Re: The non-existence of "dark numbers"

On 16.03.2025 21:08, Alan Mackenzie wrote:

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

No. The definable natural numbers strive to the smallest infinite number
ω. But they are never there.

 Whatever "strive" is supposed to mean.

That is supposed to mean what Cantor said. It is potential infinity.

N is defined as the smallest
inductive set.

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

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.

 

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 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.

 

JB who claims that lossless exchange causes losses but refuses to
explain this in detail, ....

 He's been trying to explain this over quite a few posts.

He answers evasive.

 

.... 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.
Regards, WM