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Sentence Logic as an Introduction to Axiomatic Systems

Published online by Cambridge University Press:  03 November 2016

J. Hooley
J. Hooley*
Affiliation:
Derby & District College of Technology
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Extract

“The moral of this story seems to be that mathematical reasoning based on an axiomatic basis is not difficult, ….” “Modern mathematics is turning more and more to axiomatic methods, and there is a great deal to be said for introducing students to them at an early stage, when, I believe, they would take to them more readily, and enjoy them.”

Type
Research Article
Information
The Mathematical Gazette , Volume 44 , Issue 347 , February 1960 , pp. 23 - 34
Copyright
Copyright © Mathematical Association 1960

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References

[1] The sequence of derived rules and theorems is taken from Hilbert and Ackermann’s “Principles of Mathematical Logic” (English version: Chelsea Pub. Co. 1949).Google Scholar
[2] The proof of the deduction theorem (originally due to Herbrand, 1928) is very like the one to be found in Kleene’s “Introduction to Metamathematics” (Amsterdam, 1952) though Kleene provides himself with an axiom schema (he uses axiom schemata rather than axioms) specially designed to carry out the induction step.Google Scholar
[3] Natural Inference methods were introduced by Gentzen, G. in Mathematische Zeitschrift, Vol. XXXIX (1934).Google Scholar
[4] Truth tables as a decision procedure were first explicitly used by Wittgenstein, L. (“Traetatus Logico-Philosophicus”) and Post, E. (Amer. Journal of Math., Vol. 43).Google Scholar
[5] The arithmetical procedures for establishing the independence of the axioms were found by Bernays, P. Google Scholar
[6] Completeness was first proved by Lukasiewicz, J. Google Scholar