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# A formulation of the simple theory of types

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The purpose of the present paper is to give a formulation of the simple theory of types which incorporates certain features of the calculus of λ-conversion. A complete incorporation of the calculus of λ-conversion into the theory of types is impossible if we require that λx and juxtaposition shall retain their respective meanings as an abstraction operator and as denoting the application of function to argument. But the present partial incorporation has certain advantages from the point of view of type theory and is offered as being of interest on this basis (whatever may be thought of the finally satisfactory character of the theory of types as a foundation for logic and mathematics).

For features of the formulation which are not immediately connected with the incorporation of λ-conversion, we are heavily indebted to Whitehead and Russell, Hilbert and Ackermann, Hilbert and Bernays, and to forerunners of these, as the reader familiar with the works in question will recognize.

The class of type symbols is described by the rules that ı and o are each type symbols and that if α and β are type symbols then (αβ) is a type symbol: it is the least class of symbols which contains the symbols ı and o and is closed under the operation of forming the symbol (αβ) from the symbols α and β.

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1 See Carnap, Rudolf, Abriss der Logistik, Vienna 1929, §9. (The simple theory of types was suggested as a modification of Russell's ramified theory of types by Leon Chwistek in 1921 and 1922 and by F. P. Ramsey in 1926.)

2 See, for example, Church, Alonzo, Mathematical logic (mimeographed), Princeton, N. J., 1936, and The calculi of lambda-conversion, forthcoming monograph.

3 Russell, Bertrand, Mathematical logic as based on the theory of types, American journal of mathematics, vol. 30 (1908), pp. 222262; Whitehead, Alfred North and Russell, Bertrand, Principia mathematica, vol. 1, Cambridge, England, 1910 (second edition 1925), vol. 2, Cambridge, England, 1912 (second edition 1927), and vol. 3, Cambridge, England, 1913 (second edition 1927).

4 Hilbert, D. and Ackermann, W., Grundzüge der theoretischen Logik, Berlin 1928 (second edition 1938).

5 Hilbert, D. and Bernays, P., Grundlagen der Mathematik, vol. 1, Berlin 1934, and vol. 2, Berlin 1939.

6 Schönfinkel, M., Über die Bausteine der mathematischen Logik, Mathematische Annalen, vol. 92 (1924), pp. 305316.

7 Kleene, S. C., A theory of positive integers informal logic, American journal of mathematics, vol. 57 (1935), pp. 153–173, 219244.

8 Devices of contextual definition, such as Russell's methods of introducing classes and descriptions (loc. cit.), are here avoided, and assertions concerning the necessity of axioms and the like are to be understood in the sense of this avoidance.

9 Cf. Hilbert and Ackermann, loc. cit.; Bernays, P., Axiomatische Untersuchung des Aussagen-Kalkuh der “Principia Mathematica,” Mathematische Zeitschrift, vol. 25 (1926), pp. 305320.

10 The same device of typical ambiguity which was employed in stating the rules of inference and formal axioms now serves us, not only to condense the statement of an infinite number of theorems (differing only in the type subscripts of the proper symbols which appear) into a single schema of theorems, but also to condense the proof of the infinite number of theorems into a single schema of proof. Of course, in the explicit formal development of the system, a stage would never be reached at which all of the theorems 12°, 12̒, 12̒̒, … (for example) had been proved, but by the device of a schema of proof with typical ambiguity we obtain metamathematical assurance that any required one of the theorems in the infinite list can be proved. Cf. the Prefatory Statement to the second volume of Principia mathematica.

11 Peano, G., Sul concetto di numero, Rivista di matematica, vol. 1 (1891), pp. 87–102, 256267.

11 The question suggests itself whether 30̒ could be used in place of Axiom 8 as the second part of the axiom of infinity. The writer has a proof (depending on the properties of P̒′̒‴) that 30̒ and 30̒′ are together sufficient, in the presence of 1–6α, to replace Axiom 8. A proof has also been carried out by A. M. Turing that, in the presence of 1–7 and 9α, 30̒ is sufficient alone to replace Axiom 8. Whether 8 is independent of 1–7 and 30̒ remains an open problem (familiar methods of eliminating descriptions do not apply here).

13 This schema employs descriptions, through the appearance in it of T α‴α″ and Tα″α′. In certain cases a formula Fα′α′ may be obtained which does not involve descriptions. In particular, for addition and multiplication of non-negative integers we may use the definitions due to J. B. Rosser:

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The Journal of Symbolic Logic
• ISSN: 0022-4812
• EISSN: 1943-5886
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