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Model Theory and the Cardinal Numbers P and T

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Model Theory and the Cardinal Numbers P and T COMMENTARY COMMENTARY Model theory and the cardinal numbers p and t Justin Tatch Moore1 fi Cornell University, Ithaca, NY 14853 Model Theory and Classi cation Model theory is a different branch of math- ematical logic that concerns itself with the Modern mathematical logic is a multifaceted Just as new models of geometry were dis- semantics and syntactics of abstract mathe- fi subject, which concerns itself with the covered and explored by Bolyani, Gauss, and matical structures, such as the eld of com- C; + ; · strengths and limitations of formal proofs Lobachevsky in the 19th century, modern set plex numbers ð Þ and linear orders Q;< and algorithms and the relationship between theory concerns itself with building and un- such as ð Þ. One of the results that ’ language and mathematical structure. Mod- derstanding models of the axioms of math- modernized the subject was Morley scate- ern mathematical logic also addresses foun- ematics. If the continuum hypothesis fails, goricity theorem (6). A fundamental result in dational issues that arise in mathematics. then there are orders of infinity that lie linear algebra is that any two vector spaces fi This commentary summarizes the ground- somewhere between ℵ1—the smallest un- that have the same dimension and same eld ℵ breaking results of Malliaris and Shelah countable cardinal—and 2 0 . Two examples of scalars are isomorphic. If the field of sca- (1), recently published in PNAS (2), relating are p and t.Thecardinalp is the minimum lars is the rational numbers, then the di- two branches of logic: model theory and cardinality of a collection F of infinite subsets mension of an uncountable vector space is set theory. the same as its cardinality. Thus, any two Malliaris and Shelah vector spaces over Q, which have the same fi Higher Orders of In nity outline a proof that uncountable cardinality, are in fact iso- At the end of the 19th century, Cantor made p t— morphic. The categoricity theorem is a vast the remarkable discovery that it was possible = a very rare generalization of this; it asserts that, for any to develop theory of the size or cardinality of T κ λ fi X Y instance of a provable theory and uncountable cardinals and , an in nite set. Two sets, and ,havethe if any two models of T of cardinality κ are same cardinality if there is a bijective corre- equality among orders fi isomorphic, then any two models of cardi- spondence between them: that is, there is a of in nity. nality λ are isomorphic. The proof proceeds pairing between the elements of X and Y N fi by showing that, even in this generality, so that each element of one corresponds of ,allofwhose nite intersections are theories that satisfy the hypothesis of the uniquely to an element of the other. Sets that fi fi in nite, such that there is no single in nite theorem admit an abstract notion of a basis, are either finite or have the same cardinality A⊆N , such that every element of F contains much as in the setting of vector spaces. as the natural numbers N = f0; 1; 2; ...g are A fi t except for a nite error. The cardinal is This rather extreme characteristic of a said to be countable; otherwise, they are un- fi fi de ned similarly, except one only quanti es theory is by no means typical. For example, countable. Cantor demonstrated that the ra- over families F which are totally ordered by ​ both R and Q∪ ð0; 1Þ are dense linear orders tional numbers are countable but the real containment modulo a finite error. Although of the same cardinality, but they are not numbers are uncountable: their cardinalities p ≤ t is immediate from the definitions, it ℵ isomorphic: every interval in R is uncount- are commonly denoted ℵ and 2 0 , respec- has been an open problem for over 50 years 0 able, whereas the interval of points between 2 tively. The distinction between the countable whether p = t is provable from the axioms ​ and 3 in Q∪ ð0; 1Þ is countable. Theories that and the uncountable is very important in of mathematics or whether, like the contin- are close to that of the algebraic structure mathematics. For example, the existence of uum hypothesis, it is undecidable based on C; + ; · a countable set of points in a manifold or the axioms. These cardinals, and p in par- ð Þ are generally regarded in model theory as being tame, whereas those similar Hilbert space, which can be used to arbi- ticular, arise in the study of more exotic to- R;< trarily approximate all other points, is cru- pological spaces in topology and analysis. For to the dense linear order ð Þ are regarded cially used in many places throughout mathe- p as being wild. In the 1970s, Shelah initiated example, is the cardinality of the smallest fi matical analysis. number of nowhere dense subsets which his classi cation program in model theory in Set theory—one of the four main branches are needed to cover a separable compact an attempt to separate the tame from the wild of modern logic—concerns itself with foun- topological space. andtostratifywhatlayinbetween(7). dational issues relating to uncountable sets. Curiously, it has long been known that if One method of stratifying theories is Keisler’s order (8), which measures how One of the earliest, and surely the most fa- p = ℵ1, the smallest possible value it can take, mous, problems in set theory asked whether then p = t.Thus,ifp < t is true in a model of easily the models of the theory become sat- ℵ there was a cardinal number that was strictly mathematics, then necessarily 2 0 is at least urated by taking ultrapowers. The ultrapower ℵ0 between ℵ0 and 2 ; the assertion that no ℵ3, the third uncountable cardinal. This is construction is a useful tool by which a such intermediate cardinal exists is known as interestingbecausewehaveamuchmore the “continuum hypothesis.” In comple- limited understanding of models of math- ℵ0 Author contributions: J.T.M. wrote the paper. mentary results, Gödel (3) and Cohen (4, 5) ematics in which 2 > ℵ2 than those in ℵ0 The author declares no conflict of interest. demonstrated that the continuum hypothesis which 2 ≤ ℵ2. Still, it was widely believed is neither provable nor refutable based on the that p < t should consistent with the axioms See companion article 10.107/pnas.1306114110. standard axioms of mathematics. of mathematics. 1E-mail: [email protected]. www.pnas.org/cgi/doi/10.1073/pnas.1310920110 PNAS Early Edition | 1of2 Downloaded by guest on September 28, 2021 structure is enlarged while preserving its decades that followed, very little progress was this same analysis, Malliaris and Shelah theory. The logical properties of the ultra- made in this direction. show that even the weak presence of a de- power are obtained by integration via a cer- finable linear ordering in the structure— Recent Developments tain binary valued measure, known as an specifically Shelah’spropertySOP—exactly fi In a watershed moment in both model theory 3 ultra lter. Such enlargements have a greater characterize the maximal class of theories in and set theory, Malliaris and Shelah solved tendency to contain solutions to large sys- — Keisler’sorder.Thisfinding suggests a wealth tems of logical equations. Those structures both of these central and seemingly un- related problems—by realizing they are in of theories of intermediate complexity. The that have solutions to any consistent system analysis is also unique in proving that there of logical equations, which are smaller than fact closely connected. In their article in are theories more complex than the stable the structure itself, are said to be saturated. PNAS (2), Malliaris and Shelah outline a p = t— theories but less complex than the maximal The tamer the theory, the more likely its proof that a very rare instance of a fi — class in Keisler’s order, and opens the door ultrapowers are to be saturated. provable equality among orders of in nity fi It was long known that the theories of by a tour de force analysis of de nability for new interactions between set theory and fi ðC; + ; · Þ and ðR;<Þ provided examples at in ultraproducts of nite linear orders. Using model theory. the opposite ends of this spectrum, and that thepresenceofadefinable ordering in the structure was an indication of high com- 1 Malliaris M, Shelah S (2012) Cofinality spectrum problems in 5 Cohen PJ (1964) The independence of the continuum hypothesis. plexity. Shelah also characterized, in model model theory, set theory, and general topology. arxiv:math.LO/ II. Proc Natl Acad Sci USA 51(1):105–110. 1208.5424. 6 Morley M (1965) Categoricity in power. Trans Am Math Soc theoretic terms, the smallest two classes in 2 Malliaris M, Shelah S (2013) General topology meets model theory, 114:514–538. Keisler’s order: these classes together are the on p and t. Proc Natl Acad Sci USA, 10.107/pnas.1306114110. 7 Shelah S (1978) Classification theory and the number of nonisomorphic “stable theories” (4). No model theoretic 3 Gödel K (1940) The consistency of the continuum hypothesis. Annals models. Studies in Logic and the Foundations of Mathematics,vol of Mathematics Studies, no. 3 (Princeton Univ Press, Princeton, NJ). 92. (North-Holland Publishing, Amsterdam, The Netherlands). characterization was known, however, of the 4 Cohen PJ (1963) The independence of the continuum hypothesis. 8 Keisler J (1967) Ultraproducts which are not saturated. J Symbolic maximal theories in Keisler’s order. Over the Proc Natl Acad Sci USA 50(6):1143–1148.
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