Re: Simple enough for every reader?

On 29.05.2025 02:25, Ben Bacarisse wrote:

WM <wolfgang.mueckenheim@tha.de> writes:

Every n that can be expressed by digits should be known to you.

 But the important fact, since it's /your/ proof, is what that means to
/you/ and I can not know that.

I have often published it.
Definition: A natural number is "identified" or (individually) "defined" or "instantiated" if it can be communicated such that sender and receiver understand the same and can link it by a finite initial segment to the origin 0. All other natural numbers are called dark natural numbers.
Communication can occur
- by direct description in the unary system like ||||||| or as many beeps, flashes, or raps,
- by a finite initial segment of natural numbers (1, 2, 3, 4, 5, 6, 7) called a FISON,
- as n-ary representation, for instance binary 111 or decimal 7,
- by indirect description like "the number of colours of the rainbow",
- by other words known to sender and receiver like "seven".
Only when a number n is identified we can use it in mathematical discourse and can determine the trichotomy properties of n and of every multiple kn or power n^k with respect to every identified number k. ℕ_def is the set that contains all defined natural numbers as elements – and nothing else.  ℕ_def is a potentially infinite set; therefore henceforth it will be called a collection.

You are actually prepared to state that N (defined by Peano) and N_def
(defined by your book) are the same and also that they are also not the
same?

You have not understood. They are the same. Both differ from Cantor's actually infinite set ℕ.

Can you even prove that 1 is in N using your definition?

 Nothing on this (of course).

The next lines show it. Aren't you ashamed?

 

1 ∈ M (4.1)
n ∈ M ⇒ (n + 1) ∈ M (4.2)
If M satisfies (4.1) and (4.2), then ℕ ⊆ M.
>
Of course no intelligent reader need be told that this ℕ = ℕ_def also
satisfies the axioms (4.1) and (4.2).

 But it seems you can't prove that 1 is in N, can you?

It requires a lot of stupidity or hate to put this question after seeing the axiom that 1 is in ℕ.

It should be
easy, should it not?

 It is simple using the correct definition, but
yours is junk.

The junk is in your head.

 

How you prove that {1} "has ℵo" successors.

>
I do not prove it

 But you need to.  It's is the base case in the proof you asked everyone
about.  You can't make a proof by induction by simply asserting things.

Of course. Based on the assumption that Cantor is right I can prove the existence of dark numbers. That is the usual way in mathematics and logic: Given A it follows B. That is called an implication.
Regards, WM