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The optimality of induction as an axiomatization of arithmetic1

Published online by Cambridge University Press:  12 March 2014

Daniel Leivant
Daniel Leivant*
Affiliation:
Department of Computer Science, Cornell University, Ithaca, New York 14853
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Extract

By induction for a formula φ we mean the schema

(where the terms in brackets are implicitly substituted for some fixed variable, with the usual restrictions). Let be the schema IAφ for φ in Πn (i.e. ); similarly for . Each instance of is Δn+2, and each instance of is Σn+1 Thus the universal closure of an instance α is Πn+2 in either case. Charles Parsons [72] proved that and are equivalent over Z0, where Z0 is essentially Primitive Recursive Arithmetic augmented by classical First Order Logic [Parsons 70].

Theorem. For each n > 0 there is a Πn formula π for whichis not derivable in Z0from

(i) true Πn+1sentences; nor even

(ii) Πn+1sentences consistent withZ0.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 48 , Issue 1 , March 1983 , pp. 182 - 184
Copyright
Copyright © Association for Symbolic Logic 1983

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Footnotes

1

Research supported in part by NSF grant 78–00418.

References

REFERENCES

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