MetaComp100701

MetaComp100701 - 1 METAMATHEMATICS OF COMPARABILITY Harvey...

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1 METAMATHEMATICS OF COMPARABILITY Harvey M. Friedman Department of Mathematics Ohio State University [email protected] http://www.math.ohio-state.edu/~friedman/ September 25, 2000 October 7, 2001 Abstract. A number of comparability theorems have been investigated from the viewpoint of reverse mathematics. Among these are various comparability theorems between countable well orderings ([2],[8]), and between closed sets in metric spaces ([3],[5]). Here we investigate the reverse mathematics of a comparability theorem for countable metric spaces, countable linear orderings, and sets of rationals. The previous work on closed sets used a strengthened notion of continuous embedding. The usual weaker notion of continuous embedding is used here. As a byproduct, we sharpen previous results of [3],[5]. 1. COMPARABILITY OF COUNTABLE WELL ORDERINGS. In this paper, we assume that the field of all linear orderings is a subset of w , and all linear orderings are reflexive. (This is an official convention that we break at the slightest provocation). We normally write £¢ , and use the usual notation < ¢ for the irreflexive part. We say that a linear ordering £¢ is a well ordering if and only if every nonempty subset of fld( £¢ ) has a < ¢ least element. A well ordering is a well founded linear ordering. THEOREM 1.1. The following are provably equivalent in RCA 0 . i) ATR 0 ; ii) For any two countable well orderings, there is a comparison map from one to the other; iii) For any two countable well orderings, there is an order preserving map from one into the other. Proof: For i) ´ ii), see [8], p. 198. By a comparison map we mean an isomorphism from one onto the other or from one onto an initial segment determined by a point in the other. For i) ´ iii), see [2]. QED
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2 2. COMPARABILITY OF COUNTABLE METRIC SPACES. We begin by proving what we call the comparability of countable metric spaces without worrying about what axioms are needed. Let C be any metric space. We allow C to be empty. Let x C and U C . An C neighborhood of x is an open subset of C that contains x. An C clopen neighborhood of x is a simultaneously open and closed subset of C that contains x. An C limit point of U is a point in C all of whose C neighborhoods contain at least two elements from U . Note that C limit points of U do not have to lie in U . Note that U C is closed if and only if every C limit point of U lies in U . For each ordinal a , we define C [ a ] as follows. C [0] = C . C [ a + 1 ] is the set of all C limit points of C [ a ]. C [ l ] is the intersection of all C [ b ], b < l . LEMMA 2.1. Let C be a metric space. For all ordinals a , C [ a+1 ] C [ a ] and C [ a ] is a closed subset of C . There is an ordinal a such that C [ a ] = C [ a+1 ]. If C [ a ] = C [ a+1 ] then for all b a , C [ a ] = C [ b ]. For U C , every element of U is an C limit point of U if and only if every element of U is a U limit point of U .
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MetaComp100701 - 1 METAMATHEMATICS OF COMPARABILITY Harvey...

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