Map of Some Sets of Expressions Peter Suber, Philosophy Department, Earlham College

D. Truths of number theory —at least À1 C. Formulas in a countable alphabet —at most À0 B. Wffs of finite length —at most À0 A. Theorems of a finitary formal system —at most À0

The purpose of this map is to show that these important classes of expression are related by subsumption. Even with a countable alphabet, which may have a denumerable infinity of characters, we run out of formulas before we run out of truths of number theory. Hence, there are truths of number theory that cannot be expressed even in an infinite alphabet.

There are at least À₁ truths of number theory (uncountably many), and at most À₀ formulas (countably many). Even if there are exactly À₀ formulas, wffs, and theorems, the set of wffs is a proper subset of the set of formulas, and the set of theorems is a proper subset of the set of wffs. (This is possible, remember, because infinite sets can be put into 1-1 correspondence with some proper subsets of themselves.)

In short, A B C D.

Wffs have only a finite number of symbols apiece, and theorems are determined by finitely many rules of inference using finitely many premises. These conditions create a "finitary" formal system. The map shows that finitary systems intended to capture number theory are doomed to incompleteness.

This file is an electronic hand-out for the course, Logical Systems.

Some of the logic symbols in this file are GIFs; see my Notes on Logic Notation on the Web. Some are HTML characters using the Symbol Font; see Alan Wood's guide to its symbols.