Summary
We show an axiom A such that there is no nontrivial interpretation of the alternative set theory (AST) inAST+A keeping ∈, sets and the class of all “standard” natural numbers. Furthermore, there is no interpretation ofAST inAST without the prolongation axiom, but there is an interpretation ofAST in the theory having the prolongation axiom and the basic set-theoretical axioms only.
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References
[Č-S-Z] Čuda, K., Sochor, A., Zlatoš, P.: Guide to alternative set theory. In: Mlček, J., Benešová, M., Vojtášková, B. (eds.), Proceedings of the 1st Symposium Mathematics in the alternative set theory, pp. 44–138. Union of Slovak Mathematicians and Physicists, Bratislava 1989
[S1] Sochor, A.: Metamathematics of the alternative set theory II. Commentat. Math. Univ. Carol.23, 55–79 (1982)
[S2] Sochor, A.: Constructibility in higher order arithmetics. Arch. Math. Logik32, 381–389 (1993)
[S3] Sochor, A.: Choices of convenient sets. (to appear)
[S-V] Sochor, A., Vopěnka, P.: Shiftings of the horizon. Commentat. Math. Univ. Carol.24, 127–136 (1983)
[V] Vopěnka, P.: Mathematics in the Alternative Set Theory. Leipzig: Teubner Texte 1979