Re: The non-existence of "dark numbers"

WM <wolfgang.mueckenheim@tha.de> wrote:

On 16.03.2025 21:08, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

[ .... ]

N is defined as the smallest inductive set.

But that definition is impossible to satisfy. Sets are fixed, inductive
"sets" are variable collections.

Wrong.  An inductive set exist by the axiom of infinity.

But just how do you think inductive sets vary?  Do they vary by the day
of the week, the phases of the moon, or what?  Can you give two
"variations" of an inductive set, and specify an element which is in one
of these variations, but not the other?

A proof exists that they are there. ℕ_def contains all numbers the
subtraction of which from ℕ does not result in the empty set.

That's not a mathematical statement.

The numbers 1, 2, 3 are such numbers. They are elements of that set.

 It's meaningless gibberish.

You clearly know the meaning of these words.

And their meaninglessness is clear.  You're doing a quantifier shift
again.  And your N_def, as you have "defined" it, is satisfied by any
proper subset of N.  Or in a different interpretation, N_def = N, since
An e N, N\{n} is non-empty.  Either you're incapable of writing
mathematically what you mean, or you're deliberately writing sloppily.

Obviously the subtraction of all numbers which cannot empty ℕ cannot
empty ℕ.

"Empty" in this sense is meaningless.

You are not unable to understand the meaning. But you are dishonest.

I refuse to discuss things expressed in sloppy meaningless language, as
"empty" used as a verb here is.  A number cannot "empty" a set, because
the number is not an agent; it is not an operator; it is not a function.
Such sloppy language allows you to reason sloppily, and possibly to
derive falsehoods as if they were facts.

I think you are capable of expressing your thoughts in a mathematical
fashion.  I wish I could be sure, though.

I spent quite a few posts to you trying to get you to explain what you
meant by a number "emptying" a set.  You never gave such an
explanation.

I informed you several times:
ℕ \ {1, 2, 3, ...} = { }
The set is emptied collectively by all natural numbers.
It cannot be emptied by definable natural numbers.

That "emptying" is not done by a number.  It is done by the set
difference operator \ operating on two sets.

All you're saying in the above point is that N \ N = { }.  Well, duh.

Am I meant to assume that "a number n emptying the set N" means that
"there exists a set M of which n is a member, and N \ M is empty"?  If
that's what you mean, then please write what you mean from now on, rather
than using sloppy notions like "a number emptying" a set.

[ .... ]

.... and you who claims that the subtraction of all numbers which
cannot empty ℕ can empty ℕ

How dare you lie about what I have written!  I have never claimed
anything involving the crankish notion of subtracting a number from a
set causing "emptying", whatever that might mean.

You understand very well. You have seen that subtracting is a regular
notion.

Subtraction is a function on two numbers mapping to a number of the same
type.  What you appear to be talking about is "removal" of an element or
subset of a set from that set.  Do you really use the word "subtrahieren"
in German for this?

And again, I insist that you do not lie by distortion about what I have
written.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).