Re: Simple enough for every reader?

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.