Re: The reality of sets, on a scale of 1 to 10 [Was: The non-existence of "dark numbers"]

Am 03.04.2025 um 16:40 schrieb Alan Mackenzie:

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

On 28.03.2025 17:32, Alan Mackenzie wrote:

 

So tell us, O wise one, how many elements are there in {1, 3, 5, 7, 9,
...}?  And how many elements in {0, 4, 8, 12, 16, ...}?  Which of these
two numbers is bigger, and why?

 

|ℕ|/2 > |ℕ|/4.

 Start out with a set of natural numbers.  Multiply each member by four,
giving a new set.  You'd have us believe that the new set contains fewer
elements than the original set.  Baloney.

Consider the sets IN = {0, 1, 2, 3, ...} and IN* = {{n} : n e IN} = {{0}, {1}, {2}, ...}.
Mückenheim agreed that then |IN| = |IN*|, since this is "obvous" (for him), due to the "rule of construction" (WM) or whatever.
Then I told him, that IN actually is defined due to Zermelo, i.e. IN = {{}, {{}}, {{{}}}, ...} with 0 = {}, 1 = {0}, 2 = {1}, etc. (in general n' = {n}).
This means that (surprise!) IN* = {1, 2, 3, ...}.
But in Mückenheim's world |{1, 2, 3, ...}| = |IN|-1 =/= |IN| = |{0, 1, 2, 3, ...}|. Hence "actually" |IN*| =/= |IN|. Contradiction (one might think)!
WM: <bla bla bla> (i.e. crank talk).

Regards, WM