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#### Bibliographic Item (1.0)

- Frank Plumpton Ramsey (ed R B Braithwaite)
- Foundations of mathematics and other Logical Essays
- Littlefield Adams, Paterson NJ 1960
- =ESSAYS PHILOSOPHY LOGIC MATHEMATICS TYPES PROBABILITY
- Essays, papers, reviews, and notes dating from the 1920s.

- Published

- The Foundations of Mathematics 1925
- Mathematical Logic 1926
- On a Problem of Formal Logic 1928
- Note on the preceeding paper 1926
- Facts and Propositions 1927

- Unpublished

- Truth and probability 1926
- Further Considerations 1928
- Last Papers 1928

- Appendix: Critical Notice of L Wittgenstein's "Tractatus Logico-Philosophicus"

- PM::="Principia Mathematica",
[WhiteheadRussell63].
- Dispenses with PM's pphilosophical
axiom of reducibility
by using Wittgenstein's Tractatus Logico-Philosophicus.

- PM gives a syntactic definition set of special predicative truth functions to avoid paradoxes and then has to assume that all sets (for example) of objects of a given type can be expressed using one of these functions.
- Ramsey uses a semantic definition from Wittgenstein that makes any truth function predicative.

- Argues for the need for a theory of types.
- Does not discard axioms of infinity and selection.

- PM's
Axiom of infinity
asserts the existence of an infinite domain, Needed to define infinite numbers.
Because, in PM, numbers measure the size of sets of objects of a particular type.
Indeed a number is a set of all sets with the same size of that type.
So the largest number for a given type is = to the size of the type.
- PM's
Axiom of selection
is equivalent to the Axiom of Choice
[ wikipedia ]

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