Re: Replacement of Cardinality

Am 15.08.2024 um 21:28 schrieb Jim Burns:

A definition isn't a claim something exists.

Yeah, if you define a "property", say,
       x is a /Mückenheim number/ iff x e IR & x * 0 = 1.
Using symols:
       WM(x) :<-> x e IR & x * 0 = 1 .
THEN we can prove:
      ~Ex(WM(x)).  (*)
But this "approach" does not work for constants (or "functions").
We may express (*) (using normal language) the following way:
      There are no Mückenheim numbers.
But this "approach" does not work for say, the "Mückenheim number" WM:
      WM = the smallest unit fraction ,
SINCE THERE IS NO /smallest unit fraction/.
Again, we may define the property /smallest unit fraction/:
      WM(x) :<-> x e SB & Ay e SB: x <= y.
     "x is a smallest unit fraction."
Then we can prove:
      ~Ex(WM(x)).
      "There is no smallest unit fraction."
HENCE we CAN'T define:
      WM = ix(x e SB & Ay e SB: x <= y)
      "WM is the smallest unit fraction."
The rules concerning "proper definitions" do not allow for such a definition.
Hope this helps.