Skip to main content Accessibility help
×
Home

Negative solution of the decision problem for sentences true in every subalgebra of 〈N, +〉

  • Ralph Mckenzie (a1)

Extract

It was shown by Taiclin [6], and independently announced by Tarski [7], that the elementary theory of commutative cancellation semigroups is hereditarily undecidable. In his proof Tarski exhibited a subsemigroup of 〈N, ·〉, the natural numbers with multiplication, whose theory is both hereditarily and essentially undecidable. (The details of his construction were published by V. H. Dyson [1].) In connection with these results, Tarski suggested to the author that it would be of interest to solve the decision problem for the theory K which consists of all elementary sentences which are true in every subalgebra (i.e. every subsemigroup) of 〈N, +〉. The object of this note is to prove that the theory K is hereditarily undecidable.

Copyright

References

Hide All
[1]Dyson, V. H., On the decision problem for theories of finite models, Israel Journal of Mathematics, vol. 2 (1964), pp. 5570.
[2]Ershov, Y. L., Lavrov, I. A., Taimanov, A. D. and Taiclin, M. A., Elementary theories, Russian mathematic surveys, vol. 20 (1965), pp. 35105.
[3]Lavrov, I. A., Effective inseparability of the sets of identically true formulae and finitely refutable formulae for certain elementary theories, Algebra i Logika Seminar 2, vol. 1 (1963), pp. 519 [Russian].
[4]Rabin, M. O., A simple method for undecidability proofs and some applications, Logic, methodology and philosophy of science, Proceedings of the 1964 International Congress, Bar-Hillel, ed., Amsterdam, 1965, pp. 5868.
[5]Rogers, H. Jr., Certain logical reduction and decision problems, Annals of Mathematics, vol. 64 (1956), pp. 264284.
[6]Taiclin, M. A., Undecidability of the elementary theory of commutative cancellation semigroups, Sibü. Matematičeski Žurnal, vol. 3 (1962), pp. 308309 [Russian].
[7]Tarski, A., Solution of the decision problem for the elementary theory of commutative semigroups. Notices of the American Mathematical Society, vol. 9 (1962), p. 205.
[8]Vaught, R. L., Cobham's theorem on undecidable theories, Logic methodology and philosophy of science, Proceedings of the 1960 International Congress, Nagel, Suppes and Tarski, eds., Stanford 1962, pp. 1425.

Negative solution of the decision problem for sentences true in every subalgebra of 〈N, +〉

  • Ralph Mckenzie (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed