Re: Simple enough for every reader?

WM <wolfgang.mueckenheim@tha.de> writes:
(AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte
des Unendlichen" and "Kleine Geschichte der Mathematik" at Technische
Hochschule Augsburg.)

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.

But I don't often read what you write.  It is not too onerous to copy
the definition where it is important.

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.

I thought it might be something cumbersome and vague like that.  I can't
even tell if this is a inductive collection, so I must decline any
request to review a proof by induction based on it.

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.

Ah.  That is just an assertion on your part.  I will accept that you
believe it to be true.

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?

Of course not.

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.

The axioms say that 1 is in M (I think you mean that it is in many
possible Ms) and that N is a subset of any M meeting the two axioms.  At
least that seems to be what you wrote.

Please prove that the subset you call N includes 1.  There are lots of
sets that are subsets of every possible M, and many don't include 1.

[You might think that 4.1 and 4.2 uniquely define a set M of which you
state N is a subset, but that does not help you show that 1 is in N.]

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.

Whatever you are trying to prove it can't be by induction with a base
case you accept can only be assumed:

I'd like to see the base case proved.

>
It cannot be proved but only assumed.

That is the usual way in mathematics and logic:
Given A it follows B. That is called an implication.

So write the proof correctly, stating the assumptions and the
consequences that follow.  That way the reader can tell if, maybe, one
or more of the assumptions need to be rejected.

--
Ben.