Re: Contradiction of bijections as a measure for infinite sets

On 04/05/2024 04:32 AM, Richard Damon wrote:

On 4/5/24 4:56 AM, WM wrote:

Le 04/04/2024 à 15:22, Richard Damon a écrit :

On 4/4/24 9:07 AM, WM wrote:

>

It doesn't, Bijections are always between two DISTINCT sets, not a
set and a piece of itself thought of as a set.
>

"In mathematics, a set A is Dedekind-infinite (named after the
German mathematician Richard Dedekind) if some proper subset B of A
is equinumerous to A. Explicitly, this means that there exists a
bijective function from A onto some proper subset B of A." Wikipedia.

>

Right, but that "Proper Subset" is considered as an independent item,
not as just pieces of the original set.

>
Nevertheless it is a piece of the original set.
>
Regards, WM

>
No, its ELEMENTS are part of the original set.
>
The set of Natural Numbers does not have as a member of it, the set of
Even numbers, only all the Even numbers as members of it.

One can contrive a model of the modularity properties of integers,
wherein according to the model of modularity, variously the parts
can have models building them according to being sets: that do
define membership as whatever relation there is, because set theory
has only one relation: and it's Elt, not Members.
Of course that's contrived, and the default most simple model
of integers is pretty simple, and the even integers are a subset
of it.
Set theory has only one relation and it's "Elt", "element-of".
The whole point of it being fundamental is that there's only
one relation, and then usually it's ordinary regular well-founded,
keeping things simple, then models of other things built on that,
implemented in their relations according to set theory's relation,
then labelling those and calling that "descriptive set theory".
Otherwise most usual sorts of objects are more proper in their
own theory where they're primary in their own theory, and for
example, model things of other relations by the relations they have.
Number theory, geometry, part theory, set/class distinction,
part/particle distinction, category theory, type theory, ...,
are a bunch of different theories with different primary objects
logically, with the idea that set theory is so simple that it's
fundamental, non-logically then those in set theory.
Function theory, ....