Abstract
For completeness sake and to enjoy the intellectual insight that these results provide, in this chapter several of the main classical results on undecidability and unsolvability are derived:
-
existence of undecidable statements in any consistent axiomatic theory for mathematics,
-
unsolvability of the halting problem,
-
nonexistence of a decision algorithm for elementary arithmetic,
-
nonexistence of a decision algorithm for predicate calculus.
These are easily proved using an elegant line of argument due to Gregory Chaitin. Then the somewhat more delicate line of argument leading to Gödel’s two incompleteness theorems is considered: this more detailed discussion continues to emphasize the basic role of set theory.
The chapter ends with a discussion on the axioms of reflections, whose addition to the axioms of set theory has direct practical interest: these in fact make the collection of proof mechanisms available to the verifier indefinitely extensible.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Aiello, L., Weyhrauch, R.W.: Using meta-theoretic reasoning to do algebra. In: Bibel, W., Kowalski, R. (eds.) Proc. of the 5th Conference on Automated Deduction, Les Arcs, France. LNCS, vol. 87, pp. 1–13. Springer, Berlin (1980)
Schwartz, J.T., Dewar, R.K.B., Dubinsky, E., Schonberg, E.: Programming with Sets: An Introduction to SETL. Texts and Monographs in Computer Science. Springer, Berlin (1986)
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
Copyright information
© 2011 Springer-Verlag London Limited
About this chapter
Cite this chapter
Schwartz, J.T., Cantone, D., Omodeo, E.G. (2011). Undecidability and Unsolvability. In: Computational Logic and Set Theory. Springer, London. https://doi.org/10.1007/978-0-85729-808-9_6
Download citation
DOI: https://doi.org/10.1007/978-0-85729-808-9_6
Publisher Name: Springer, London
Print ISBN: 978-0-85729-807-2
Online ISBN: 978-0-85729-808-9
eBook Packages: Computer ScienceComputer Science (R0)