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Modern mathematical logic is a multifaceted subject, which concerns itself with the strengths and limitations of formal proofs and algorithms and the relationship between language and mathematical structure. Modern mathematical logic also addresses foundational issues that arise in mathematics. This commentary summarizes the groundbreaking results of Malliaris and Shelah (1), recently published in PNAS (2), relating two branches of logic: model theory and set theory.
Higher Orders of Infinity
At the end of the 19th century, Cantor made the remarkable discovery that it was possible to develop theory of the size or cardinality of an infinite set. Two sets, X and Y, have the same cardinality if there is a bijective correspondence between them: that is, there is a pairing between the elements of X and Y so that each element of one corresponds uniquely to an element of the other. Sets that are either finite or have the same cardinality as the natural numbers 
are said to be countable; otherwise, they are uncountable. Cantor demonstrated that the rational numbers are countable but the real numbers are uncountable: their cardinalities are commonly denoted 
and 
, respectively. The distinction between the countable and the uncountable is very important in mathematics. For example, the existence of a countable set of points in a manifold or Hilbert space, which can be used to arbitrarily approximate all other points, is crucially used in many places throughout mathematical analysis.
Set theoryā€”one of the four main branches of modern logicā€”concerns itself with foundational issues relating to uncountable sets. One of the earliest, and surely ā€¦
ā†µ1E-mail: justin{at}math.cornell.edu.
Model theory and the cardinal numbers š¯” and š¯”±
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