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“A. Whitehead and B. Russell, Principia Mathematica, 2nd edition,Cambridge 1925. In particular, we also reckon among the axioms of PMtheaxiom of infinity (in the form: there exist denumerably many individuals),and the axioms of reducibility and of choice (for alltypes)”

Ramsey saysSuch an axiom has no place in mathematics, and anything which cannot beproved without using it cannot be regarded as proved at all.This axiom there is no reason to suppose true; and if it were true, thiswould be a happy accident and not a logical necessity, for it is not atautology. (THE FOUNDATIONS OF MATHEMATICS* (1925) by F. P. RAMSEY

]n the Introduction to the second edition of Principia, Russell repudiated Reducibility as 'clearly not the sort of axiom with which we can rest content'

Drawing on the ideas of Wittgenstein, Russell hoped to avoid an axiom of reducibility for elementary number theory

Ramsey says Such an axiom has no place in mathematics, and anything which cannot be proved without using it cannot be regarded as proved at all. This axiom there is no reason to suppose true; and if it were true, this would be a happy accident and not a logical necessity, for it is not a tautology. (THE FOUNDATIONS OF MATHEMATICS* (1925) by F. P. RAMSEY

(1) Godel uses the axiom of reducibility axiom 1V of his system is the axiom of reducibility "As Godel says "this axiom represents the axiom of reducibility (comprehension axiom of set theory)" (K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965,p.12-13) . Godel uses axiom 1V the axiom of reducibility in his formula 40 where he states "x is a formula arising from the axiom schema 1V.1 ((K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965,p.21 " [40. R-Ax(x) ≡ (∃u,v,y,n)[u, v, y, n <= x & n Var v & (n+1) Var u & u Fr y & Form(y) & x = u ∃x {v Gen [[R(u)*E(R(v))] Aeq y]}] x is a formula derived from the axiom-schema IV, 1 by substitution "