數學想象力與數學創造力：非標準自然數的發現
    遙望天空，看著星星閃爍。誰知天上星星有多少？
上世紀30年代，數學家在書房裡發揮數學想象力與數學創造力，嚴格地證明了非標準自然數的存在性。由此，天上的星星有多少？就有了新的說法。
   說明：有了非標準算術，非標準分析離我們就不遠了。
請見本文附件，可知一斑。
袁萌   陳啟清   12月8日
 附件：非標準自然數的存在證明
The existence of non-standard models of arithmetic can be demonstrated by an application of the compactness theorem（緊緻性定理）. 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