Adopey.1985 net.math utcsrgv!utzoo!decvax!duke!unc!dopey.bts Fri Mar 19 10:46:33 1982 0.9999... infinity Then, the real numbers are the equivalence classes of these rational sequences. The rational numbers we started with are "embedded" in the reals by identifying them with con- stant sequences. Hence, the real number 1/2 is the equivalence class of the rational sequence 1/2,1/2,1/2,1/2,... To construct a simple "non-standard" number system, start with sequences of reals and an ultrafilter U on the natural numbers, the indices for our sequences. (Here's all you need to know about the ultrafilter we'll use. It's a collection of sets of indices. Every set of indices with finite complement is in U. Every set of indices is in U or its complement is in U. And, if a set of indices is in U, then any super-set of indices of that set is in U as well.) Now we say that two sequences s and t are equivalent, s ~ t, if { n : s(n) = t(n) } is in U. Now, as you'd expect, the non-standard reals are just the equivalence classes of the sequences of reals. The standard real numbers are embedded in the non-standard universe by identifying them with constant sequences. For instance, the real number pi is associated with the equivalence class of the sequence pi,pi,pi,pi,... , a typical infinitesimal might be the equivalence class of the sequence 0.1, 0.01, 0.001, 0.0001, 0.00001, ... and a typical infinite element might be the equivalence class of the sequence 1, 10, 100, 1000, 10000, 100000, ... Finally-- and this is the first deviation from what you'll find in a text on non-standard analysis-- let's agree on the following interpretation of non-terminating decimal fractions. If x is a non-terminating decimal fraction, let x(n) be the n-th symbol of x, read from left to right. Associate x with the equivalence class of the sequence sx, where sx(n) = x(1)x(2) ... x(n) This means that 0.9999... will be associated with the equivalence class of the sequence 0, 0., 0.9, 0.99, 0.999, 0.9999, ... There's one more technical detail, then I'll get back to 0.9999... and 1. In general, a formula about two ele- ments of the non-standard universe is true if, when you take a sequence from each equivalence class the set of indices for which the reals in the sequences satisfy the formula is in U. Let s be the sequence of associated with 0.9999... and t the constant sequence 1. Then 0.9999...
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