I see you persist in using your own private meaning for 'function' -- quite unnecessarily, too, but what of that -- and you persist in misusing '1-to-1' and 'onto'.
It is an expanded meaning for 'function', where in addition to the term that for any input there must an output, there is also the term that there is input without any output.
This expansion is quite unnecessary in order to not missing what's really going in Hilbert's Hotel framework.
Let's put that aside, though. Instead, I'd like to know what you mean by 'cardinality', since that appears to be a private doronism as well. In particular, how do you make relative comparisons of the cardinality of two sets?
Cardinality is the size of a given set in a way that does not take into account its structure.
The relative comparisons of the cardinality of two sets, is done by the expanded function, whether a set has bounded amount of members or unbounded amount of members.
Once again:
By using the traditional definition of function ("some x in X has
exactly one y in Y"), we conclude (or we discover) that , for example, {1,2,3,4,5} and {9,7,6} do not have the same finite cardinality because there is no function from {1,2,3,4,5} to {9,7,6} in two cases, as follows:
Code:
{1,2,3,4,5}
↓ ↓ ↓
{9,7,6}
By using the non-traditional definition of function ("x in X has
at most one y in Y"), we conclude (or we discover) that , for example, {1,2,3,4,5} and {9,7,6} do not have the same finite cardinality because there are two functions from {1,2,3,4,5} to {9,7,6} that do not return any value from {9,7,6}, as follows:
Code:
{1,2,3,4,5}
↓ ↓ ↓ ↓ ↓
{9,7,6}
So in the case of finite cardinality, the traditional and non-traditional definition of function, provide the same results (or the same discoveries).
Now let us use the non-traditional definition of function in case of the set of all natural numbers (which has an unbounded amount of members), as used in Hilbert's Hotel.
The names of the rooms and the names of the visitors in that hotel are actually all the members of the set of all natural numbers (notated by N), and in order to see what really happens in Hilbert's Hotel let's play with the pairs' game, by using an expression of the form (
x,
y) as follows:
The outer "(" and ")" define Hilbert's hotel environment.
x defines the name of a given room in that environment.
y defines a room in that environment such that it can be without any visitor (notated by ()) OR with exactly one visitor (notated by
, where n is a placeholder for some visitor's name).
In the following pairs' game framework, where there is a function from rooms' names and visitors' names (such that both names are in 1-to-1 and onto from the names to the set of all natural numbers)
(1,(1))
(2,(2))
(3,(3))
(4,(4))
(5,(5))
...
is expressed by
1 → 1
2 → 2
3 → 3
4 → 4
5 → 5
...
which shows that |N| = |N|
---------------------------------------------
In the following pairs' game framework, where there is a function from rooms' names and visitors' names (such that both names are in 1-to-1 and onto from the names to the set of all natural numbers)
(1,(1))
(2,( ))
(3,(2))
(4,(3))
(5,(4))
...
is expressed by
1 → 1
2 →
3 → 2
4 → 3
5 → 4
...
which shows that |N| > |N|
---------------------------------------------
|N| > |N| is deducible only if the definition of function is:
x in X has
at most one y in Y
Such definition enables to define function even if it has an input but not any output (as seen, for example, in 2 → ).
---------------------------------------------
|N| > |N| is not deducible if the definition of function is:
x in X has
exactly one y in Y
Such definition does not enable to define function unless it has input and output (for example, 2 → is not deducible by such definition).
--------------------------------------------------
In other words, |N| > |N| is deducible only if the non-traditional definition of function is used.
