You have no argument as long as you insist on inventing your own meaning for things.
EDIT:
In order to follow the philosophical view of traditional mathematicians, we do not use the term "inventing", but we use the term "discovering".
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 these definitions in case of the set of all natural numbers (which has an unbounded amount of members), as used in Hilbert's Hotel.
It is clear that the names of the rooms and the names of the visitors in that hotel have 1-to-1 and onto with all of the members of the set of all natural numbers.
Yet it is clear that there are rooms with no members, in the following case
(1,(1))
(2,())
(3,(2))
(4,(3))
(5,(4))
...
of the pair's game (as explained in
http://www.internationalskeptics.com/forums/showpost.php?p=9717505&postcount=2905), where no visitor left the hotel.
The traditional definition of function ("some x in X has
exactly one y in Y") can't provide a sufficient tool in order to conclude (or to discover) what really happens in the provided case above.
On the contrary, the non-traditional definition of function ("x in X has
at most one y in Y") provides a sufficient tool in order to conclude (or to discover) that there are more rooms than visitors (the function at room 2, in the example above, does not return any visitor's name, yet such a function is rigorously defined by the non-traditional definition of function).
Since both rooms and visitors are actually two aspects of the same thing, which is all the members of the set of natural numbers, we are able to conclude (or to discover) that |N| > |N|, or in other words (and without loss of generality) the cardinality of sets with unbounded amount of members, is not (mathematically) well-defined.
laca.