GeomAxioms1[1].8.00

GeomAxioms1[1].8.00 - Geometry Axioms by Harvey M. Friedman...

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Geometry Axioms by Harvey M. Friedman January 8, 2000 This regards the Special FOM e-Mail Posting number 79. A key lemma for the geometry axioms is the following. Let F be an ordered field in which *every polynomial in one variable assumes a maximum value over any nonempty closed interval.* Then F is a real closed field. Of course, the hypothesis is equivalent to *every polynomial in one variable assumes a maximum and a minimum value over any nonempty closed interval.* We will actually prove the following. If a polynomial assumes a positive and a negative value then it has a zero. To prove this, we fix P(x) to be any polynomial of degree 1 with a positive and negative value. We define a critical interval to be any nonempty open interval on which P is strictly monotone and where P is not strictly monotone on any larger open interval. Here an open interval may not have endpoints in F, and may be infinite on the left or right or both sides. Obviously, the critical intervals are pairwise disjoint. LEMMA 1. There are finitely many critical intervals. Proof: Suppose there are infinitely many critical intervals. In particular, let I 1 ,I 2 ,...,I 2p+1 be critical intervals from left to right, where p = deg(P). Look at J 1 ,...,J p = I 2 ,I 4 ,...,I 2p , which must be finite intervals. We can stretch
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GeomAxioms1[1].8.00 - Geometry Axioms by Harvey M. Friedman...

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