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 31.05.2025 02:02, Ben Bacarisse wrote:

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

On 30.05.2025 03:08, Ben Bacarisse wrote:

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

>

I thought it might be something cumbersome and vague like that.  I can't
even tell if this is a inductive collection,

>
It is obvious and clear. Do you know a case where a natural number can be
in it and cannot be in it? No. You can only curse. It is the same as
Peano's set. If you can't understand blame it on yourself.

Can you prove it is an inductive set/collection?

>
See my book. The set is defined by induction.

You are losing the plot.  Here is the definition you offered for the
"obvious and clear" idea of N_def:

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".

But if you really want to talk more about your junk definition of N...

If n is in it, then also n+1 is in it.

... then you fail.  Because that's true of all the M but not of N which
is an unspecified subset of them.  Neither the waffle nor the
technical-looking junk defines anything that is obviously inductive.

so I must decline any
request to review a proof by induction based on it.

>
Of course. There is no counter argument. So you must decline.

No, I decline because I don't know if it is an inductive set.  Do you?

>
Every mathematician knows that the definable natural numbers are an
inductive set.

Yes, and some can even write the definition of N correctly!  The point
at issue is that you can't.  Your two offers are rubbish.  One is a
dozen lines of waffle and the other is technical-looking nonsense.

(I note you deleted the cumbersome and vague definition.

>
It has been given to be understood. Now you have or have not understood. If
not, the further presence would not help, I assume.

I think it helps the reader to see your sleight of hand.  I put it back
so they can see your switched from my talking about one definition --
the waffle that obviously can't be used to prove anything -- to your
referring to the one that you think (wrongly) is technically correct.

If it really
were obvious and clear, I would have left it in to show the world how
wrong I was to call it cumbersome and vague.)

>
I can give you a simpler and shorter definition: Every n that can be
expressed by digits is definable.

So now we just have the problem that 1 is not provably in N as you
define it.

I see you've cut the incorrect definition and the claim that the axioms
directly say that 1 is in N because, presumably, you now see that they
don't.

>

As I said that requires an intelligent reader recognizing that without ℕ
obeying the axioms too the paragraph would be nonsense.

That's funny!  Yes, an intelligent reader will see you've written a junk
definition

>
You are a dishonest liar.

Then just prove that 1 is in N as you define it.  To help you, and so that
other readers can see the junk definition you published, here it is again:

 1 ∈ M (4.1)
 n ∈ M ⇒ (n + 1) ∈ M (4.2)
 If M satisfies (4.1) and (4.2), then ℕ ⊆ M.

How hard can it be to prove that 1 ∈ ℕ?  You can't, can you?  I know you
can't because you said the definition was written with intelligent
students in mind who will see that it's wrong and just assume that ℕ
meets the required axioms!

But that is not relevant.

But it is /very/ relevant.  Your published definition is junk and you
know it, so calling me a liar is all you have left.  You could, if the
definition were not junk, prove me wrong by using it to show that 1 ∈ ℕ
but you can't.

What definition of N do you want your intelligent readers to assume?

>
ℕ is Cantor's infinite set.

Surely that can't be right.  I thought your book is about potential
infinity, not actual infinity.  You pretended to be happy with your
incorrect definition because your intelligent readers would assume the
correct definition, but you don't want then to assume Cantor's infinite
set in your textbook, do you?

And just so readers can see how you duck and dive my line:

What definition of N do you want your intelligent readers to assume?

was /immediately/ followed by:

Presumably you don't want [them] to assume one Cantor uses

so you could have just said "yes" but then you would not have been able
to switch the discourse away from your textbook's error to the junk you
are pushing here.

--
Ben.