On 17.03.2025 11:51, FromTheRafters wrote:
WM presented the following explanation :
On 17.03.2025 02:29, FromTheRafters wrote:
WM brought next idea :
I informed you several times:
ℕ \ {1, 2, 3, ...} = { }
When two sets are the same set, the difference set is the empty set.
When two sets are not the same set, the difference is not empty. This
is the case when the set ℕ_def of all numbers which cannot empty ℕ, is >> subtracted from ℕ
ℕ \ ℕ_def =/= { }
That is false if N_def is the same as N
True.
, which it is.
False.
In fact, every
element in a set is "defined" or else you could not find out whether or
not it qualifies for set inclusion.
ℕ is a set which is attained by the natural numbers. "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. Let Google
translate.
You seem to think that there is N_def followed by a fairy dust of
undefined elements which get left behind after some small sequence of
finite members (FISON) are 'removed' from N (leaving fairy dust behind
as a set) instead of 'constructing' a new set from 'the elements of' the
set of naturals.
So it is! Remember the harmonic series. It is infinite although all
defined numbers are separately collected in converging subseries.
Regards, WM
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