Editing Foundations of mathematics (section)

=== Mathematical logic ===

In 1847, [[Augustus De Morgan|De Morgan]] published his [[De Morgan's laws|laws]] and [[George Boole]] devised an algebra, now called [[Boolean algebra]],   that allows expressing [[Aristotle's]] logic in terms of formulas and [[algebraic operation]]s. Boolean algebra is the starting point of mathematization  [[logic]] and the basis of [[propositional calculus]]

Independently, in the 1870's, [[Charles Sanders Peirce]] and [[Gottlob Frege]] extended propositional calculus by introducing  [[Quantifier (logic)|quantifiers]], for building [[predicate logic]].

Frege pointed out three desired properties of a logical theory:{{Citation needed|date=April 2020|reason=Please provide a reference as these ideas are normally attributed to David Hilbert}}[[consistency]] (impossibility of proving contradictory statements), [[Completeness (logic)|completeness]] (any statement is either provable or refutable; that is, its negation is provable), and [[Decidability (logic)|decidability]] (there is a decision procedure to test every statement).

By near the turn of the century, [[Bertrand Russell]] popularized Frege's work and discovered [[Russel's paradox]] which implies that the phrase ''"the set of all sets"'' is self-contradictory. This paradox seemed to make the whole mathematics inconsistent  and is one of the major causes of the foundational crisis of mathematics.