公理集合

For certain infinite sets ''X'', it is also possible to avoid the axiom of choice.

对于特定的无限集合''X''，也有可能避免选择公理。

The problem then becomes constructing such an ordering, and it turns out that every set can be well-ordered if and only if the axiom of choice is true.

问题就变成构造这样一个排序，而它使得所有集合可以是良序的，当且仅当选择公理为真。

To take just one example, consider the set theorist's axiom of extensionality : For any sets A and B , if A and B have the same members, then A is the same set as B .

举例说，考虑集合论的外延公理：对任何集合A和B，如果A和B有相同的成员，那么A作为集合和B同一。

It has been proved in [2] that a normal fuzzy set structure is a kind of non-standard model of the axiomatic set theory with urelements.

在文献[2]中我们证明了正规弗晰集合结构是带本元的公理集合论的一类非标准模型。

The fuzzy set theory is a rapidly growing branch of applied mathematics whereas the Boolean-valued model is a nonstandard model of the axiomatic set theory.

在文献[1]中我们曾引进正规弗晰集合结构的概念，这种正规弗晰集合结构是对弗晰集合论的一种本质的刻划与推广；在文献[1]中还证明了任一正规弗晰集合结构都是带本元的集合论公理系统ZFa的一个布尔值模型。

The research on rough set axiomatic system plays an important role in the rough set theory and its applications.

粗糙集合理论是一种新的有效处理不精确、不确定、含糊信息的理论，粗集公理系统研究是粗集理论及应用研究的基础。

As an application of the above proof, we obtain at once that there is an open and dense set in the set of Ω-stable systems such that every system in the set satisfies Axiom A and the no-cycle condition (Axiom A and the strong transversality condition, respectively).

作为上述方法的一个运用，马上可以得出在Ω稳定系统的集合中有一开稠集满足公理 A 和无环性条件(对应的，公理 A 和强横截性条件)。

The contents of the abstract of the short Communication is:"The paper gives a complete differential partition to a Euclidean straight line with a fixed frame, and three axioms for the integral of infinitesimals indexed by real numbers; proves in standard mathematics there are positive infinitesimals outside of real number set; Gives cosmic, macro and micro counterexamples to two axioms in Jordan, Carathéodory, and Lebesgue measure theory; transforms Weierstrass limit into Huang limit, Cantor continuum into Huang continuum, and Newton-Leibniz formula into Huang formula."

这个短的发言的摘要的内容如下："此文对一条确定了固定标架的欧几里德直线给出了完整的微分分拆，并对以实数为标号的无穷小的积分给出了三条公理；在标准数学中证明了在实数集合之外存在正的无穷小；对若当，卡拉特欧多里和勒贝格测度论中的两条公理给出了宇观的，宏观的和微观的反例；将外尔斯特拉斯极限改进为黄氏极限，将康托连续统改进为黄氏连续统，和将牛顿－莱布尼茨公式改进为黄氏公式。"

The contents of the abstract of the short Communication is:"The paper gives a complete differential partition to a Euclidean straight line with a fixed frame, and three axioms for the integral of infinitesimals indexed by real numbers; proves in standard mathematics there are positive infinitesimals outside of real number set; Gives cosmic, macro and micro counterexamples to two axioms in Jordan, Carathéodory, and Lebesgue measure theory; transforms Weierstrass limit into Huang limit, Cantor continuum into Huang continuum, and Newton-Leibniz formula into Huang formula."

这个短的发言的摘要的内容如下：&此文对一条确定了固定标架的欧几里德直线给出了完整的微分分拆，并对以实数为标号的无穷小的积分给出了三条公理；在标准数学中证明了在实数集合之外存在正的无穷小；对若当，卡拉特欧多里和勒贝格测度论中的两条公理给出了宇观的，宏观的和微观的反例；将外尔斯特拉斯极限改进为黄氏极限，将康托连续统改进为黄氏连续统，和将牛顿－莱布尼茨公式改进为黄氏公式。&

Breath, muscle contraction of the buttocks; arch body, as far as possible to hold his head, right leg straight towards the ceiling (peg-leg knee in order to avoid muscle tension).

呼气，收缩臀部肌肉；拱起身体，尽量抬起头来，右腿伸直朝向天花板（膝微屈，以避免肌肉紧张）。

The cost of moving grain food products was unchanged from May, but year over year are up 8%.

粮食产品的运输费用与5月份相比没有变化，但却比去年同期高8％。