Ramified theory of types | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-31

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Principia_Mathematica


00:03:02 1 Scope of foundations laid
00:03:47 2 Theoretical basis
00:05:26 2.1 Contemporary construction of a formal theory
00:08:23 2.2 Construction
00:14:09 2.3 Primitive ideas
00:15:21 2.4 Primitive propositions
00:19:54 3 Ramified types and the axiom of reducibility
00:24:54 4 Notation
00:28:36 4.1 An introduction to the notation of "Section A Mathematical Logic" (formulas ✸1–✸5.71)
00:33:56 4.2 An introduction to the notation of "Section B Theory of Apparent Variables" (formulas ✸8–✸14.34)
00:44:14 4.3 Introduction to the notation of the theory of classes and relations
00:50:30 5 Consistency and criticisms
00:50:55 5.1 Gödel 1930, 1931
00:53:04 5.2 Wittgenstein 1919, 1939
00:55:10 5.3 Gödel 1944
00:59:36 6 Contents
01:00:51 6.1 Part I Mathematical logic. Volume I ✸1 to ✸43
01:01:00 6.2 Part II Prolegomena to cardinal arithmetic. Volume I ✸50 to ✸97
01:01:23 6.3 Part III Cardinal arithmetic. Volume II ✸100 to ✸126
01:01:45 6.4 Part IV Relation-arithmetic. Volume II ✸150 to ✸186
01:02:40 6.5 Part V Series. Volume II ✸200 to ✸234 and volume III ✸250 to ✸276
01:03:26 6.6 Part VI Quantity. Volume III ✸300 to ✸375
01:04:09 7 Comparison with set theory
01:04:36 8 Differences between editions
01:08:50 9 See also
01:10:30 10 Footnotes



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SUMMARY
=======
The Principia Mathematica (often abbreviated PM) is a three-volume work on the foundations of mathematics written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1925–27, it appeared in a second edition with an important Introduction to the Second Edition, an Appendix A that replaced ✸9 and all-new Appendix B and Appendix C.
PM was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy, being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them.
One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, which Russell discovered allowed for the construction of paradoxical sets. PM sought to avoid this problem by ruling out the unrestricted creation of arbitrary sets. This was achieved by replacing the notion of a general set with the notion of a hierarchy of sets of different 'types', a set of a certain type only allowed to contain sets of strictly lower types. Contemporary mathematics, however, avoids paradoxes such as Russell's in less unwieldy ways, such as the system of Zermelo–Fraenkel set theory.
PM is not to be confused with Russell's 1903 The Principles of Mathematics. PM states: "The present work was originally intended by us to be comprised in a second volume of Principles of Mathematics... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions."
PM has long been known for its typographical complexity. Famously, several hundred pages of 
PM precede the proof of the validity of the proposition 1+1=2. The Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentie ...