Q的可数非标准模型的完备化
Completions of countable non-standard models of Q
摘要：
本文研究了具有可数多个元素的有理数的非标准模型。这些都是序域，因此使用非标准柯西序列来完成它们是有意义的。本文的主要结果表明，这些完备化是实封闭的的，即每一个正数都是平方，每一个奇次多项式都有根.这样，我们给出了Hauschild定理的一个推论的直接证明。在本说明的前一个版本中，由于没有意识到这些结果，我们没有提到这一参考文献。我们感谢Matthias Aschenbrenner指出这一点和相关工作。我们还提供了一些信息的一套实部的有限元等完成-关于更有趣的结果沿着这一点，我们已经告知马蒂亚斯Aschenbrenner。实现这些结果的主要思想依赖于一种用一阶逻辑描述多项式实零点的方法。这是通过仔细使用这种多项式的符号变化来实现的。
英文摘要：
In this note, we study non-standard models of the rational numbers with countably many elements. These are ordered fields, and so it makes sense to complete them, using non-standard Cauchy sequences. The main result of this note shows that these completions are real closed, i.e. each positive number is a square, and each polynomial of odd degree has a root. This way, we give a direct proof of a consequence of a theorem of Hauschild. In a previous version of this note, not being aware of these results, we missed to mention this reference. We thank Matthias Aschenbrenner for pointing out this and related work. We also give some information about the set of real parts of the finite elements of such completions -about the more interesting results along this we have been informed by Matthias Aschenbrenner. The main idea to achieve the results relies on a way to describe real zeros of a polynomial in terms of first order logic. This is achieved by carefully using the sign changes of such a polynomial.
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