ONT Re: Set Theory
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Note 2
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| Appendix
|
| Elementary Set Theory
|
| This appendix is devoted to elementary set theory.
| The ordinal and cardinal numbers are constructed
| and the most commonly used theorems are proved.
| The non-negative integers are defined and
| Peano's postulates are proved as theorems.
|
| A working knowledge of elementary logic is assumed, but acquaintance with
| formal logic is not essential. However, an understanding of the nature
| of a mathematical system (in the technical sense) helps to clarify and
| motivate some of the discussion. Tarski's excellent exposition [1]
| describes such systems very lucidly and is particularly recommended
| for general background.
|
| This presentation of set theory is arranged so that it may be
| translated without difficulty into a completely formal language. †
| In order to facilitate either formal or informal treatment the
| introductory material is split into two sections, the second of
| which is essentially a precise restatement of part of the first.
| It may be omitted without loss of continuity.
|
| The system of axioms adopted is a variant of systems of Skolem and of
| A.P. Morse and owes much to the Hilbert-Bernays-von Neumann system as
| formulated by Gödel. The formulation used here is designed to give
| quickly and naturally a foundation for mathematics which is free
| from the more obvious paradoxes.
|
| For this reason a finite axiom system is abandoned
| and the development is based on eight axioms and
| one axiom scheme ‡ (that is, all statements of a
| certain prescribed form are accepted as axioms).
|
| It has been convenient to state as theorems many
| propositions which are essentially preliminary
| to the desired results. This clutters up the
| list of theorems, but it permits omission of
| many proofs and abbreviation of others.
| Most of the devices used are more or
| less evident from the statements of
| the definitions and theorems.
|
| † That is, it is possible to write the theorems in terms of
| logical constants, logical variables, and the constants of
| the system, and the proofs may be derived from the axioms
| by means of rules of inference. Of course, a foundation
| in formal logic is necessary for this sort of development.
| I have used (essentially) Quine's meta-axioms for logic [1]
| in making this kind of presentation for a course.
|
| ‡ Actually, an axiom scheme for definition is also assumed without explicit
| statement. That is, statements of a certain form, which in particular
| involve one new constant and are either an equivalence or an identity,
| are accepted as definitions and are treated in precisely the same
| fashion as theorems. The axiom scheme of definition is in the
| fortunate position of being justifiable in the sense that,
| if the definitions conform with the prescribed rules,
| then no new contradictions and no real enrichment
| of the theory results. These results are due
| to S. Lésniewski.
|
| JLK, Gen Top, pages 250-251.
|
| Bibliography, pages 282-291.
|
| W.V.O. Quine [1],
|'Mathematical Logic',
| Cambridge (USA), 1947.
|
| A. Tarski [1],
|'Introduction to Modern Logic',
| 2nd American Ed., New York, 1946.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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