数学想象力与数学创造力：非标准自然数的发现

    遥望天空，看着星星闪烁。谁知天上星星有多少？

上世纪30年代，数学家在书房里发挥数学想象力与数学创造力，严格地证明了非标准自然数的存在性。由此，天上的星星有多少？就有了新的说法。

   说明：有了非标准算术，非标准分析离我们就不远了。

请见本文附件，可知一斑。

袁萌   陈启清   12月8日

 附件：非标准自然数的存在证明

The existence of non-standard models of arithmetic can be demonstrated by an application of the compactness theorem（紧致性定理）. To do this, a set of axioms P* is defined in a language including the language of Peano arithmetic together with a new constant symbol x. The axioms consist of the axioms of Peano arithmetic P together with another infinite set of axioms: for each numeral n, the axiom x > n is included. Any finite subset of these axioms is satisfied by a model that is the standard model of arithmetic plus the constant x interpreted as some number larger than any numeral mentioned in the finite subset of P*. Thus by the compactness theorem there is a model satisfying all the axioms P*. Since any model of P* is a model of P (since a model of a set of axioms is obviously also a model of any subset of that set of axioms), we have that our extended model is also a model of the Peano axioms. The element of this model corresponding to x cannot be a standard number, because as indicated it is larger than any standard number.

…….省略