When axioms collide

(first published ca. 2002)


When axioms collide, a new truth emerges.

Here we discuss the Zermelo-Fraenkel infinite set axiom and its collision with the euclidean one-line-per-point-pair axiom. The consequence is another 'euclidean' (as opposed to 'non-euclidean') geometry that uses another, and equally valid, axiom, permitting an infinite number of lines per planar point pair.The ZF infinity axiom establishes a prototype infinite set, with the null set as its 'initial' element that permits a set x'' = {x' u {x'}}. Infinite recursion requires a denumerable set.

Fraenkel appears to have wanted the axiom in order to justify the set N. (From the set N, it is then possible to justify nondenumerable infinite sets.) So the axiomatic infinite set permits both denumerable and nondenumerable infinite sets composed of elements with a property peculiar to the set. The ZF infinite set does not of itself imply existence of, say, N. But the axiom, along with the recursion algorithm f(n) = n + 1, which is a property that can be made one-to-one with the axiomatic set elements, does imply N's existence.

So now let us graph a summation formula, such as zeta(-2), and draw a line L through each partial sum height y parallel to the x axis. That is, f(n) is asymptotic to (lim n->inf.)f(n).In other words, the parallels drawn through consecutive values of f(n) squeeze closer and closer together. At the limit, infinite density of lines is reached, where the distance between parallels is 0.Such a scenario, which might be called a singularity, is not permitted by the euclidean and pseudo-euclidean axiom of one line per point pair.

Yet the set J = {a parallel to the x axis through zeta(-2)} is certainly bijective with the axiomatic infinite set.However, by euclidean axiom, the set's existence must be disregarded. The fact that ZF permits the set J to exist and that it takes another axiom to knock it out means that J exists in another 'euclidean' geometry where the one-line-per-point-pair axiom is replaced. We can either posit a newly found geometry or we can modify the old one. We retain the parallel postulate, but say that an infinitude of lines runs through two planar points.

Two infinite sets of lines would then be said to be distinct if there is non-zero distance between the sets of parallels positioned at (lim x-garbled inf.)f(x) and (lim x->inf.)g(x).In addition, it is only the infinite subset of J with elements 0 distance apart that overlays two specific points.So we may say that an infinitude of lines runs through two planar points which are (geometrically) indistinct from a single line running through those two points.

So we axiomatically regard, in this case, an infinitude of lines to be (topologically?) equivalent to one line.