Re: Contradiction of bijections as a measure for infinite sets

On 04/03/2024 07:43 PM, Ross Finlayson wrote:

On 04/03/2024 08:59 AM, FromTheRafters wrote:

WM presented the following explanation :

Le 02/04/2024 à 17:51, Jim Burns a écrit :

On 4/2/2024 3:36 AM, WM wrote:

>

If your assumption leads to "no bijection",
but there is a bijection,
then your assumption is wrong.

>
My trick proves that there is no bijection.
Or could you explain why first bijecting n and n/1 should destroy an
existing bijection?

>
Your 'trick' only fails to demonstrate a bijection. Failing to
demonstrate a bijection does not mean that there is no bijection, only
that your 'trick' doesn't work to that end.

>
The only luck he's going to have is with something
like the Equivalency Function, the Natural/Unit Equivalency Function,
which only exists as the continuum limit of a very least amount
of numerical resources that involve the integer continuum,
and that as a function, isn't a Cartesian function in the usual
sense of being re-orderable, so that Cantor's proofs about the
existence and lack thereof of functions, become refined to
specifically being about Cartesian functions, quite simply.
>
The definition of function, while it deserves its own theory
altogether as for matters of relation, is among the most fluid
of the mathematical concepts, while that the special and extra-
ordinary formalism like EF, N/U EF, sweep, for matters of functions,
makes for that it's a nice result in axiomatic set theory's usual
descriptive set theory's usual world.
>
What do you think about that?
>
>

One might wonder "wouldn't that require some axiom or definition
about functions besides the usual expansion of comprehension of
the definition of functions as being any subset as from the
Cartesian product of two domains the any of the one or many
to one or many?" Yet, it's just sort of demonstrated and
just following the definition of infinite limit as is quite
usual already, then that it also arrives at a model of a
continuous domain [0,1] is also quite a spectacle.
Especially that it furthermore falls out of otherwise
the arguments for uncountability of the complete ordered
field without being contradicted, of course is required.