Abstract
Quine’s system of set theory, New Foundations (NF), can be conveniently formalized as a first-order theory containing two predicates ≡ (identity) and ε (set membership). One of the most attractive features of NF is its simplicity. Apart from the rules and axioms of first-order identity logic, we need only two specifically set theoretical axiom schemes: the extensionality axiom: (Ext) \(\wedge z(z \in x \leftrightarrow z \in y) \to x \equiv y,\) and the set abstraction schema: (Abst) \(\vee y\, \wedge x(x \in y \leftrightarrow \mathfrak{A}),\) where y is not free in \(\mathfrak{A}\) and \(\mathfrak{A}\) is stratified (as usual, we call a formula stratified if indices can be assigned to its variables in such a manner that it becomes a formula of simple type theory).
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Bibliography
A. Ehrenfeucht, and A. Mostowski, ‘Models of Axiomatic Theories admitting Automorphism’, Fundament a Mathematica 43 (1956) 50–68.
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References
Cf. [6].
The unprovability of Inf can also be demonstrated directly by exhibiting a model, as will be apparent from the proof of Theorem 1.
Written out in full S2-S4 read: \(\vee z\, \wedge u(u \in z \leftrightarrow u = x\, \vee \,u = y)\) \(\vee z\, \wedge u(u \in z \leftrightarrow \vee y: \in x\,\,\,\,u \in y)\) \(\vee z\, \wedge u(u \in z \leftrightarrow u \subset x).\)
We write ⊧ M \(\mathfrak{A}\) [u 1,…, u n] to mean: <u 1,…,u n> satisfies \(\mathfrak{A}\) in the model M. Frequently we use an even more abbreviated notation; e. g. ⊧ M u 0 ≡ ϕ for ⊧ M (x ≡ ϕ) [u 0].
i.e. U 0* = {x∈U 2∣ ⊧ N (x is a finite cardinal)}
Thi.s method of proof appears to have been first used by Ehrenfeucht and Mostowski in [1]
R0is provable as a schema in elementary number theory.
If Ind held, we could define the notion W n(x) of truth for prenex NFU statements with ⩽ n quantifiers. We could also define the notion P(x) of provability in first-order predicate logic. By Herbrand’s theorem, if a statement with ⩽ n quantifiers is provable in predicate logic, then there exists a proof in which no formula has more than n quantifiers. Hence, by Ind, we could prove in NFU:P(\(\mathfrak{A}\)) ↔ W n(\(\mathfrak{A}\)) for all NFU statements \(\mathfrak{A}\) with ⩽ n quantifiers. In particular, taking \(\mathfrak{A}\) as an axiomatization of NFU, we have: W n(\(\mathfrak{A}\)), therefore: \(\left. {\overline {\, {} \,}}\! \right| P(\left. {\overline {\, {} \,}}\! \right| \mathfrak{A})\), which would give us a consistency proof for NFU within NFU itself, thus violating Gödel’s second incompleteness theorem.
i.e. Ext, the pair set axiom, the sum set axiom, the replacement axiom, the power set axiom, and the axiom of infinity.
More precisely, there is an r such that, ⊧ M (r is a linear ordering) and {xy| ⊧ M (<x,y>εr)} is of type α.
We identify cardinals with their corresponding initial ordinals.
cf(δ) is the smallest ordinal which is cofinal with δ. Thus cf(δ) > λ says that any partition of δ into λ parts must have at least one part of cardinality δ.
i.e. Vx = \(\begin{array}{*{20}{c}} { \cup \mathfrak{B}{V_v}} \\ {v < x} \end{array}\)
Morley has used similar methods to obtain ω-models with collections of indiscernibles.
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© 1969 D. Reidel Publishing Company, Dordrecht, Holland
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Jensen, R.B. (1969). On the Consistency of a Slight (?) Modification of Quine’s New Foundations . In: Davidson, D., Hintikka, J. (eds) Words and Objections. Synthese Library, vol 21. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1709-1_16
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DOI: https://doi.org/10.1007/978-94-010-1709-1_16
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