弱集合论中Voppennka原理的结果
Consequences of Vopěnka's Principle over weak set theories
摘要：
证明了Voppennka原理（VP）可以在ZF的一个弱片段上恢复几乎整个ZF，即如果EST是由可拓性、空集、配对、并、笛卡尔积、$Δ_0$ -分离和沿$ω$的归纳公理组成的理论，则${\rm EST+VP}$证明了无穷、替换（因而也是分离）和幂集公理.这一结果受到了\cite{Tz14}和H.弗里德曼的\cite{Fr05}，其中区分了各种形式的VP。作为推论，${\rm EST}+$基金会$+{\rm VP}$ = ${\rm ZF+VP}$和${\rm EST}+$基金会$+{\rm AC+VP}={\rm ZFC+VP}$。同时证明了Foundation公理与ZF-\{Foundation\}+ ${\rm VP}$无关.选择公理是否独立于${\rm ZF+VP}$是一个开放的问题。一个非常弱的形式的选择遵循VP和一些类似的其他形式的选择。
英文摘要：
It is shown that Vopěnka's Principle (VP) can restore almost the entire ZF over a weak fragment of it. Namely, if EST is the theory consisting of the axioms of Extensionality, Empty Set, Pairing, Union, Cartesian Product, $Δ_0$-Separation and Induction along $ω$, then ${\rm EST+VP}$ proves the axioms of Infinity, Replacement (thus also Separation) and Powerset. The result was motivated by previous results in \cite{Tz14}, as well as by H. Friedman's \cite{Fr05}, where a distinction is made among various forms of VP. As a corollary, ${\rm EST}+$Foundation$+{\rm VP}$=${\rm ZF+VP}$, and ${\rm EST}+$Foundation$+{\rm AC+VP}={\rm ZFC+VP}$. Also it is shown that the Foundation axiom is independent from ZF--\{Foundation\}+${\rm VP}$. It is open whether the Axiom of Choice is independent from ${\rm ZF+VP}$. A very weak form of choice follows from VP and some similar other forms of choice are introduced.
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