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From: Ben Bacarisse <ben@bsb.me.uk>
Newsgroups: sci.logic
Subject: Re: Simple enough for every reader?
Date: Tue, 27 May 2025 00:57:45 +0100
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WM <wolfgang.mueckenheim@tha.de> writes:

> 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?

Not without knowing what the set N_def is, since the argument starts
"For all n in N_def".  I can't verify even the simplest statement that
might follow without knowing what N_def is.

> The resulting set is ℕ_def. (According to set theory however it is not a
> set but a potentially infinity collection.)

So you are not asking me to verify a proof at all but rather to accept a
definition?  One that starts from claims about the thing being defined?
And you think this is how maths is done?

>> Your textbook defies N
>
> It defines ℕ_def.

It claims to define N.  It's very poor form to tell students you are
defining N when you are not.

In another reply (please don't split threads -- you may have time to
discuss this stuff endlessly but I don't) you say:

>> Your textbook defies N (incorrectly)
>
> My textbook defines the classical natural numbers, ℕ, meanwhile more
> precisely called ℕ_def, correctly.

So when you write N and N_def you are referring to the same thing?  I
thought you were claiming there was some difference when you use those
symbols.  Please don't use N unless you mean the N that mathematicians
define.

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

Since you now claim that (contrary to what the textbook states) this is
not intended to be a definition of N (as real mathematicians use the
term) but rather of something you call N_def, I can't really argue with
it.  But is not very useful.  Can you prove that 1 is in N_def using
this definition?

-- 
Ben.