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Single Idea 13428

[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility ]

Full Idea

The Axiom of Reducibility asserted that to every non-elementary function there is an equivalent elementary function [note: two functions are equivalent when the same arguments render them both true or both false].

Gist of Idea

Reducibility: to every non-elementary function there is an equivalent elementary function

Source

Frank P. Ramsey (The Foundations of Mathematics [1925], §2)

Book Ref

Ramsey,Frank: 'Philosophical Papers', ed/tr. Mellor,D.H. [CUP 1990], p.191

A Reaction

Ramsey in the business of showing that this axiom from Russell and Whitehead is not needed. He says that the axiom seems to be needed for induction and for Dedekind cuts. Since the cuts rest on it, and it is weak, Ramsey says it must go.

The 23 ideas from Frank P. Ramsey

3750 "It is true that x" means no more than x [Ramsey]
18818 Sentence meaning is given by the actions to which it would lead [Ramsey]
13427 Either 'a = b' vacuously names the same thing, or absurdly names different things [Ramsey]
13426 Formalists neglect content, but the logicists have focused on generalizations, and neglected form [Ramsey]
13425 Formalism is hopeless, because it focuses on propositions and ignores concepts [Ramsey]
13428 Reducibility: to every non-elementary function there is an equivalent elementary function [Ramsey]
13430 Infinity: there is an infinity of distinguishable individuals [Ramsey]
13334 Contradictions are either purely logical or mathematical, or they involved thought and language [Ramsey]
22328 I just confront the evidence, and let it act on me [Ramsey]
22325 A belief is knowledge if it is true, certain and obtained by a reliable process [Ramsey]
19724 Belief is knowledge if it is true, certain, and obtained by a reliable process [Ramsey]
6894 Mental terms can be replaced in a sentence by a variable and an existential quantifier [Ramsey]
10993 Ramsey's Test: believe the consequent if you believe the antecedent [Ramsey, by Read]
9418 All knowledge needs systematizing, and the axioms would be the laws of nature [Ramsey]
9420 Causal laws result from the simplest axioms of a complete deductive system [Ramsey]
14279 Asking 'If p, will q?' when p is uncertain, then first add p hypothetically to your knowledge [Ramsey]
19143 Ramsey gave axioms for an uncertain agent to decide their preferences [Ramsey, by Davidson]
13766 'If' is the same as 'given that', so the degrees of belief should conform to probability theory [Ramsey, by Ramsey]
8494 Obviously 'Socrates is wise' and 'Socrates has wisdom' express the same fact [Ramsey]
8495 The distinction between particulars and universals is a mistake made because of language [Ramsey]
8493 We could make universals collections of particulars, or particulars collections of their qualities [Ramsey]
6409 The 'simple theory of types' distinguishes levels among properties [Ramsey, by Grayling]
3212 Beliefs are maps by which we steer [Ramsey]