Bacle said:
Now, in our case, the standard reals R are M1, and the hyperreals are M2, so every
1st-order statement that holds in M1, would hold in hyperreals. If I understood you
well, you are going in the opposite direction.
That is right -- elementary equivalence is an "if-and-only-if" statement. This follows from the one-directional version because if any particular statement S holds in M2, then "not S" cannot hold in M1.
Now, "reals" and "hyperreals" are, in some sense, the
simple version. The grander method of non-standard analysis is to construct a formal theory that let's you describe not only algebra, but enough of set-theory to describe analysis too.
Also: aren't the standard reals the only _Archimedean_ ordered field ( and not just the
only ordered field).
Yep -- and the hyperreals do, in fact, satisfy the transfer of the Archimedean property as they should. For reference, the transfer is:
For every hyperreal number x, there exists a hypernatural number n such that x < n
For reference, one typical formulation of the language is as follows. Let S
0 be the set of natural numbers. Then define
S_{n+1} = \mathcal{P}(S_n) \cup S_n
and let S_\omega be the union of all the S_i. Roughly speaking, S_\omega contains anything you can construct set-theoretically from the natural numbers using the "power set" operation only a finite number of times.
You can then define a first-order formal theory where each element of S_\omega is a constant symbol, set membership is a binary relation symbol, and every true set-theoretic statement involving elements of S_\omega is taken as an axiom. This is clearly complete, so any two models are elementarily equivalent.
The neat trick is that you can choose the non-standard model so that the interpretation of \in coincides with the standard set-theoretic meaning. However {}^\star \mathcal{P}(A) \neq \mathcal{P}({}^\star A) whenever A is an infinite set.
Anyways, since
C is an element of S_\omega, it has an interpretation in the non-standard model. And if in the standard model
C was defined as
Rx
R, the same must be true in the non-standard model. (execise: check that {}^\star A \times {}^\star B = {}^\star A \mathop{ {}^\star \times} {}^\star B)
JungleJesus said:
A standard complex number can be conceptualized as a pair of real numbers (x,y) of the form "x + iy" where i is the square-root of -1. I am wondering if/how the non-standard complex numbers can be constructed.
Any of the usual ways to construct the complexes transfers, so you really do just repeat verbatim any of the standard constructions you're familiar with.
For the record, the calculus of hyperreal numbers can be defined by transfer too. For example, the statement
\lim_{x \to a} f(x) = L
is true for non-standard a, L, and f if and only if
[iindent]\forall \epsilon > 0 \exists \delta > 0 : 0 < |x - a| < \deta \implies |f(x) - L| < \epsilon[/indent]
where \epsilon, \delta are varying over positive hyperreal numbers.
It just that this is somewhat less efficient way to go about things.
JungleJesus said:
Does anyone know where I could find an online book or something along those lines? I know Robinson briefly mentioned the complex extension in his book, but that's about as much as I have seen.
Unfortunately, most of what I've learned I've picked up from scattered reading, and I can't really recommend or even recall anything.
Keisler's textbook goes into multi-variable calculus; in some sense, complex analysis is a manifestation of the properties of differential forms -- and differential geometry is built upon multi-variable calculus. This might be a reasonable line to pursue?
