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 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?

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?

(I note you deleted the cumbersome and vague definition.  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.)

Can you even prove that 1 is in N using your definition?

Nothing on this (of course).

>
The next lines show it. Aren't you ashamed?

Of course not.

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.

The axioms say that 1 is in M (I think you mean that it is in many
possible Ms) and that N is a subset of any M meeting the two axioms.  At
least that seems to be what you wrote.
Please prove that the subset you call N includes 1.  There are lots of
sets that are subsets of every possible M, and many don't include 1.

>
I told you already that I have written my book for intelligent
students. That means not to repeat the obvious. If ℕ should not obey the
conditions put on M, then the two axioms would be ado about nothing. An
intelligent reader understands that.

...

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 and will assume you can't have meant what you wrote.  That's
an odd strategy for an introductory textbook.  Are there any other
places where you have written nonsense for the reader to spot?

Anyway, now you know you can't even prove that 1 is N as you define it.

I am curious, though, why you did not ask one of your intelligent
students to write a correct definition for one of the revised editions.
Do you think it's a good plan to have nonsense definitions in a textbook
and hence you kept it in, or did you not know it required the reader to
assume what you should have defined until just now?  Surely someone has
pointed this out before now.

That is the usual way in mathematics and logic:
Given A it follows B. That is called an implication.

So write the proof correctly, stating the assumptions and the
consequences that follow.  That way the reader can tell if, maybe, one
or more of the assumptions need to be rejected.

>
All natural numbers of Cantor's set ℕ can be manipulated collectively, for
instance subtracted: ℕ \ {1, 2, 3, ...} = { }. Here all have
disappeared.

What definition of N do you want your intelligent readers to assume?
Presumably you don't want then to assume one Cantor uses or every number
in {1, 2, 3, ...} could be proven to be in N and therefore none would be
in the difference.

(I know it's hard because you've told me that WMaths can't define set
membership, difference and equality rigorously.  Otherwise you could
prove WMaths most surprising conjecture: the existence of E and P such
that E ∈ P and P \ {E} = P.)

--
Ben.