Re: how

Am 25.04.2024 um 02:17 schrieb Moebius:

Am 24.04.2024 um 22:36 schrieb Chris M. Thomasson:
 

I can say that, well let create a symbol... (XYZ) hold all of the natural numbers. Therefore (XYZ) + 1 is already in (XYZ), fair enough?

 I have to admit that I don't know what you are talking about.
 We have the symbol "IN" to denote the set of (all) natural numbers.
 Now: For all n e IN: n + 1 e IN.
 Is that what you mean? :-)
 How about:
 "I can say that, well let introduce a symbol... [that denotes the set of all natural number, say "IN".] [The set] IN hold all of the natural numbers. Therefore n + 1 (with n in IN) is already in IN, fair enough?"
 ?
 Yeah, fair enough. :-)
 Though it's not about the symbol (say "IN"), but about the set (it denotes).

On second thought... We might DEFINE
     IN + k := {n + k : n e IN}  (k e IN).
In this case we might indeed state:
     IN + 1 c IN
(since it's true).
So we might state: "I can say that, well let introduce a symbol that denotes the set of all natural number, say "IN". Then the set IN contains all natural numbers and the set IN + 1 (sse definition above) is a subset of IN, fair enough?"
Yeah.