Skolem's paradox
It consists of a domain (a set of objects) and an interpretation of the symbols and formulas in the language, such that the axioms of the theory are satisfied within this structure.
One line of inquiry questions whether it is accurate to claim that any first-order sentence actually states "there are uncountable sets".
More recently, scholars such as Hilary Putnam have introduced the paradox and Skolem's concept of relativity to the study of the philosophy of language.
[5] When Zermelo proposed his axioms for set theory in 1908, he proved Cantor's theorem from them to demonstrate their strength.
[6] In 1915, Leopold Löwenheim gave the first proof of what Skolem would prove more generally in 1920 and 1922, the Löwenheim–Skolem theorem.
[4] In 1922, 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.
The definition of countability requires that a certain one-to-one correspondence between a set and the natural numbers must exist.
[11] Though Skolem gave his result with respect to Zermelo's axioms, it holds for any standard first-order theory of sets,[12] such as ZFC.
[10] Contemporary set theorists describe concepts that do not depend on the choice of a transitive model as absolute.
[15] From their point of view, Skolem's paradox simply shows that countability is not an absolute property in first-order logic.
[16][17] 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.
[18]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.
Very much aware of Skolem's 1922 paper, von Neumann investigated countable models of his axioms in detail.
[21] 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,[22] as Zermelo was a mathematical Platonist who opposed intuitionism and finitism in mathematics.
[24] Zermelo argued that his axioms should instead be studied in second-order logic,[25] a setting in which Skolem's result does not apply.
[26] 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.
[27] The surprise with which set theorists met Skolem's paradox in the 1920s was a product of their times.
[28] 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.
[12] 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.
[32] 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".
[33] After surveying Skolem's argument that the result is not contradictory, Kleene concluded: "there is no absolute notion of countability".
[34] 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.
[35] Other mathematicians such as Reuben Goodstein and Hao Wang have gone so far as to adopt what is called a "Skolemite" view: that not only does the Löwenheim-Skolem theorem prove that set-theoretic notions of countability are relative to a model, but that every set is countable from some "absolute" perspective.
Paul Cohen's method for extending set theory, forcing, is often explained in terms of countable models, and was described by Akihiro Kanamori as a kind of extension of Skolem's paradox.
[41] 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; Jean van Heijenoort described it as "not a paradox...[but] a novel and unexpected feature of formal systems".
[42] Hilary Putnam considered Skolem's result a paradox, but one of the philosophy of language rather than of set theory or formal logic.
[44] Timothy Bays argued that Putnam's argument applies the downward Löwenheim-Skolem theorem incorrectly,[45] while Tim Button argued that Putnam's claim stands despite the use or misuse of the Löwenheim-Skolem theorem.
