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The Principia Mathematica (often abbreviated PM) is a three-volume work on the foundations of mathematics written by the mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913.
A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems ; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms ...
Together with the axiom of choice (see below), these are the de facto standard axioms for contemporary mathematics or set theory.They can be easily adapted to analogous theories, such as mereology.
Gödel's second incompleteness theorem shows that, under general assumptions, this canonical consistency statement Cons(F) will not be provable in F. The theorem first appeared as "Theorem XI" in Gödel's 1931 paper "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I".
The union of the assumption sets at lines m and n, excluding k (the denied assumption). [17] From a sentence and its denial [b] at lines m and n, infer the denial of any assumption appearing in the proof (at line k). [17] Double arrow introduction [17] Biconditional definition (Df ↔), [22] biconditional introduction: m, n ↔ I [17]
An axiomatic system is said to be consistent if it lacks contradiction.That is, it is impossible to derive both a statement and its negation from the system's axioms. Consistency is a key requirement for most axiomatic systems, as the presence of contradiction would allow any statement to be proven (principle of explo
Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica, it has been claimed that most (or even all) mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second-order logic.
Taking Bertrand Russell's predicate logic in his Principia Mathematica as standard, one replaces universal instantiation, , with universal specification (!). Thus universal statements, like "All men are mortal," or "Everything is a unicorn," do not presuppose that there are men or that there is anything.