Home > Principia Mathematica

The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Bertrand Russell and Alfred North Whitehead and published in 1910- 1913. It is an attempt to derive all mathematical truths from a well-defined set of axioms and inference rules in symbolic logic. The main inspiration and motivation for the Principia was Frege's earlier work on logic, which had led to some contradictions discovered by Russell. These were avoided in the Principia by building an elaborate system of types: a set has a higher type than its elements and one can not speak of the "set of all sets" and similar constructs which lead to paradoxes (see Russell's paradox).

The Principia only covered set theory, cardinal numbers, ordinal numbers and real numbers; deeper theorems from real analysisReal analysis is that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. It can be seen as a rigorous version of calculus and studies concepts such as sequences and their limits, continuity, differentiation were not included, but by the end of the third volume it was clear that all known mathematics could in principle be developed in the adopted formalism.

The questions remained whether a contradiction could be derived from the Principia's axioms, and whether there exists a mathematical statement which could neither be proven nor disproven in the system. These questions were settled by Gödel's incompleteness theoremIn mathematical logic, Godel's incompleteness theorems are two celebrated theorems proved by Kurt Godel in 1930. Somewhat simplified, the first theorem states: In any consistent formalization of mathematics that is sufficiently strong to define the concep in 19311931 is the common year starting on Thursday. see link for calendar) Events January January 4 Female aviator Elly Beinhorn begins her flight to Africa January 6 Thomas Edison submits his last patent application. January 22 Sir Isaac Isaacs sworn in as the. Gödel's second incompleteness theorem shows that basic arithmetic cannot be used to prove its own consistency, so it certainly cannot be used to prove the consistency of anything stronger.