It was in the spring of 1939 that he gave his first Cambridge lecture course.
1939年春天，艾伦第一次在剑桥开课了，
He started with fourteen Part III students, but 'no doubt the attendance will drop off as the term advances,' he wrote home.
他的班上有14名第三阶段的学生，但他给家里写信说，到课的人数肯定会越来越少。
He must have kept at least one, for he had to set questions on his course for the examination in June.
但他至少要留住一个学生，这样他才能为6月的课程考试出题。
One of these asked for a proof of the result of Computable Numbers.
其中的一道题，就是关于可计算数的证明。
It must have been very pleasing to be able to set as an examination problem in 1939, the question that Newman had posed as unanswered only four years before.
能把这个问题作为试题，是很令人愉快的，仅仅在四年前，纽曼还认为这个问题是无法解决的呢。
But at the same time, Alan joined Wittgenstein's class on Foundations of Mathematics.
与此同时，艾伦还参加了维特根斯坦的数学基础课程。
Although this had the same title as Alan's course, it was altogether different.
虽然这与艾伦的课程名称相同，但它们的内容完全不一样。
The Turing course was one on the chess game of mathematical logic; extracting the neatest and tightest set of axioms from which to begin,
图灵的课，是一场数学逻辑的游戏，由最初的公理，推导出整洁而严密的定理，
making them flower according to the exact system of rules into the structures of mathematics, and discovering the technical limitations of that procedure.
按照精密的规则系统，发展出数学体系，并发现这个过程中的局限性。
But Wittgenstein's course was on the philosophy of mathematics; what mathematics really was.
而维特根斯坦讲的是数学哲学，他的问题是，数学到底是什么东西。