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Set Theory

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Logical Foundations of Mathematics and Computational Complexity

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Abstract

We present axiomatizations of the two main set theories: the Theory of Types and Zermelo-Fraenkel Set Theory. We present some basic facts about the arithmetic of infinite cardinal and ordinal numbers. We introduce the concept of large cardinals, which are cardinal numbers whose existence is not provable from the axioms of Zermelo-Fraenkel Set Theory, and present some applications of axioms postulating the existence of large cardinals. We discuss two important, but slightly controversial axioms: the Axiom of Choice and the Axiom of Determinacy. In the last section we present Quine’s version of the type theory and two approaches based on non-standard analysis.

Meaning! Listen to the mathematician talk. Great space, man, what has mathematics to do with meaning? Mathematics is a tool and as long as it can be manipulated to give proper answers and to make correct predictions, actual meaning has no significance.

Isaac Asimov, The Imaginary

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Notes

  1. 1.

    In English: Basic Laws of Arithmetic.

  2. 2.

    More precisely, we have only to worry about the variables that are quantified, as they talk about the ‘totality of all sets of a given order’, while parameters do not matter, as they talk about particular elements.

  3. 3.

    Item No. AB.1.1165.

  4. 4.

    Even if it were complete, it would not guarantee that there was a unique model.

  5. 5.

    Book 10, Proposition 20 [117].

  6. 6.

    Born and lived in the Kingdom of Bohemia.

  7. 7.

    The history of the invention of the diagonal argument is not quite clear. Some historians attribute the idea to Paul du Bois-Reymond.

  8. 8.

    See also Theorem 2 on page 41.

  9. 9.

    I am using the standard convention that \(x^{y^{z}}\) means x raised to the power y z; it should not be confused with (x y)z which is equal to x yz.

  10. 10.

    In theories without the Axiom of Foundation we have to add explicitly that the elements of α are well-ordered by the membership relation ∈.

  11. 11.

    The explanation of the game by Kenneth Kunen is a little different: “…to try to completely demolish your ego by transcending your number via some completely new principle”, [172] page 396.

  12. 12.

    Recall that in fact the ordinal ω α and the cardinal ℵ α are represented by the same set.

  13. 13.

    More precisely, it is consistent with ZFC to assume that it is.

  14. 14.

    More precisely, he defined an interpretation of the axioms of ZFC in the theory without the Axiom of Choice.

  15. 15.

    More precisely, either one player has a winning strategy, or both players have strategies that guarantee them to win or to tie.

  16. 16.

    I am using he for the first player and she for the second.

  17. 17.

    See [147], page 132.

  18. 18.

    S. Shelah proved that even if we only want the Lebesgue measurability, the inaccessible cardinal is needed [266].

  19. 19.

    Note, however, that nonstandard analysis uses models, whereas here we are talking about a theory.

  20. 20.

    An important exception is high energy physics. Physicist hope to discover new particles by using more powerful accelerators.

  21. 21.

    However, experimental computations with natural numbers are perhaps as old as mathematics itself.

  22. 22.

    See page 212.

  23. 23.

    See page 175.

  24. 24.

    It is not clear that the two numbers can be ∗ π and π; to this end one would have to prove that the strings 222090721 and 17012003 occur infinitely often in the decimal expansion of π.

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Authors and Affiliations

  1. ASCR, Prague, Czech Republic

    Pavel Pudlák

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Main Points of the Chapter

Main Points of the Chapter

  • The two major axiomatic systems for set theory are Bertrand Russell’s Theory of Types and Zermelo-Fraenkel Set Theory. Today the latter one is accepted by most mathematicians as a theory describing the true universe of sets.

  • Though based on different ideas, the Theory of Types and Zermelo Set Theory, a basic part of Zermelo-Fraenkel Set Theory, are essentially the same. Zermelo-Fraenkel set theory is conceptually simpler, it is stronger, and it can be naturally extended to much stronger systems.

  • Infinite sets are defined as sets satisfying a certain property that is not satisfied by finite sets.

  • In Zermelo-Fraenkel Set Theory there are infinitely many different infinite cardinalities and there is no largest one.

  • The cardinality of the real numbers is bigger than the cardinality of natural numbers, but we do not know, and, maybe, never will, whether there are other infinite cardinalities in between.

  • Large cardinal axioms are new axioms that serve to make Zermelo-Fraenkel Set Theory stronger. Using these axioms we can decide some important statements that are undecidable in pure Zermelo-Fraenkel Set Theory.

  • The problem with such axioms is that the stronger they are the greater the danger is that they are inconsistent. There is no way to secure their consistency; we can only rely on experience that a lot of research has been done and no contradiction has been found.

  • The Axiom of Choice has some consequences which look paradoxical, but it does not introduce an inconsistency into Zermelo-Fraenkel Set Theory.

  • It is possible to replace the Axiom of Choice by axioms that make sets of real numbers look better. In particular, it is consistent to assume that all subsets of the real numbers are measurable, provided that we abandon the unrestricted Axiom of Choice and use only its weaker version.

  • The particular set of axioms that we are using may be the result of a historical accident. It is conceivable that if the Axiom of Determinacy had been discovered before the Axiom of Choice, it may have become the preferred one.

  • Various other axiom systems for set theory have been proposed, but only the Zermelo-Fraenkel system has been accepted by working mathematicians. A likely reason is that it is the strongest available theory.

  • New Foundations is an interesting system because we still do not know whether it is consistent. More precisely, we do not have proof of contradiction in the system, nor are we able to prove its consistency using Zermelo-Fraenkel Set Theory (even with the help of large cardinal axioms).

  • Nonstandard analysis offers a different way of developing the foundations of calculus and in some cases has helped solve open problems.

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Pudlák, P. (2013). Set Theory. In: Logical Foundations of Mathematics and Computational Complexity. Springer Monographs in Mathematics. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00119-7_3

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