Enlargements and topology

Abstract

The notion of an ultrapower of a higher order structure is formulated and used in an independent proof that the enlargement of a consistent set of sentences in the sense of ROBINSON has a model. The implications of this result in foundations are discussed and the Hahn-Banach Theorem and the Finiteness Principle of the predicate calculus are shown to follow. The Characterisation of topological properties in terms of monads of standard points in the non-standard theory of topological spaces is shown to be directly related to the theories of nets and of filters. The notions of monad of a filter and monad of a net are introduced and their place in the above relationships is determined.