions of continuity in sight.

To say that you can order things implies a concept of a continuum.

What?

Given the definition above, we _define_ x < y for real numbers
x, y by saying x < y if and only if y is a proper subset of x.

But you cannot communicate any of that with out some appeal to my a priori
sense of continuum. The very simple fact that I see these symbols ordered
on my computer screen involves my a priori concept of continuum.


As regards the axioms typically assumed for analysis, they are, by
implication, those of axiomatic set theory which is isomorphic to
symbolic logic.

Uh no, set theory is symbolic logic _plus_ some axioms.

Which can be stated in terms of boolean algebra.

Uh, no.

Don't you get the feeling that just _maybe_ you should _learn_
something about these things, before proclaiming that mathemtaticians
are all wrong?

Perhaps 'boolean algebra' may not be the best term to use. The distinctions
between 'boolean algebra', formal logic, propositional logic, etc. are not
always clearly maintained in computer science literature, and I would need
to carefully review the literature to draw clear distinctions.

What I am saying is that set theory begins with concepts which are identical
to those found expressed in terms of the primary logical connectives.

http://baldur.globalsymmetry.com/open-source/org/sth/math/logic.xhtml

It's really quite simple. If your system is logically consistent and
sufficiently powerful, then it provides a means of transitioning from any
given state (collection of statements forming an aggregate statement) to
another state defined by the rules for manipulating symbols. Any such
system can be codified in binary.

What is the geometry of a locally tangent space?



--
Nil conscire sibi
.

Relevant Pages

  • Re: Cantor Confusion
    ... The "number" pi is definitely a merely fictitious element of continuum. ... naturals, integers, rationals, irrationals, or reals. ... intergers and naturals are genuine. ... genuine numbers to the reals is tempting but not justified. ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... continuum is something every part of which has parts ... the so called 4th logical possibility is valid: The reals must ... not quantitatively compared with the rationals. ... Cantor ascribed two different infinite numbers, so called cardinalities, ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... continuum is something every part of which has parts ... the so called 4th logical possibility is valid: The reals must ... not quantitatively compared with the rationals. ... Cantor ascribed two different infinite numbers, so called cardinalities, ...
    (sci.math)
  • Re: Should oo+a=oo be abandoned?
    ... Fictive elements of the continuum cannot be reached ... so you've decided togo for the two-point compactification ... of the reals. ... DA1 is used to prove that the rationals are countable. ...
    (sci.math)
  • Re: Do reals really have to be genuine numbers?
    ... They form a continuum with the ... impossibililty to correctly illustrate all rationals. ... A second still free injection of the unit 1 ... (also like an infinite string 1.000...) ...
    (sci.math)