本文审查了 Gale 1960 年关于显示偏好弱公理需求函数候选者的论文，指出其证明缺口，并给出三个完整证明，确认其发现满足弱公理但非需求函数的候选者的贡献。
摘要：
我们研究了 Gale 于 1960 年发表的重要论文。该论文给出了一个满足显示偏好弱公理的需求函数候选者的例子，但令人怀疑它是否是某个弱序的需求函数。我们审查了这篇论文，首先仔细审查了 Gale 证明了什么。然后我们发现了 Gale 证明中的一个缺口，并表明他未能证明该需求函数候选者不是一个需求函数。接着，我们给出了 Gale 论断的三个完整证明。第一，我们利用 Gale 自己证明的一个事实，构造了一个在 1960 年即可构造的证明。第二，我们使用 Shephard 引理构造了一个现代且简单的证明。第三，我们按照 Gale 最初设想的方向构造了一个证明。我们的结论如下：尽管在 1960 年 Gale 无法证明他构造的需求函数候选者不是一个需求函数，但他实质上证明了这一点，因此，发现满足弱公理但不是需求函数的需求函数候选者的功劳应归于 Gale，这是公平的。
英文摘要：
We investigate Gale's important paper published in 1960. This paper contains an example of a candidate of the demand function that satisfies the weak axiom of revealed preference and that is doubtful that it is a demand function of some weak order. We examine this paper and first scrutinize what Gale proved. Then we identify a gap in Gale's proof and show that he failed to show that this candidate of the demand function is not a demand function. Next, we present three complete proofs of Gale's claim. First, we construct a proof that was constructible in 1960 by a fact that Gale himself demonstrated. Second, we construct a modern and simple proof using Shephard's lemma. Third, we construct a proof that follows the direction that Gale originally conceived. Our conclusion is as follows: although, in 1960, Gale was not able to prove that the candidate of the demand function that he constructed is not a demand function, he substantially proved it, and therefore it is fair to say that the credit for finding a candidate of the demand function that satisfies the weak axiom but is not a demand function is attributed to Gale.
