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Definition by induction in quine's new foundations for mathematical logic
Published online by Cambridge University Press: 12 March 2014
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In Quine's New foundations the axiom of infinity does not appear to be provable. In a certain stronger system, very closely related to Quine's New foundations, the axiom of infinity is provable. One of the peculiarities of this latter system is that even unstratified propositions can be proved by induction (this is used in the proof of the axiom of infinity). It would seem that definition by induction should also be possible quite irrespective of any conditions of stratification in this latter system. In this paper it is shown that such is the case.
Theorem. If α, α1, α2, …, αs are any classes, and if there is no confusion of bound variables in {Sxp}(v) or {Sxq}(v) and neither z nor v occurs free in p or q, and if
⊦ (y1, y1, … ys):. y1ϵα1 . y2ϵα2 . … . ysϵαs : ⊃ : (Ez) : zϵα : (v) : v . ≡ . {Sxp}(v),
⊦ (y, n, y1, y2, …, ys) :. yϵα . n ϵ Fin . y1ϵα1 . y2ϵα2. … . ysϵαs : ⊃ : (Ex) :zxα : (v) : v=z . ≡ . {Sxq}(v),
then there is an r such thai there is no confusion of bound variables in {Sxr}(v) and neither z nor v occurs free in r, and
⊦ (n, y1, y2, …, ys) :. nϵ Fin . y1ϵα1 . y2ϵα2 . … . ysϵαs : ⊃ : (Ez) : zϵα : (v) : v=z . ≡ . {Sxr}(v),
⊦ (y1, y2, …, ys) : y1ϵα1 . y2ϵα2 . … . ysϵαs . ⊃ . ιx{Snr}(0) = ιxp,
⊦ (n, y1, y2, …, ys) : nϵ Fin . y1ϵα1 . y2ϵα2 . … . ysϵαs . ⊃ . ιx{Snr}(n+1) = ιx{Syq}(ιxr).
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- Copyright © Association for Symbolic Logic 1939
References
1 Quine, W. V., New foundations for mathematical logic, The American mathematical monthly, vol. 44 (1937), pp. 70–80CrossRefGoogle Scholar.
2 The system Q, suggested by Rosser, Barkley, On the consistency of Quine's New Foundations for Mathematical Logic, this Journal, vol. 4 (1939), pp. 15–24Google Scholar.
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