Principia Mathematica: Difference between revisions
elaborate on the conclusions of the incompleteness theorem as the "answer" to the consistency/completeness questions in PM |
Frege's work was only one of several motivations. |
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:''For [[Isaac Newton]]'s [[1687]] book containing basic laws of physics, see ''[[Philosophiae Naturalis Principia Mathematica]]''.'' |
:''For [[Isaac Newton]]'s [[1687]] book containing basic laws of physics, see ''[[Philosophiae Naturalis Principia Mathematica]]''.'' |
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The '''''Principia Mathematica''''' is a three-volume work on the foundations of [[mathematics]], written by [[Alfred North Whitehead]] and [[Bertrand Russell]] and published in [[1910]]-[[1913]]. It is an attempt to derive all mathematical truths from a well-defined set of [[axiom]]s and [[inference rule]]s in [[symbolic logic]]. |
The '''''Principia Mathematica''''' is a three-volume work on the foundations of [[mathematics]], written by [[Alfred North Whitehead]] and [[Bertrand Russell]] and published in [[1910]]-[[1913]]. It is an attempt to derive all mathematical truths from a well-defined set of [[axiom]]s and [[inference rule]]s in [[symbolic logic]]. |
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One of the main inspirations and motivations for the Principia was [[Gottlob Frege|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]]). |
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The ''Principia'' only covered [[set theory]], [[cardinal numbers]], [[ordinal numbers]] and [[real numbers]]; deeper theorems from [[real analysis]] 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 ''Principia'' only covered [[set theory]], [[cardinal numbers]], [[ordinal numbers]] and [[real numbers]]; deeper theorems from [[real analysis]] 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. |
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Revision as of 05:35, 16 October 2005
- For Isaac Newton's 1687 book containing basic laws of physics, see Philosophiae Naturalis Principia Mathematica.
The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell 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. One of the main inspirations and motivations 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 analysis 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 theorem in 1931. 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. In other words, the statement "there are no contradictions in the Principia system" cannot be proven true or false in the Principia system unless there are contradictions in the system (in which case it can be proven both true and false).
Quote from the book
- "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." – page 362.
See also
External links
- Stanford Encyclopedia of Philosophy: