Re: Simple enough for every reader?

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.

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.

 

You are actually prepared to state that N (defined by Peano) and N_def
(defined by your book) are the same and also that they are also not the
same?

>
You have not understood. They are the same. Both differ from Cantor's
actually infinite set ℕ.

 Ah.  That is just an assertion on your part.  I will accept that you
believe it to be true.

Find a natural number that belongs to ℕ_def but not to Peano's set or vice versa.

 

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.

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.

 [You might think that 4.1 and 4.2 uniquely define a set M

No, they define many sets M.

of which you
state N is a subset, but that does not help you show that 1 is in N.]

As I said that requires an intelligent reader recognizing that without ℕ obeying the axioms too the paragraph would be nonsense.

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.
Could all natural numbers of Cantor's set ℕ be distinguished, then this subtraction could also happen but, caused by the well-order, a last natural number would disappear.
Regards, WM