Re: Simple enough for every reader?

On 2025-06-05 20:36:45 +0000, WM said:

On 05.06.2025 09:32, Mikko wrote:

On 2025-06-04 17:32:28 +0000, WM said:
 

On 04.06.2025 08:43, Mikko wrote:

On 2025-06-03 13:17:57 +0000, WM said:

 

I had only to show that in Cantor's set theory proofs by arithmetic induction are possible.

 Which you didn't show.

 Cantor shows it.

 

That confirms my proof:
ℵo - 1 = ℵo
P[1]: {1} has infinitely many (ℵo) successors.
P[n]: {1, 2, 3, ..., n} has infinitely many (ℵo) successors.
P[n+1]: {1, 2, 3, ..., n, n+1} has infinitely many (ℵo) successors.

 So far good. But no P[n] -> P[n+1] and no induction.

 ℵo - 1 = ℵo.

 No -> there.

 If {1, 2, 3, ..., n} has infinitely many (ℵo) successors, then {1, 2, 3, ..., n, n+1} has one successor less, that is infinitely many (ℵo) successors.

 It is relevant to the extent that you cannot learn form it how a direct
proof should be presented.

 I need not learn it, because I did it.
 

The expression "subtracting them individually" should be represented
mathematically, e.g. a sequence. Informal expressions tend to lead
to bad proofs.

 This expression is represented by
 ((((ℕ \ {1}) \ {2}) \ {3}) ...) = { }

That may be good enough when you want to prove something that we already
believe anyway. But for a sufficiently rigorous proof of something else
the "..." should be replaced with something more mathematicsl.
--
Mikko