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**General Discussion / Re: Maths thread.**

« **on:**March 19, 2017, 10:30:07 am »

An example that I've found useful for understanding the distinctions around axioms and models is to think about group theory. The definition of a group is essentially a list of axioms. A model of the group axioms is any collection of things which satisfy the group axioms -- in other words, a group! There are lots of statements which are undecidable from the group axioms. For instance, the statement "there is a nonidentity element which is its own inverse" is an undecidable statement, since there are some models where it is true (Z mod 2) and some where it is false (Z mod 3).

Similarly for rings. The definition of a ring lays out the axioms, and each ring is a model of those axioms. The statements "multiplication is commutative" and "each nonzero element has a multiplicative inverse" are independent of the ring axioms. If we choose to make these new axioms, then the models we are left with are called fields.

In a real analysis class, you probably laid out the axioms for the real numbers, namely that they form a complete ordered field. Any two such fields are isomorphic, but there are still different models. Two of the most common are Dedekind Cuts and equivalence classes of Cauchy sequences.

And of course no discussion of axioms is complete without mentioning Euclid's postulates. For a long time people wonder if his fifth postulate, known as the parallel postulate, could be deduced from the first four. This was finally settled a few centuries ago by producing alternative models of the first four axioms: flat, spherical, and hyperbolic geometry. The parallel postulate holds in flat but fails in the others, demonstrating that it is independent of the other axioms.

Similarly for rings. The definition of a ring lays out the axioms, and each ring is a model of those axioms. The statements "multiplication is commutative" and "each nonzero element has a multiplicative inverse" are independent of the ring axioms. If we choose to make these new axioms, then the models we are left with are called fields.

In a real analysis class, you probably laid out the axioms for the real numbers, namely that they form a complete ordered field. Any two such fields are isomorphic, but there are still different models. Two of the most common are Dedekind Cuts and equivalence classes of Cauchy sequences.

And of course no discussion of axioms is complete without mentioning Euclid's postulates. For a long time people wonder if his fifth postulate, known as the parallel postulate, could be deduced from the first four. This was finally settled a few centuries ago by producing alternative models of the first four axioms: flat, spherical, and hyperbolic geometry. The parallel postulate holds in flat but fails in the others, demonstrating that it is independent of the other axioms.