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Skolem's paradox

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Thoralf Skolem, after whom the paradox is named

In mathematical logic and philosophy, Skolem's paradox is the seeming contradiction that a first-order model of set theory could prove the existence of uncountable sets, but be itself countable. The paradox arises from part of the Löwenheim–Skolem theorem; Thoralf Skolem was the first to discuss the seemingly contradictory aspects of the theorem, and to discover the relativity of set-theoretic notions now known as non-absoluteness. Although it is not an actual antinomy like Russell's paradox, the result is typically called a paradox and was described as a "paradoxical state of affairs" by Skolem.[1]

Formally, Skolem's paradox is that every countable axiomatization of set theory in first-order logic, if it is consistent, has a model that is countable. This appears contradictory because it is possible to prove, from those same axioms, a sentence that intuitively says (or that precisely says in the standard model of the theory) that there exist sets that are not countable. Thus the seeming contradiction is that a model that is itself countable, and which therefore contains only countable sets, satisfies the first-order sentence that intuitively states "there are uncountable sets".

A mathematical explanation of the paradox, showing that it is not a true contradiction in mathematics, was first given by Skolem. Skolem's work was harshly received by Ernst Zermelo, who argued against the limitations of first-order logic, but the result quickly came to be accepted by the mathematical community.

The philosophical implications of Skolem's paradox have received much study. One line of inquiry questions whether it is accurate to claim that any first-order sentence actually states "there are uncountable sets". This line of thought can be extended to question whether any set is uncountable in an absolute sense. More recently, the paper "Models and Reality" by Hilary Putnam, and responses to it, led to renewed interest in the philosophical aspects of Skolem's result.

Background

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One of the earliest results in set theory, published by Georg Cantor in 1874, was the existence of uncountable sets, such as the powerset of the natural numbers, the set of real numbers, and the Cantor set. An infinite set X is countable if there is a function that gives a one-to-one correspondence between X and the natural numbers, and is uncountable if there is no such correspondence function.[2][3] When Zermelo proposed his axioms for set theory in 1908, he proved Cantor's theorem from them to demonstrate their strength.[4]

In 1915, Löwenheim gave the first proof of what Skolem would prove more generally in 1920 and 1922, the Löwenheim–Skolem theorem.[5][6] The downward form of this theorem shows that if a countable first-order collection of axioms is satisfied by an infinite structure, then the same axioms are satisfied by some countable structure. Since the first-order versions of Zermelo's axioms of set theory are a countable collection of axioms, this implies that if the first-order versions of Zermelo's axioms are satisfiable, they are satisfiable in some countable model.[7]

The result and its implications

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Skolem pointed out the seeming contradiction between the Löwenheim–Skolem theorem, which implies that there is a countable model of Zermelo's axioms, and Cantor's theorem, which states that uncountable sets exist, and which is provable from Zermelo's axioms. "So far as I know," Skolem wrote, "no one has called attention to this peculiar and apparently paradoxical state of affairs. By virtue of the axioms we can prove the existence of higher cardinalities... How can it be, then, that the entire domain B [a countable model of Zermelo's axioms] can already be enumerated by means of the finite positive integers?"[1]

Formally, consider some formalization of set theory, such as Zermelo's axioms. Then by the Löwenheim–Skolem theorem, there exists some model B of Zermelo's axioms which is countable, meaning we can put each set in the model B in relation with some natural number. However, using Cantor's theorem it follows that the set of all subsets of the model B must be uncountably large, meaning we cannot put each set of the model B in relation with some natural number.[8]

However, this is only an apparent paradox. In the context of a specific model of set theory, the term "set" does not refer to an arbitrary set, but only to a set that is actually included in the model. The definition of countability requires that a certain one-to-one correspondence between a set and the natural numbers must exist. This correspondence itself is a set. Skolem resolved the paradox by concluding that the existence of such a set cannot be proven in a countable model; that is, countability is "relative" to a model, and countable, first-order models are incomplete.[8]

Skolem used the term "relative" to describe when the same set could be countable in one model of set theory and not countable in another: relative to one model, no enumerating function can be proven to put some set into correspondence with the natural numbers, but relative to another model, it may be possible to prove the existence of this function.[9] He described this as the "most important" result in his 1922 paper. Contemporary set theorists describe concepts that do not depend on the choice of a transitive model as absolute. From their point of view, Skolem's paradox simply shows that countability is not an absolute property in first-order logic.[10][11]

Skolem described his work as a critique of (first-order) set theory, intended to illustrate its weakness as a foundational system:

I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come for a critique.[12]

— Thoralf Skolem, Some remarks on axiomatized set theory (1923)

Reception by the mathematical community

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Ernst Zermelo, who argued against the significance of Skolem's paradox

A central goal of early research into set theory was to find a first-order axiomatization for set theory which was categorical, meaning that the axioms would have exactly one model, consisting of all sets.[13] Skolem's result showed that this is not possible, creating doubts about the use of set theory as a foundation of mathematics. It took some time for the theory of first-order logic to be developed enough for mathematicians to understand the cause of Skolem's result; no resolution of the paradox was widely accepted during the 1920s. In 1928, Abraham Fraenkel still described the result as an antinomy:

Neither have the books yet been closed on the antinomy, nor has agreement on its significance and possible solution yet been reached.[12]

— Abraham Fraenkel, Introduction to set theory (1928)

In 1925, John von Neumann presented a novel axiomatization of set theory, which developed into NBG set theory. Very much aware of Skolem's 1923 paper, von Neumann investigated countable models of his axioms in detail.[14][15] In his concluding remarks, von Neumann commented that there is no categorical axiomatization of set theory, or any other theory with an infinite model. Speaking of the impact of Skolem's paradox, he wrote:

At present we can do no more than note that we have one more reason here to entertain reservations about set theory and that for the time being no way of rehabilitating this theory is known.[14]

— John von Neumann, An axiomatization of set theory (1925)

Zermelo at first considered the Skolem paradox a hoax and spoke against it starting in 1929.[14] Skolem's result applies only to what is now called first-order logic, but Zermelo argued against the finitary metamathematics that underlie first-order logic,[16] as Zermelo was a mathematical Platonist who opposed intuitionism and finitism in mathematics.[17] Zermelo argued that his axioms should instead be studied in second-order logic, a setting in which Skolem's result does not apply. Zermelo published a second-order axiomatization in 1930 and proved several categoricity results in that context. Zermelo's further work on the foundations of set theory after Skolem's paper led to his discovery of the cumulative hierarchy and formalization of infinitary logic.[18]

The surprise with which set theorists met Skolem's paradox in the 1920s was a product of their times. Gödel's completeness theorem and the compactness theorem, theorems which illuminate the way that first-order logic behaves and established its finitary nature, were not first proved until 1929.[19] Leon Henkin's proof of the completeness theorem, which is now a standard technique for constructing countable models of a consistent first-order theory, was not presented until 1947.[20] Thus, in the 1920s, the particular properties of first-order logic that permit Skolem's paradox were not yet understood.[21] It is now known that Skolem's paradox is unique to first-order logic; if set theory is studied using higher-order logic with full semantics, then it does not have any countable models, due to the semantics being used. By the time that Zermelo was writing his final refutation of the paradox in 1937, the community of logicians and set theorists had largely accepted the incompleteness of first-order logic.[22]

Later opinions

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Later mathematical logicians did not view Skolem's paradox a fatal flaw in set theory. Stephen Cole Kleene described the result as "not a paradox in the sense of outright contradiction, but rather a kind of anomaly".[23] After surveying Skolem's argument that the result is not contradictory, Kleene concluded: "there is no absolute notion of countability".[23] Geoffrey Hunter described the contradiction as "hardly even a paradox".[24] Fraenkel et al. claimed that contemporary mathematicians are no more bothered by the lack of categoricity of first-order theories than they are bothered by the conclusion of Gödel's incompleteness theorem: that no consistent, effective, and sufficiently strong set of first-order axioms is complete.[25]

Countable models of Zermelo–Fraenkel set theory have become common tools in the study of set theory. Cohen's method for extending set theory, forcing, is often explained in terms of countable models, and was described by Kanamori as a kind of extension of Skolem's paradox.[26] The fact that these countable models of Zermelo–Fraenkel set theory still satisfy the theorem that there are uncountable sets is not considered a pathology; van Heijenoort described it as "a novel and unexpected feature of formal systems".[27]

See also

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References

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  1. ^ a b Skolem 1967, p. 295.
  2. ^ Cantor 1874. English translation: Ewald 1996, pp. 839–843.
  3. ^ Bays 2007, p. 616.
  4. ^ Zermelo 1967, p. 203.
  5. ^ Löwenheim 1967, p. 228.
  6. ^ Skolem 1967, p. 290.
  7. ^ Bays 2000, p. 616.
  8. ^ a b Goodstein 1963, p. 209.
  9. ^ Resnik 1966, pp. 426–427.
  10. ^ Kunen 1980.
  11. ^ Enderton 2001, p. 152.
  12. ^ a b van Dalen & Ebbinghaus 2000, p. 147.
  13. ^ Kanamori 2004, p. 505.
  14. ^ a b c van Dalen & Ebbinghaus 2000, p. 148.
  15. ^ von Neumann 1925.
  16. ^ Kanamori 2004, p. 519.
  17. ^ van Dalen & Ebbinghaus 2000, pp. 158–159.
  18. ^ van Dalen & Ebbinghaus 2000, p. 151.
  19. ^ Dawson 1993, p. 17.
  20. ^ Baldwin 2017, pp. 6–8.
  21. ^ Moore 1980, p. 96.
  22. ^ van Dalen & Ebbinghaus 2000, p. 145.
  23. ^ a b Kleene 1967, p. 324.
  24. ^ Hunter 1971, p. 208.
  25. ^ Fraenkel et al. 1973, p. 304.
  26. ^ Kanamori 2012, pp. 47–48.
  27. ^ van Heijenoort 1967, p. 290.

Bibliography

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Further reading

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