538 MATHEMATICS: W. V. QUINE PROC. N. A. S.

ON THE CONSISTENC Y OF "NEW FOUNDA TIONS" By W. V. QUINE HARVARD UNIVERSITY Communicated by Hassler Whitney, June 12, 1951 The question of the consistency of the system of set theory in my "New Foundations,"' briefly NF, has long been mooted.2 3 The question takes on added interest with Wang's new revision4 of the system of my Mathe- matical Logic;5 for Wang has proved that this revision, which is to be used in the forthcoming third printing of Mathematical Logic, is consistent if NF is consistent. The purpose of the present note is to point out that NF is consistent if and only if the system of Whitehead and Russell's Principia Mathematica,6 briefly PM, is consistent in a certain one of its natural ver- sions. PM may be conceived in a form in which the variables carry indices indi- cating logical type. Alternatively PM may be conceived under a conven- tion which Whitehead and Russell called typical ambiguity; under this con- vention the indices are suppressed but still only those formulae are ad- mitted as meaningful which are stratified, i.e., so constituted that type in- dices could be inserted everywhere without violating the theory of types. Another point on which we have a choice, in framing a version of PM, is the status of relations and relational predication: we may assume relational predication as primitive on a par with set-membership, and decree a sprawling annex to the theory of types to accommodate relations, as was done in effect by Whitehead and Russell; or alternatively we may, follow- ing Wiener,7 reduce relations to certain sets of sets of sets (at the cost of giving up relations between things of different types). The four systems ob- tained by superimposing these two pairs of alternatives will be called PM1, PM2, PM3, and PM4, thus: PM1 uses type indices and admits the relational apparatus as primitive; PM2 uses typical ambiguity and admits the relational apparatus as primitive; PM3 uses type indices and assumes only the truth functions, quantification, and membership ('e') as primitive; and PM4 uses typical ambiguity and assumes only the truth functions, quantification, and membership as primitive. All four systems are to be understood as not including the axiom of infinity or the axiom of choice. PM2 is inconsistent, as Whitehead and Russell knew; cf. vol. 3, pp. 75, 80. On the other hand PM1 and PM3 are easily proved consistent.8 What will concern us is PM4. The axioms of PM4 are the extensionality axiom:

(x)(y)(z)a(w)(w xco hw e losures (y univeraqz], and the abstraction axioms, which comprise the closures (by universal quan- Downloaded by guest on September 25, 2021 VOL. 37, 1951 MA THEMA TICS: W. V. Q UINE 539

tification of any free variables) of all stratified formulas r(ax) (y) (y e x O)' such that 4 has no free 'x'. The theorems of PM4 are all the stratified formulas quantificationally implied by axioms; i.e., all the stratified formulas 4& such that, for some conjunction x of axioms, rx * 1V becomes a valid schema of quantification theory (or first-order predicate calculus) when e' is thought of as a schematic predicate letter. Note that stratification of 6t assures that of rX D *t; for the axioms in x are stratified and devoid of free variables, and are hence freely combinable without breach of over-all stratification. It is still conceivable, however, that the proof of rx v IVP in quantification theory might itself be irremedi- ably unstratified; and in this event one might object that the proof cannot occur in PM4 (where stratification is essential to significance), and hence that yC has no business among the theorems of PM4. But this objection will probably be withdrawn when one reflects that the curious phenomenon which we are imagining is a function of whatever particular systematization is chosen for quantification theory. Many systematizations of quantifica- tion theory are available, all of which are known to be complete and, in any ordinary sense, logically equivalent to one another; and it is only the trivial detail of our choice among these systems that decides whether cer- tain stratified formulas, or certain different ones, or none at all, are depend- ent for their proofs upon excursions through unstratified formulas. Thus it seems more reasonable to specify the theorems of PM4 absolutely and independently of any particular systematization of quantification theory, by appeal to quantificational implication as in the preceding paragraph. NF differs from PM4 in that stratification is not required for meaningful- ness. The axioms of NF are the same as those of PM4, but the theorems of NF comprise all the formulas (stratified or not) which are quantificationally implied by axioms. Thus the stratified theorems of NF are exactly the theorems of PM4. It follows that NF is consistent if and only if PM4 is consistent. For, if NF is inconsistent then every formula is a theorem of NF, and therefore both 'x E y' and its negation are theorems of PM4. Conversely, if PM4 is inconsistent then every stratified formula is a theorem of PM4, and there- fore both 'x e y' and its negation are theorems of NF. Incidentally the observation that every stratified theorem of NF is a theorem of PM4 has a certain interest over and above the proof of relative consistency. A so-called axiom of infinity, to the effect that A does not be- long to all classes which contain 0 and are closed with respect to successor, has been discussed a good deal3' 9 with regard to its demonstrability or in- demonstrability in NF. Since it is a stratified formula, we can now say that its demonstrability in NF is only as likely as its demonstrability in PM4. More generally, given any stratified formula, the question of its demon- strability in NF or consistency with NF reduces to the same question Downloaded by guest on September 25, 2021 540 ZOOLOG Y: WEISS AND ROSSETTI PROC. N. A. S.

relative to PM4. 1 Quine, W. V., "New Foundations for Mathematical Logic," American Mathematical Monthly, 44, 70-80 (1937). 2 Curry, H. B., review of same, Zentr. Mathematik, 16, 193 (1937). 3 Rosser, Barkley, "On the Consistency of Quine's 'New Foundations for Mathe- matical Logic,' "J. Symbolic Logic, 4, 15-24 (1939); Rosser, Barkley, and Wang, Hao, "Non-Standard Models for Formal Logics," Ibid., 15, 113-129 (1950). 4 Wang, Hao, "A Formal System of Logic," Ibid., 25-32. ' Quine, W. V., Mathematical Logic, New York, 1940; Cambridge, Mass., 1947, 1951. 6 Whitehead, A. N., and Russell, Bertrand, Principia Mathematica, Cambridge, England, 1910-1913; 3 vols. 7 Wiener, Norbert, "A Simplification of the Logic of Relations," Proc. Camb. Phil. Soc., 17, 387-390 (1912-1914). 8 Tarski, Alfred, "Einige Betrachtungen uber die Begriffe der w-Widerspruchsfreiheit und der w-Vollstindigkeit," Monatsh. Mathematik u. Physik, 40, 97-112 (1933); Beth, E. W., "Une d&nonstration de la non-contradiction de la logique des types au point de vue fini," Nieuw Archief voor Wiskunde, 19, 59-62 (1936). 9 Rosser, Barkley, "Definition by Induction in Quine's 'New Foandations for Mathe- matical Logic,' " J. Symbolic Logic, 4, 80-81 (1939); "The Axiom of Infinity in Quine's New Foundations," Ibid., at press; Quine, W. V., "On the Axiom of Infinity, c-Incon- sistency, and Non-Standard Models," Ibid., pending.

GROWTH RESPONSES OF OPPOSITE SIGN AMONG DIFFERENT NEURON TYPES EXPOSED TO THYROID HORMONE* By PAUL WEISS AND FIAMMETTA RossETTi DEPARTMENT OF ZOOLOGY, UNIVERSITY OF CHICAGO Read before the Academy, April 23, 1951 Introduction.-The metamorphic changes transforming the tissues of the tadpole into those of the mature frog have been shown to be under the "control" of the thyroid hormone. That is, they fail to occur, if the hor- mone is absent. However, it is becoming increasingly clear that "control" does not mean the determination of the specific character of the ensuing changes, but refers merely to the reactivation and further sustenance of different chains of morphogenetic events, temporarily arrested in the larval stage, then continuing in each tissue reacting according to its own charac- teristic properties. Hormone action does not initiate heterogeneity in homogeneous tissues; it merely leads to the realization, including visuali- zation, of latent differences based on pre-existing heterogeneity. The mosaic of terminal hormone effects is anticipated by a corresponding latent mosaic of differential susceptibility and response among the various re- acting tissues (Weiss, 1924). In brief, the morphogenetic action of a hormone is not too unlike the action of the photographic developer in Downloaded by guest on September 25, 2021