Re: Cantor Confusion

On 16 Nov 2006 11:32:41 -0800, mueckenh@xxxxxxxxxxxxxxxxx wrote:


David Marcus schrieb:

Indeed. If people *object* to an axiom, that is philosophy.

But if people choose a set of axioms, that is what?

Everyone is welcome to choose their own axioms.

That's mathematics?

Of course.

And if people not decide to use an axiom, that is what?

Mathematics obviously. David says so.

But if people decide not to use an axiom, that is philosophy?

Of course. Philosophy(x)=disagree(David).

Would like to do. Please le me know which words are available in your
universe of discourse.

I told you several times that the terminology in any modern textbook is
fine. For some reason you do not like this answer.

I told you the terminology used in a modern textbook to show that
finished infinity is used there. For some reason you do not like to
understand it.

"Some mathematicians object to the Axiom of Infinity on the grounds
that a collection of objects produced by an infinite process (such as
N) should not be treated as a finished entity."

Regards, WM


What would be something that is "actually infinite"?

Read Cantor, he can explain it better than me.

e. An "infinite number" is a number other than the natural numbers.

An "infinite number" would be a number other than a natural number.

Are you agreeing or disagreeing?

I am astonished that you cannot understand simplest sentences. You seem
to have difficulties with conditional constructs. Should you ever
intend to study mathematics be prepared that such constructs will
appear quite frequently.

If an actually infinite set of numbers existed, and if neighbouring
elements had a fixed distance from each other, then the set must
contain an infinite number.

Is that a "no" or a "yes"?

Read again, simplified: If neighbouring elements have a fixed distance,
the answer is yes.
If neighbouring elements have not a fixed distance like the rational
numbers: the answer is no

Regards, WM

~v~~
.

Relevant Pages

  • Re: Ultimate debunking of Cantors Theory
    ... without the axiom of infinity. ... laid the foundations of the geometry associated with his name. ... mathematics lays it foundations with the axioms of set theory. ...
    (sci.math)
  • Re: Ultimate debunking of Cantors Theory
    ... But this thing about the axiom of infinity is ... The point is that if one wants to talk about such things rigorously, one has to set down axioms, in the same way Euclid laid the foundations of the geometry associated with his name. ... Modern mathematics lays it foundations with the axioms of set theory. ...
    (sci.math)
  • Re: Cantor Confusion
    ... especially the axiom of infinity. ... concrete in criticism. ... It is when people announce that the axiom ... Everybody is free to develop mathematics without the axiom of infinity, ...
    (sci.math)
  • Re: Implementable Set Theory and Consistency of ZFC
    ... This is still not correct as the Axiom of Infinity is an axiom of ZFC ... Set Theory, even less when the latter is supposed to allow the Naturals ... consistent within common mathematics. ...
    (sci.math)
  • Re: Cantor Confusion
    ... But if people decide not to use an axiom, ... axioms, that is mathematics. ... "Some mathematicians object to the Axiom of Infinity on the grounds ... If neighbouring elements have a fixed distance, ...
    (sci.math)