Math Forum Discussions

Ben Bacarisse

Posts: 1,972
Registered: 7/4/07

Re: ? 533 Proof
Posted: Aug 4, 2014 11:33 AM

Plain Text

mueckenh@rz.fh-augsburg.de writes:

> On Sunday, 3 August 2014 23:48:21 UTC+2, Zeit Geist wrote:
>
>> You have Shown this For Sets of All Finite Cardinalities.
>> You have Not Shown that this is True of N itself.
>
> Nobody can show anything for N itself. All proofs in set theory apply
> all natural numbers only.

No, that is a distinction you've made up to get the results you want.
In set theory, properties can be proved about N itself. For example
whether it is or is not equal to some other set:

{1,2,3} =/= N
{ n c N | n > 3 \/ n < 4 } = N
image(f) = N where f(x) = x

You reject this (you've said that WMaths can't prove properties of such
sets) but at the same time you assert the functions like f(x) = x + 1
(from Z to Z) and g(x) = x^3 (from R to R) are bijections. This is why
you must never show how you prove that these functions are, indeed,
bijections. (Well, at least until you can re-define "bijection" so as
to avoid having to show that no elements are missing from the image.)

In set theory, any of the often cited enumerations of Q+ have the
provable property that image(e) = Q+ and domain(e) = N.

<snip>
--
Ben.