On 1/11/07, Zefram <[EMAIL PROTECTED]> wrote:
Strictly, what we want for the case of zero AGAINST votes and more than
zero FOR votes is the limit of 1/n as n approaches zero from above.
I think that's aleph-0, but I'm not 100% sure. This does seem to match
your definition of an infinite hyperreal.
YDAFI, BYGIA.
Let R denote the real numbers and N the natural numbers. One way to
construct the hyperreal numbers is as follows. Consider the structure
R^N, the collection of all sequences of real numbers. This has a
natural ring structure as a product of copies of R, but unfortunately
it isn't a field, since it is full of zero divisors. The hyperreals
are a special quotient ring of R^N. In other words we say that two
sequences are ``the same'' if they satisfy a particular rule.
The next paragraph is quite technical, so skip it if you want.
The rule we choose is the following. First select a nonprincipal
ultrafilter on the natural numbers. An ultrafilter on N is a maximal
family of subsets of N which doesn't include the empty set, is closed
under taking supersets, and has the finite intersection property. It
is nonprincipal if the intersection of all the subsets is the empty
set. Zorn's lemma must be used to show the existence of such an
ultrafilter. With the nonprincipal ultrafilter selected, we now say
that two sequences (x_n), (y_n) are to be identified provided that the
set of natural numbers n such that x_n = y_n is exactly an element of
the ultrafilter.
Okay, so we've got a rule ~ for identifying sequences.
The structure *R := R^N/~ we get by identifying sequences of real
numbers according to this rule is called an ultrapower of R. There is
a theorem by Los (in TeX, \L o\'s) that says that the new structure *R
satisfies all the *first-order* axioms satisfied by the original
structure R. Since the axioms for a field are first-order, this
implies that *R is a field. Moreover, there is an embedding of R into
*R; a real number r is mapped to the (equivalence class of the)
constant sequence (r, r, r, ...).
The only axiom of R that is not first-order is the greatest lower
bound axiom. This is the axiom that says that any nonempty subset of
R which has a lower bound has a greatest lower bound. And in fact *R
doesn't satisfy this axiom. (If it did, it would be isomorphic to R.)
Unfortunately, Los's theorem doesn't guarantee this, it has to be
shown explicitly.
Luckily, this is not difficult to do. All we have to do is find a
number which is positive but smaller than any positive real number.
Here is one: the (equivalence class of the) sequence
(1, 1/2, 1/3, 1/4, 1/5, ..., 1/n, ...).
Since the sequence is decreasing, it's smaller than any constant
sequence of the form
(r, r, r, ...)
where r is positive. But since all its terms are positive, it's
bigger than zero. Hence it defines an infinitesimal, which we call h.
Since h is not zero and *R is a field, we can then define a rather
large hyperreal H by H = h^{-1}.
The original construction of *R was performed in the 1960s by Abraham
Robinson. The definition of Aleph-nought is classical and is due to
Georg Cantor. Briefly, Aleph-nought is the smallest infinite
cardinal. In general, a cardinal is the smallest ordinal of a
particular cardinality, although for Aleph-nought there's really only
one obvious choice anyway.
--
Michael Slone
