Mathematical logic and foundations

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• ConceptFoundations of mathematics

  The foundations of mathematics study the basic framework in which all of mathematics is built. They ask which language mathematics is written in, which axioms it starts from, and what counts as a proof. Today the usual answer is predicate logic together with the axioms of Zermelo–Fraenkel set theory. The field also studies the limits of this framework, such as the incompleteness theorems.

• ConceptLogicism

  Logicism is the view that mathematics, or at least arithmetic, is a part of logic. Its claim is that mathematical notions can be defined in purely logical terms and that mathematical truths can be proved from logical laws alone. Gottlob Frege, and later Bertrand Russell and Alfred North Whitehead in Principia Mathematica, tried to carry out this program. It is one of the classical answers in the foundations of mathematics.

• ConceptFormalism

  Formalism is the view that mathematics is about formal systems: strings of symbols and the rules that transform them. On this view, a theorem is a string derived from the axioms by the rules. It need not describe any abstract objects. David Hilbert is the name most linked with formalism, through Hilbert's program. It is one of the classical answers in the foundations of mathematics.

• ConceptIntuitionism

  Intuitionism is the view, founded by L. E. J. Brouwer, that mathematics is a construction of the human mind. A mathematical object exists only when it can be constructed, and a statement is true only when it has a proof. As a result, intuitionists do not accept the law of excluded middle in general. Its logic, intuitionistic logic, is a central example of constructivism.

• ProblemHilbert's program

  Write all of classical mathematics as a formal system with finitely describable axioms and rules of inference. Then prove, by finitary means only, that this formal system is consistent. Here a proof is finitary when it reasons only about finite strings of symbols in a concrete, checkable way. Consistency means that no formula $\varphi$ is derivable together with its negation $\neg \varphi$.

• CounterexampleBurali-Forti paradox

  Use von Neumann ordinals: an ordinal is a transitive set that is well-ordered by the membership relation $\in$. Suppose that the collection $\Omega$ of all ordinals is a set. Every element of an ordinal is an ordinal, so $\Omega$ is transitive. The ordinals are well-ordered by $\in$, so $\Omega$ is a transitive set well-ordered by $\in$. Hence $\Omega$ is an ordinal, and so $\Omega \in \Omega$. This contradicts the fact that no ordinal is an element of itself. So the collection of all ordinals is not a set.