By E. Askwith

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**Sample text**

FED. D is an ultrafilter on 1, if in addition, (d) E c;;, 1::::>. E ED V (1 - E) E D. An example of a filter is the family of those sets of natural numbers whose complements are finite. This is known as the Frechet filter. It is obvious that this family of subsets of IN is in fact a filter. It is not an ultrafilter however: it contains neither the set of even numbers nor the set of odd numbers. Nevertheless, the following general theorem shows that the Frechet filter can be extended, non-constructively, to an ultrafilter.

J ABC;:::: A'BC. The axioms marked * are missing in Hilbert; they become necessary because of the way in which the concept (angle congruence as a 6-ary relation between points) is formulated. One could now use this fully-fashioned axiom system of Hilbert as the basis for the strict derivation of the results of elementary Euclidean geometry, for example the congruence theorems for triangles, Pythagoras' Theorem and so on. We do not wish to dwell on this but simply remark: Suppose that in the Euclidean plane £ there are two lines g,g' meeting at a point 0, and points E, E' on g, g' , respectively, so chosen that 9 and g' are perpendicular and 0 E ~ 0 E' , then the vector space V constructed as above from points of the plane is isomorphic to £, the two-dimensional vector space over IR.

Will be extended appropriately to apply to the equivalence classes thus defined. The details appear in the following box. , -,-1,0,1) . 1N = {O, 1,2, ... } , D is an ultrafilter over 1N that contains all cofinite sets. nr; = (JRN / D, ::;D, +D, ·D, -D,r}, OD, ID), is defined by {aihEN/D = {{bihEN I {i E 1Nlai = bi} ED}; JRN /D = {{ai}iEN/D I ai E JR,i E 1N}; {aihEN/D::;D {bi}iEIN/ D == {i E 1Nlai::; bi} ED; {ai}iEN/D +D {bi}iEl.. ; OD = {O,O,O, ... }/D; ID = {I,I,I, ... }/D. We have to show that this definition of ultrapower is legitimate, and in particular that the operations +D, etc, are well-defined on the equivalence classes.

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