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Notes by RBJ on

Mathematical Logic as Based on
The Theory of Types

by Bertrand Russell

Russell's magnum opus (with A.N.Whitehead) Principia Mathematica cost him over 10 years of sustained and exhausting intellectual effort. Of this the most difficult part was that of devising a logical system which was sound (free from contradictions), sufficiently powerful for mathematics, and philosophically acceptable to Russell. In this paper Russell first presents the system on which he finally settled, his Theory of Types. I The Contradictions
II All and any
III The meaning and range of generalised propositions
IV The hierarchy of types
V The axiom of reducibility
VI Primitive ideas and propositions of symbolic logic
VII Elementary theory of classes and relations
VIII Descriptive functions
IX Cardinal numbers
X Ordinal numbers

I. The Contradictions

Since the main object of his work has been to devise a logic which is free from contradictions Russell begins by enumerating a selection of the most important known contradictions and explaining how they are avoided by his Theory of Types. The contradictions he mentions are:
  1. The Liar: "this sentence is not true" can be neither true nor false
  2. Russell's Paradox: R={ x | ¬ x x } (hence R R ¬(R R))
  3. T(R,S) ¬R(R,S) (hence T(T,T) ¬T(T,T))
  4. The least integer not definable in fewer than nineteen syllables is here defined in eighteen symbols.
  5. The least indefinable ordinal is thus defined.
  6. Richard's Paradox: the finitely definable decimals are countable but cannot be enumerated
  7. Buralli-Forti's contradiction: the series of all ordinals is itself an ordinal, but does not contain itself.
All of these paradoxes involve self-reference or reflexiveness. Russell goes through these in detail and concludes that in each case a totality is assumed such that if it were legitimate it would at once be enlarged by new members defined in terms of itself. Hence he adopts the rule:

Whatever involves all of a collection must not be one of the collection.
Which is more fully explicated as:

If, provided a certain collection had a total, it would have members only definable in terms of that total, then the said collection has no total.

This is the "vicious circle" principle, and its consequences for the development of logic are considerable. Though Russell has undoubtedly put his finger on the spot, the spot is a great deal smaller than the finger, and his rule obliterates much of importance.

Russell immediately points out some of the problems which it raises for him.

II. All and any

III. The meaning and range of generalised propositions

IV. The hierarchy of types

V. The axiom of reducibility

VI. Primitive ideas and propositions of symbolic logic

VII. Elementary theory of classes and relations

VIII. Descriptive functions

IX. Cardinal numbers

X. Ordinal numbers

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