Interpretations of the alternative set theory.

*(English)*Zbl 0792.03036The paper is another one in a series of papers concerning the author’s main interest: the metamathematics of alternative set theory (AST) and higher-order arithmetics (here, the third-order arithmetic and some modifications are considered). The author shows the strength of the full comprehension scheme (the quantification of class variables is allowed in formulas defining classes) and the axiom of prolongation (being a form of expressing \(\omega_ 1\)-saturation). Adding only extensionality and existence of sets \((\text{Set}(0)\&(\forall x,y)(\exists z)(z= x\cup\{y\}))\) we obtain a theory strong enough for interpreting the whole AST or third-order arithmetic. None of the four given axioms can be deleted.

Another interesting result of the paper is that adding a suitable axiom (an adaptation of constructibility) of AST we obtain a theory in which the system of all classes is fully described by the system of all sets and the class FN (the class of all natural numbers having the property that every subclass is a set, by prolongation there are natural numbers without this property).

The principal technical means used in the paper are an adaption of the constructible process (described formerly), the ultrapower construction and the fact that saturated elementary equivalent models are isomorphic.

Note that in a previous paper of the author and J. Sgall the consistency of AST + there is a nonconstructible class is proved by an adaptation of the forcing method.

Another interesting result of the paper is that adding a suitable axiom (an adaptation of constructibility) of AST we obtain a theory in which the system of all classes is fully described by the system of all sets and the class FN (the class of all natural numbers having the property that every subclass is a set, by prolongation there are natural numbers without this property).

The principal technical means used in the paper are an adaption of the constructible process (described formerly), the ultrapower construction and the fact that saturated elementary equivalent models are isomorphic.

Note that in a previous paper of the author and J. Sgall the consistency of AST + there is a nonconstructible class is proved by an adaptation of the forcing method.

Reviewer: K.Čuda (Praha)

##### MSC:

03E70 | Nonclassical and second-order set theories |

03E35 | Consistency and independence results |

03F35 | Second- and higher-order arithmetic and fragments |

##### Keywords:

interpretation; metamathematics of alternative set theory; higher-order arithmetics; third-order arithmetic; full comprehension scheme; axiom of prolongation; ultrapower
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\textit{A. Sochor}, Arch. Math. Logic 32, No. 6, 391--398 (1993; Zbl 0792.03036)

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##### References:

[1] | [?-S-Z] ?uda, K., Sochor, A., Zlato?, P.: Guide to alternative set theory. In: Ml?ek, J., Bene?ová, M., Vojtá?ková, B. (eds.), Proceedings of the 1st Symposium Mathematics in the alternative set theory, pp. 44-138. Union of Slovak Mathematicians and Physicists, Bratislava 1989 |

[2] | [S1] Sochor, A.: Metamathematics of the alternative set theory II. Commentat. Math. Univ. Carol.23, 55-79 (1982) · Zbl 0493.03030 |

[3] | [S2] Sochor, A.: Constructibility in higher order arithmetics. Arch. Math. Logik32, 381-389 (1993) · Zbl 0785.03034 · doi:10.1007/BF01270463 |

[4] | [S3] Sochor, A.: Choices of convenient sets. (to appear) · Zbl 0805.03025 |

[5] | [S-V] Sochor, A., Vop?nka, P.: Shiftings of the horizon. Commentat. Math. Univ. Carol.24, 127-136 (1983) · Zbl 0524.03045 |

[6] | [V] Vop?nka, P.: Mathematics in the Alternative Set Theory. Leipzig: Teubner Texte 1979 · Zbl 0693.03034 |

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