Re: cantorian algebra

G. Frege wrote:

On Tue, 28 Aug 2007 09:26:11 +0200, Han de Bruijn
<Han.deBruijn@xxxxxxxxxxxxxx> wrote:

A proof that relies on things that cannot exist, like aleph's or omega's,
should be considered as invalid.

Then -following your dogma- every proof in mathematics should be
considered as invalid. After all, NO mathematical object exists (in a
spacio-temporal sense).

I'm not rigourous as that ..

"If we don't see infinity in the physical world around us, then where
do we see it? Why, in our heads, of course. Actually, we see all of
mathematics in our heads. We may see three airplanes or three apples
in the physical world, but the abstract notion of "3" does not exist
in the physical world -- it only exists in our minds. The notion of
"3" is simple enough, and is an abstraction of enough concrete
objects, that there is little chance of our disagreeing on the notion.
Our conversations seem to suggest that the "3" in my head is very much
like the "3" in your head (though we will never be 100% certain of
that). But more complicated notions such as infinity, less grounded in
physical reality, are harder to explain; it is harder to be sure that
we are successfully conveying a concept from the inside of one head to
the inside of another."

http://www.math.vanderbilt.edu/~schectex/courses/thereals/potential.html

Good references anyway. Thanks!

Back to your idiotic claim from above:

Actually, Cantor proved by using his concept of and equinumerousity
(of infinite sets) that there are (uncountable) infinitely many
transcendental numbers. ~~~~~~~~~~~~~~~

"Cantor's first results on cardinality appear in an 1874 paper where
he introduces the 1-1 correspondences, and uses them to show that the
algebraic numbers can be put in 1-1 correspondence with the natural
numbers; and in the same paper he proves that such a correspondence
between any interval of reals and the natural numbers is not possible.
Thus he has a new proof of Liouville's 1844 result on the existence of
[...] transcendental numbers [...]."

http://www.math.uwaterloo.ca/~snburris/htdocs/scav/cantor/cantor.html

Han de Bruijn

.

Relevant Pages

  • Re: Nested Accordion Menu
    ... Let's hope that all this is just invalid HTML and not cluelessly written ... I would recommend not to omit those two tags. ... it may not be obvious where the `head' element ends and the ... may be transformed client-side with XSLT. ...
    (comp.lang.javascript)
  • Re: Nested Accordion Menu
    ... I strongly recommend AGAINST nested menus ... Let's hope that all this is just invalid HTML and not cluelessly written ... I would recommend not to omit those two tags. ... it may not be obvious where the `head' ...
    (comp.lang.javascript)
  • Re: cantorian algebra
    ... should be considered as invalid. ... "If we don't see infinity in the physical world around us, ... mathematics in our heads. ... Our conversations seem to suggest that the "3" in my head is very much ...
    (sci.math)
  • Re: A case for HTML as a programming language
    ... >> The validator.w3.org does not consider the empty file valid HTML. ... suspect that not only is NOT required, it's actually invalid ... put <head> outside an section. ...
    (comp.programming)
  • Re: Are Midland Mainline staff instructed to lie about their names?
    ... Roughly half to two-thirds of the correspondence I produce at work goes ... out with my head of department's printed signature on it, ...
    (uk.railway)