Re: The non-existence of "dark numbers"

On 17.03.2025 12:56, Alan Mackenzie wrote:

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.

The set of FISONs is an inductive set. But it is not ℕ because
∀n ∈ UF: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
The subtraction of the set of all FISONs all of which cannot empty ℕ cannot empty ℕ.

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?

"Es ist sogar erlaubt, sich die neugeschaffene Zahl ω als Grenze zu denken, welcher die Zahlen ν zustreben, wenn darunter nichts anderes verstanden wird, als daß ω die erste ganze Zahl sein soll, welche auf alle Zahlen ν folgt, d. h. größer zu nennen ist als jede der Zahlen ν." E. Zermelo (ed.): "Georg Cantor – Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 195.

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.

 

You clearly know the meaning of these words.

 And their meaninglessness is clear.

Your following statements prove that you understand the meaning.

You're doing a quantifier shift

Of course. Here it is justified since the subtraction of all FISONs which cannot empty ℕ cannot empty ℕ.

 And your N_def, as you have "defined" it, is satisfied by any
proper subset of N.

No, it is the union of FISONs.

Or in a different interpretation, N_def = N,

No. ℕ_def is a proper subset.

since
An e N, N\{n} is non-empty.

Here we use
∀n ∈ UF: |ℕ \ {1, 2, 3, ..., n}| = ℵo

Either you're incapable of writing
mathematically what you mean,

I did it frequently:
∀n ∈ UF: |ℕ \ {1, 2, 3, ..., n}| = ℵo
|ℕ \ {1, 2, 3, ...}| = 0

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;

Subtraction is 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.

You are unable to read or to understand. You criticise your reading or your incoherent thinking, not my writing.

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

Simply read what I write. It is ridiculous. I write: Obviously the subtraction of all numbers which cannot empty ℕ cannot empty ℕ. And you reply that a number is not an agent. No comment necessary.

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

Contrary to ℕ \ ℕ_def.

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?

Set subtraction is also used in English.
Regards, WM