The multiplicative rationals

Additive and Multiplicative Ramsey Theory in the Reals and the Rationals

Let a finite partition F of the real interval (0, 1) be given. We show that if every member of F is measurable or if every member of F is a Baire set, then one member of F must contain a sequence with all of its finite sums and products (and, in the measurable case, all of its infinite sums as well). These results are obtained by using the algebraic structure of the Stone-Čech compactification ...

Embedding trees in the rationals.

An example is presented of a simple algebraic statement whose truth cannot be decided within the framework of ordinary mathematics, i.e., the statement is independent of the usual axiomatizations of set theory. The statement asserts that every tree-like ordering of power equal to or less than the first uncountable cardinal can be embedded homomorphically into the rationals.

Recounting the Rationals: Twice!

We derive an algorithm that enables the rationals to be efficiently enumerated in two different ways. One way is known and is credited to Moshe Newman; it corresponds to a deforestation of the so-called Calkin-Wilf tree of rationals. The second is new and corresponds to a deforestation of the Stern-Brocot tree of rationals. We show that both enumerations stem from the same simple algorithm. In ...

Functional Pearl: Enumerating the rationals

Recounting the Rationals

It is well known (indeed, as Paul Erdős might have said, every child knows) that the rationals are countable. However, the standard presentations of this fact do not give an explicit enumeration; rather they show how to construct an enumeration. In this note we will explicitly describe a sequence b(n) with the property that every positive rational appears exactly once as b(n)/b(n+1). Moreover, ...