Re: The non-existence of "dark numbers"

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

On 16.03.2025 18:23, Alan Mackenzie wrote:

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

The definable numbers are  potentially infinite sequence. With n also
n+1 and n^n^n belong to it.

So, no answer.

But let's take the implications of your second sentence there, ".... with
n, also n+1 ....".  Since I think we'd agree that 0 is a definable
number, then we've just defined the "definable numbers" as the inductive
set.

Yes.

Thus the set of "definable numbers", N_def is N.

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

Whatever "strive" is supposed to mean.  N is defined as the smallest
inductive set.  I think your definition of N_def is also that.  So the
two must be the same.

This is another proof that "dark numbers" don't exist, one you cannot
disagree with without contradicting what you've previously written.

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.  It's meaningless gibberish.

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

"Empty" in this sense is meaningless.  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.  Clearly you have none.

I doubt I can be bothered any more to address the other falsehoods,
contradictions, and delusions in your last post.  But it's clear that
what you write about such topics is wrong in general, if not totally.

Obviously both of you are cranks.

You are the crank, the one who is trying to insist on the existence of
non-existent mathematical entities.  Both Jim and myself subscribe to
standard mathematics.  Both Jim and myself have degrees in mathematics.
You do not.

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.

.... 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.

It would seem an apology from you is called for.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).