ISBN-10: 0821812076

ISBN-13: 9780821812075

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Additional info for A Representation Theory for Commutative Topological Algebra

Example text

Let r = ◦ t. e. f (r) = 0. 7. (Extreme Value Theorem) Let f : [0, 1] → R be continuous, then f attains a maximum on [0, 1]. Proof. Let m = max{ ∗f (t) : t ∈ T}. Then since T is hyperfinite, we get by transfer m = ∗f (t) for some t ∈ T. Let r = ◦ t. By continuity, f (r) = ◦ m and we claim that it is the maximum attained by f on [0, 1] : For each s ∈ [0, 1], take u ∈ T such that u ≈ s. Then m ≥ ∗f (u). By continuity, ∗f (u) ≈ f (s), so ◦ m ≥ f (s). We end this section with the following existence result.

1 Notes and exercises Under ZF, the requirement that every filter extends to some ultrafilter is strictly weaker than AC ([Moore (1982)]). Construction of a non-Archimedean ordered field similar to the ultrapower of R dates back as early as more than a half century ago in [Hewitt (1948)]. The measure corresponds to a nonprincipal ultrafilter over a countable set is never σ-additive. But over an uncountable set, the σ-additivity of such measure is equivalent to the existence of a large cardinal called the measurable cardinal, which is much larger than inaccessible cardinals.

N<ω We define ∗ X := π(X), where X ∈ V (R). ∗ Hence π Vn (R) = Vn (R) and ∗ ∗ : V (R) → Vn (R). n<ω So we have sets such as ∗N, ∗Q, ∗R, ∗C, . . as well as the ∗Vn (R), n < ω. This is the uniform κ-saturated elementary extension of all sets in V (R), hence of all ordinary mathematical objects that we have referred to earlier. The above is summarized by the commutative diagram Fig. 1. dom(π) ⊂V 5 lll l l l ⊂ ll ll π lll l l l  l l l ∗ ∗ / Vn (R) ⊂ V ( ∗R) V (R) n<ω Fig. 1 The ∗-embedding into the nonstandard universe.