mac (01/29/83)
Hebrew Cardinal Numbers
The number Aleph[2] is the next cardinal greater than Aleph[1], and so
on for all the Alephs. The next higher infinity (infinite cardinal)
after integers (Aleph[0]) is Aleph[1]. The cardinality of reals is
sometimes called Beth[1]. It can be shown to be 2^Aleph[0] (using the
definition of infinite exponents) = cardinality(powerset of integers).
Beth[1] >= Aleph[1]. However, it cannot be shown that Beth[1] =
Aleph[1]. Nor than the contrary be shown. This is independent of the
axioms of set theory. The assumption that Beth[1] = Aleph[1] is the
"Continuum Hypothesis".
In general, defining Beth[0] =: Aleph[0] and Beth[i+1] =: 2^Beth[i],
the "General Continuum Hypothesis" ("G.C.H") states that (i>=0) ->
Beth[i] = Aleph[i].
Cardinality of n-Space
While it's true as Mr. King demonstrated that there are maps
R <-> R^n, there are no continuous maps. In other words, all such maps
involve similar trickery.
Ham Sandwiches
An anectode about Russel comes to mind. He once noted that a
contradiction anywhere in mathematics coule be used to prove anything
(use proof by contradiction). Someone challanged him "if one and one
are one, prove you're the pope". Proof was as follows:
(i) I am one.
(ii) The pope is one.
(iii) If one and one are one, the pope and I are one.
(iv) I must be the pope.