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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: Simple enough for every reader?
Date: Tue, 27 May 2025 15:27:38 +0300
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On 2025-05-26 13:44:06 +0000, WM said:

> On 26.05.2025 12:44, Mikko wrote:
>> On 2025-05-26 10:17:27 +0000, WM said:
>> 
>>> On 26.05.2025 02:52, Ben Bacarisse wrote:
>>>> WM <wolfgang.mueckenheim@tha.de> writes:
>>> 
>>>>> With pleasure:
>>>>> For every n ∈ ℕ that can be defined, i.e., ∀n ∈ ℕ_def:
>>>> 
>>>> I can't comment on an argument that is based on a set you have not
>>>> defined.
>>> 
>>> Can you understand my proof by induction?
>>> The resulting set is ℕ_def. (According to set theory however it is not 
>>> a set but a potentially infinity collection.)
>> 
>> A proof by inductin does not define. It proves.
> 
> Here it proves that the natural numbers accessible by induction are not 
> all natural numbers.
>> 
>> For example: if we assume that
>>  0 ∈ ℕ_def
>> and that
>>  ∀n (n ∈ ℕ_def → (n + 1) ∈ ℕ_def)
>> then we can apply induction and prove that
>>  ℕ ⊆ ℕ_def .
> 
> Then you get a contradiction because by induction the natural numbers 
> accessible by induction are not all natural numbers.
> {1} has infinitely many (ℵo) successors.
> If {1, 2, 3, ..., n} has infinitely many (ℵo) successors, then {1, 2, 
> 3, ..., n, n+1} has infinitely many (ℵo) successors. For every n that 
> can be defined.
> 
> Do you accept this proof?

No. A proof should start with a clear presentation of the premises.
Then a sequence of sentences should follow, each with an indication
of how they follow from the previous sentence, and which earlier
sentences of the proof are also needed for the inference. It is
easier to refer those earlier sentences if the sentences of the
proof are numbered or otherwise labelled. The last one of these
sentences should be what was intended to be proven.

-- 
Mikko