So, Doronetics has its own non-standard definition for mapping, domain, and range. Would you please post those unique definitions so that we might better understand your posts?
jsfisher,
Again you wish to get things only in terms of verbal_symbolic skills.
But in order to understand what is going on in Hilbert's Hotel argument, also visual_spatial skills have to be used.
By using verbal_symbolic AND visual_spatial skills on the following mapping
1 → 1
2 →
3 → 2
4 → 3
5 → 4
...
one enables to understand that there are more rooms than visitors even if the names of the rooms are identical to the names of the visitors (where emptiness is no-visitor (and therefore no-name), where no-visitor can't be considered as some form of visitor, exactly as no-member (for example in the case of the empty set) can't be considered as some form of member)).
By using verbal_symbolic AND visual_spatial skills at Hilbert's Hotel argument, one enables to conclude that the exact cardinality of an infinite set like N is not well-defined, because there is difference between the amount of the rooms and the amount of the visitors, even if there is 1-to-1 and onto between the names of the rooms and the names of the visitors.
More things about mapping, take for example the mapping between the members of N and the members of P(N), as done in |P(N)| > |N| Cantor's proof.
During this proof we check the identity between some N member and the members of some P(N) member, or in other words, the mapping from some N member to some P(N) member penetrates into the checked P(N) member in order to find an identity. The non-matched cases (where one of the cases is the mapping between some N member and the "content" of {}, where in this case the mapping is of the form n → ) enable us to define some P(N) member that is not in the range of any one of the N members, which enables us to conclude that |P(N)| > |N|.
So as you see, the mapping of the form x → is used also by Traditional Mathematics,
so you are invited to provide its verbal_symbolic-only definition.
Edit: Here is the standard formal definition (
http://en.wikipedia.org/wiki/Domain_of_a_function):
Given a function f:X→Y, the set X is the domain of f; the set Y is the codomain of f. In the expression f(x), x is the argument and f(x) is the value. One can think of an argument as an input to the function, and the value as the output.
The image (sometimes called the range) of f is the set of all values assumed by f for all possible x; this is the set {f(x) | x ∈ X}. The image of f can be the same set as the codomain or it can be a proper subset of it. It is in general smaller than the codomain; it is the whole codomain if and only if f is a surjective function.
A well-defined function must carry every element of its domain to an element of its codomain.
Please explain how this definition enables to conclude that there is no copy of n (which is a member of N) in {} (which is a member of P(N)), without using a mapping of the form n → , which actually enables to penetrate into {} in order to conclude that there is no copy of n in {}.
EDIT: A room does not disappear because it is empty, and so is the case about mapping 2 → , which is understood only if one uses also his\her visual_spatial skills.
)) Useful Maths.