Editing Skolem's paradox

{{Short description|Mathematical logic concept}}
{{good article}}
[[Image:ThoralfSkolem-OB.F06426c.jpg|thumb|upright=0.8|[[Thoralf Skolem]], after whom the paradox is named]]
In [[mathematical logic]] and [[philosophy]], '''Skolem's paradox''' is the apparent contradiction that a [[countable set|countable]] [[model theory|model]] of [[first-order logic|first-order]] [[set theory]] could contain an [[uncountable]] [[Set (mathematics)|set]]. 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 (mathematical logic)|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.{{sfn|Skolem|1967|p=295}}

In model theory, a model corresponds to a specific interpretation of a [[formal language]] or theory. 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. The Löwenheim–Skolem theorem shows that any model of set theory in [[first-order logic]], if it is [[consistent]], has an equivalent [[structure (mathematical logic)|model]] that is countable. This appears contradictory, because [[Georg Cantor]] proved that there exist sets which are [[Cantor's diagonal argument|not countable]]. Thus the seeming contradiction is that a model that is itself countable, and which therefore contains only countable sets, [[Satisfiability|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 in 1922 by Skolem. He explained that the countability of a set is not absolute, but relative to the model in which the [[cardinality]] is measured. Skolem's work was harshly received by [[Ernst Zermelo]], who argued against the limitations of first-order logic and Skolem's notion of "relativity," 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, scholars such as [[Hilary Putnam]] have introduced the paradox and Skolem's concept of relativity to the study of the [[philosophy of language]].

== Background ==

One of the [[Cantor's first set theory article|earliest results]] in [[set theory]], published by Cantor in 1874, was the existence of different sizes, or cardinalities, of infinite sets.{{sfn|Kanamori|1996|p=3}} An infinite set <math> X </math> is called [[countable]] if there is a function that gives a [[one-to-one correspondence]] between <math> X </math> and the [[natural numbers]], and is [[uncountable]] if there is no such correspondence function.<ref>{{Harvnb|Cantor|1874}}. English translation: [[#Ewald|Ewald 1996]], pp. 839{{ndash}}843.</ref>{{sfn|Bays|2007|p=2}} In 1874, Cantor proved that the [[real number]]s were uncountable; in 1891, he proved by his [[Cantor's diagonal argument|diagonal argument]] the more general result known as [[Cantor's theorem]]: for every set <math> S </math>, the [[power set]] of <math> S </math> cannot be in [[bijection]] with <math> S </math> itself.{{sfn|Kanamori|1996|p=7}} When Zermelo proposed [[Zermelo set theory|his axioms for set theory]] in 1908, he proved Cantor's theorem from them to demonstrate their strength.{{sfn|Zermelo|1967|p=200}}

In 1915, [[Leopold Löwenheim]] gave the first proof of what Skolem would prove more generally in 1920 and 1922, the [[Löwenheim–Skolem theorem]].{{sfn|van Heijenoort
|1967|p=232}}{{sfn|Skolem|1967|p=290}} Löwenheim showed that any [[first-order logic|first-order]] sentence with a [[Model theory|model]] also has a model with a countable domain; Skolem generalized this to infinite sets of sentences. The downward form of the Löwenheim–Skolem theorem shows that if a [[countable set|countable]] first-order collection of [[axiomatization|axioms]] is satisfied by an infinite [[structure (mathematical logic)|structure]], then the same axioms are satisfied by some countably infinite structure.{{sfn|Nourani|2014|pp=160{{ndash}}162}} Since the first-order versions of standard axioms of set theory (such as [[Zermelo–Fraenkel set theory]]) are a countable collection of axioms, this implies that if these axioms are satisfiable, they are satisfiable in some countable model.{{sfn|Bays|2007|p=2}}

== The result and its implications ==

In 1922, Skolem pointed out the seeming contradiction between the Löwenheim–Skolem theorem, which implies that there is a [[countable]] [[Model theory|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?"{{sfn|Skolem|1967|p=295}}

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 such a set does not necessarily exist in a countable model; that is, countability is "relative" to a model,{{sfn|Skolem|1967|p=300}} and countable, first-order models are [[Completeness (logic)|incomplete]].{{sfn|Goodstein|1963|p=209}}

Though Skolem gave his result with respect to Zermelo's axioms, it holds for any standard first-order theory of sets,{{sfn|Eklund|1996|p=153}} such as [[ZFC]].{{sfn|Bays|2007|p=2}} Consider Cantor's theorem as a long formula in the [[ZFC#Formal language|formal language of ZFC]]. If ZFC has a model, call this model <math>\mathrm{M}</math> and its [[domain of discourse|domain]] <math>\mathbb{M}</math>. The interpretation of the [[Element (mathematics)|element symbol]] <math> \in </math>, or <math>\mathcal{I} ( \in )</math>, is a set of ordered pairs of elements of <math>\mathbb{M}</math>{{emdash}}in other words, <math>\mathcal{I} ( \in )</math> is a subset of <math>\mathbb{M} \times \mathbb{M}</math>. Since the Löwenheim–Skolem theorem guarantees that <math>\mathbb{M}</math> is countable, then so must be <math>\mathbb{M} \times \mathbb{M}</math>. Two special elements of <math>\mathbb{M}</math> model the [[natural numbers]] <math>\mathbb{N}</math> and the [[power set]] of the natural numbers <math>\mathcal{P} ( \mathbb{N} ) </math>. There is only a countably infinite set of ordered pairs in <math>\mathcal{I} ( \in )</math> of the form <math>\langle x , \mathcal{P} ( \mathbb{N} ) \rangle</math>, because <math>\mathbb{M} \times \mathbb{M}</math> is countable. That is, only countably many elements of <math>\mathbb{M}</math> model members of the uncountable set <math>\mathcal{P} ( \mathbb{N} ) </math>. However, there is no contradiction with Cantor's theorem, because what it states is simply that no element of <math>\mathbb{M}</math> models a [[bijective function]] from <math>\mathbb{N}</math> to <math>\mathcal{P} ( \mathbb{N} ) </math>.{{sfn|Bays|2007}}

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 put some set into correspondence with the natural numbers, but relative to another model, this correspondence may exist.{{sfn|Resnik|1966|pp=426{{ndash}}427}} He described this as the "most important" result in his 1922 paper.{{sfn|Skolem|1967|p=300}} Contemporary set theorists describe concepts that do not depend on the choice of a [[transitive model]] as [[absoluteness (mathematical logic)|absolute]].{{sfn|Kunen|1980|pp=117{{ndash}}118}} From their point of view, Skolem's paradox simply shows that countability is not an absolute property in first-order logic.{{sfn|Kunen|1980|p=141}}{{sfn|Nourani|2014|p=161}}

Skolem described his work as a critique of (first-order) set theory, intended to illustrate its weakness as a foundational system:
{{Blockquote
|text=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.{{sfn|van Dalen|Ebbinghaus|2000|p=147}}
|author=Thoralf Skolem
|source=''Some remarks on axiomatized set theory'' (1922)<ref group=note>Translated from the original German ''Einige Bemerkungen zur axiomatischen Begriindung der Mengenlehre''</ref>
}}

== Reception by the mathematical community ==

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]]:
{{Blockquote
|text= Neither have the books yet been closed on the antinomy, nor has agreement on its significance and possible solution yet been reached.{{sfn|van Dalen|Ebbinghaus|2000|p=147}}
|author= Abraham Fraenkel
|source=''Introduction to set theory'' (1928)<ref group=note>Translated from the original German ''Einleitung in die Mengenlehre''</ref>
}}

In 1925, [[John von Neumann]] presented a novel axiomatization of set theory, which developed into [[NBG set theory]]. Very much aware of Skolem's 1922 paper, von Neumann investigated countable models of his axioms in detail.{{sfn|van Dalen|Ebbinghaus|2000|p=148}}{{sfn|von Neumann|1925}} 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:
{{Blockquote
|text=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.{{sfn|van Dalen|Ebbinghaus|2000|p=148}}
|author=John von Neumann
|source=''An axiomatization of set theory'' (1925)<ref group=note>Translated from the original German ''Eine Axiomatisierung der Mengenlehre''</ref>
}}

Zermelo at first considered Skolem's paradox a hoax, and he spoke against Skolem's "relativism" in 1931.{{sfn|van Dalen|Ebbinghaus|2000|p=153}} Skolem's result applies only to what is now called [[first-order logic]], but Zermelo argued against the [[finitism|finitary]] [[metamathematics]] that underlie first-order logic,{{sfn|Kanamori|2004|pp=519{{ndash}}520}} as Zermelo was a [[Philosophy of mathematics#Platonism|mathematical Platonist]] who opposed [[intuitionism]] and [[finitism]] in mathematics.{{sfn|van Dalen|Ebbinghaus|2000|pp=158{{ndash}}159}} Zermelo believed in a kind of infinite [[Platonic ideal]] of logic, and he held that mathematics had an inherently infinite character.{{sfn|van Dalen|Ebbinghaus|2000|p=149}} Zermelo argued that his axioms should instead be studied in [[second-order logic]],{{sfn|van Dalen|Ebbinghaus|2000|p=151}} a setting in which Skolem's result does not apply.{{sfn|Eklund|1996|p=153}} Zermelo published a second-order axiomatization of set theory in 1930.{{sfn|Haaparanta|2009|p=352}} 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]].{{sfn|van Dalen|Ebbinghaus|2000|p=152}}

<!-- Fraenkel et al. (1973, pp. 303–304) explain why Skolem's result was so surprising to set theorists in the 1920s. --> 
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.{{sfn|Dawson|1993|p=17}} [[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.{{sfn|Baldwin|2017|pp=5}}{{sfn|Hodges|1985|p=275}} Thus, in the 1920s, the particular properties of first-order logic that permit Skolem's paradox were not yet understood.{{sfn|Moore|1980|p=96}} 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.{{sfn|Eklund|1996|p=153}} 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. Zermelo left this refutation unfinished.{{sfn|van Dalen|Ebbinghaus|2000|p=145}}

== Later opinions ==
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".{{sfn|Kleene|1967|p=324}} After surveying Skolem's argument that the result is not contradictory, Kleene concluded: "there is no absolute notion of countability".{{sfn|Kleene|1967|p=324}} [[Geoffrey Hunter (logician)|Geoffrey Hunter]] described the contradiction as "hardly even a paradox".{{sfn|Hunter|1996|p=208}} 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.{{sfn|Fraenkel|Bar-Hillel|Levy|van Dalen|1973|pp=304{{ndash}}305}}

Other mathematicians such as [[Reuben Goodstein]] and [[Hao Wang (academic)|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.{{sfn|Resnik|1966|pp=425-426}} [[L. E. J. Brouwer]] was another early adherent to the idea of absolute countability, arguing from the vantage of [[mathematical intuitionism]] that all sets are countable.{{sfn|Kneale|Kneale|1962|p=673}} Both the Skolemite view and Brouwer's intuitionism stand in opposition to mathematical Platonism,{{sfn|Klenk|1976|p=475}} but [[Carl Posy]] denies the idea that Brouwer's position was a reaction to earlier set-theoretic paradoxes.{{sfn|Posy|1974|p=128}} Skolem was another mathematical intuitionist, but he denied that his ideas were inspired by Brouwer.{{sfn|Shapiro|1996|p=407}}

Countable models of [[Zermelo–Fraenkel set theory]] have become common tools in the study of set theory. [[Paul Cohen|Paul Cohen's]] method for extending set theory, [[Forcing (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.{{sfn|Kanamori|1996|pp=40{{ndash}}42}} 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 (mathematics)|pathology]]; [[Jean van Heijenoort]] described it as "not a paradox...[but] a novel and unexpected feature of formal systems".{{sfn|van Heijenoort|1967|p=290}}

[[Hilary Putnam]] considered Skolem's result a paradox, but one of the [[philosophy of language]] rather than of set theory or formal logic.{{sfn|Putnam|1980|p=464}} He extended Skolem's paradox to argue that not only are set-theoretic notions of membership relative, but [[semantic]] notions of language are relative: there is no "absolute" model for terms and predicates in language.{{sfn|Putnam|1980|p=466}} Timothy Bays argued that Putnam's argument applies the downward Löwenheim-Skolem theorem incorrectly,{{sfn|Bays|2001|p=336}} while Tim Button argued that Putnam's claim stands despite the use or misuse of the Löwenheim-Skolem theorem.{{sfn|Button|2011|pp=325-327}} Appeals to Skolem's paradox have been made several times in the [[philosophy of science]], with scholars making use of Skolem's idea of the relativity of model structures.{{sfn|Hanna|2024|pp=105{{ndash}}108}}{{sfn|Penchev|2020|p=1}}

==See also==

* {{annotated link|Paradoxes of set theory}}

==Notes==
{{reflist|group=note}}

==References==
{{Reflist|20em}}

==Bibliography==
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* {{cite book |editor-last=van Heijenoort |editor-first= Jean |editor-link=Jean van Heijenoort |year=1967|title=From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 |publisher=Harvard University Press}}
** {{harvc |last=Löwenheim |first=Leopold |author-link=Leopold Löwenheim |others=Translated by Stefan Bauer-Mengelberg |year=1967 |orig-year=1922 |c=On Possibilities in the Calculus of Relatives|pages=228{{ndash}}251 |in=van Heijenoort}} 
** {{harvc |last=Skolem |first=Thoralf |author-link=Thoralf Skolem |others=Translated by Stefan Bauer-Mengelberg |year=1967 |orig-year=1922 |c=Some Remarks on Axiomatized Set Theory |pages=290{{ndash}}301 |in=van Heijenoort}}
** {{harvc |last=Zermelo |first=Ernst|author-link=Ernst Zermelo|c=Investigations in the Foundations of Set Theory I |in=van Heijenoort |year=1967 |orig-year=1908|pages=199{{ndash}}215 |others=Translated by Stefan Bauer-Mengelberg}}
* {{cite journal |last=von Neumann |first= John|author-link= John von Neumann|title=Eine Axiomatisierung der Mengenlehre |year=1925 |journal= Journal für de Reine und Angewandte Mathematik |volume= 154 |pages=219{{ndash}}240 |doi= 10.1515/crll.1925.154.219|url= https://doi.org/10.1515/crll.1925.154.219 |language=de}}
{{refend}}

==Further reading==
* {{cite thesis |type=Ph.D. thesis |last=Bays |first=Timothy |year=2000 |title=Reflections on Skolem's Paradox |institution=UCLA Philosophy Department |url=http://www.nd.edu/~tbays/papers/pthesis.pdf}}
* {{cite journal |last=Moore |first=A.W. |author-link=Adrian William Moore |year=1985 |title=Set Theory, Skolem's Paradox and the Tractatus |journal=Analysis |doi=10.2307/3327397 |jstor=3327397 |volume=45 |issue=1 |pages=13–20 }}

==External links==
* [http://boole.stanford.edu/skolem Vaughan Pratt's celebration of his academic ancestor Skolem's 120th birthday]

{{Mathematical logic}}

[[Category:Inner model theory]]
[[Category:Mathematical paradoxes]]
[[Category:Model theory]]

[[de:Löwenheim-Skolem-Theorem#Das Skolem-Paradoxon]]