Question

Consider an axiomatic system that consists of elements in a set S and a set P...

Consider an axiomatic system that consists of elements in a set S and a set P of pairings of elements (a, b) that satisfy the following axioms:

A1 If (a, b) is in P, then (b, a) is not in P.

A2 If (a, b) is in P and (b, c) is in P, then (a, c) is in P.

Given two models of the system, answer the questions below.

M1: S= {1, 2, 3, 4}, P= {(1, 2), (2, 3), (1, 3)}

M2: Let S be the set of real numbers and let P

consist of all pairs (x, y) where x < y.

Q. Find another independent axiom A3 of the system, which is true in M1, but not in M2. Use this result to argue that the original system is not complete.

An example of such independent axiom:

A3 There is an element x in S such that (y, x) is not in P for any y in S. Then, A3 is true in M1 since there is 4 in S with no pairs in P containing. But, every element in S of M2, namely, every real number has infinitely many smaller numbers in S. Thus, M1 is a model of the augmented system containing A1, A2, A3, where as M2 is not. It concludes that the given system with A1 and A2 is not complete.

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