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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.logic
Subject: Re: Simple enough for every reader?
Date: Mon, 2 Jun 2025 13:36:27 +0200
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On 02.06.2025 03:56, Ben Bacarisse wrote:
> 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:

The following is the explanation of being definable. All numbers which 
ca be reached by induction are definable. To know this property is 
necessary in order to distinguish them from dark numbers.
> 
>> 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.

It is clear that ℕ has the same properties as all the M because 
otherwise ℕ could be the empty set. Then the axioms would be void. Every 
intelligent reader would recognize that this cannot be the object of the 
paragraph.

> 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

Not by stupid or envious readers. All others can distinguish definable 
and undefinable numbers, and, by the FISON, can relate it to inductive 
construction.

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

You are wrong.

>> You are a dishonest liar.
> 
> Then just prove that 1 is in N as you define it.

See above.
>> 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?

The reason is clear: Not all natural numbers of Cantor's set can be 
individually defined:
Since all natural numbers can be reduced to the empty set by subtracting 
them collectively,
ℕ \ {1, 2, 3, ...} = { }
they could also be reduced to the empty set by subtracting them 
individually - if this was possible. But then the well-order would force 
the existence of a last one. Contradiction.

Therefore the bijection is only possible for the potentially infinite 
Peano-set ℕ_def. They cannot all be manipulated because every subtracted 
subset has successors.

Regards, WM