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# Foundational Branches of Mathematics

These areas consider the framework in which mathematics itself is carried out. To the extent that these consider particular mathematical topics, they border on other areas of the Mathematics Subject Classification; to the extent that these consider the nature of proof and of mathematical reality, they border on philosophy!

• 03: Mathematical logic, or Symbolic Logic, lies at the heart of the discipline, but a good understanding of the rules of logic came only after their first use. Besides basic propositional logic used formally in computer science and philosophy as well as mathematics, this field covers general logic and proof theory, leading to Model theory. Here we find celebrated results such as the Gödel incompleteness theorem and Church's thesis in recursion theory. Applications to set theory include the use of forcing to determine the independence of the Continuum hypothesis. Applications to analysis include Nonstandard analysis, an alternate perspective for calculus. Undecidability issues permeate algebra and geometry as well. This heading includes Set Theory as well: axiomatizations of sets, cardinal and ordinal arithmetic, and even Fuzzy Set theory.
• 18: Category theory, a comparatively new field of mathematics, provides a universal framework for discussing different domains of study. Here the emphasis is not so much on the underlying sets (the groups, manifolds, or whatever) as on the functions between them and the relations which characterize them. One may attempt to base much of mathematics on fundamental themes in this area (e.g. topoi rather than sets).
• 08: General algebraic systems include those structures with a very simple axiom structure, as well as those structures not easily included with groups, rings, fields, or the other algebraic systems. In this field one may consider the general nature of algebraic axioms and how the different classes of them are related.

(Also shown in the map is a section 04 (Set Theory) which has been folded into section 03 with the 2000 version of the MSC.)

Considerations in logic of complexity and provability lead to topics in theoretical computer science; questions of decidability arise in a natural way in number theory and group theory. The apparent paradoxes of set theory (particularly from use of the Axiom of Choice) lead to foundational issues in topology and measure theory. An emphasis on axioms leads to the development of abstract algebra and (synthetic) geometry.

You might want to continue the tour with a trip through algebra.