This paper is concerned with learners who aim to learn patterns in infinite binary sequences: shown longer and longer initial segments of a binary sequence, they either attempt to predict whether the next bit will be a 0 or will be a 1 or they issue forecast probabilities for these events. Several variants of this problem are considered. In each case, a no-free-lunch result of the following form is established: the problem of learning is a formidably difficult one, in that no matter what method is pursued, failure is incomparably more common that success; and difficult choices must be faced in choosing a method of learning, since no approach dominates all others in its range of success. In the simplest case, the comparison of the set of situations in which a method fails and the set of situations in which it succeeds is a matter of cardinality (countable vs. uncountable); in other cases, it is a topological matter (meagre vs. co-meagre) or a hybrid computational-topological matter (effectively meagre vs. effectively co-meagre).

本文与旨在学习无限二进制序列模式的学习者有关：显示二进制序列的越来越长的初始片段，他们试图预测下一位是0还是1，或者发出预测概率这些事件。考虑了此问题的几种变体。在每种情况下，都建立了以下形式的免费午餐结果：学习问题是一个非常困难的问题，因为无论采用哪种方法，失败都比成功更为普遍；在选择一种学习方法时，必须面对困难的选择，因为在成功的道路上，没有任何一种方法能主宰所有其他方法。在最简单的情况下，方法失败的情况与成功的情况的比较是基数（可数与不可数）的问题；在其他情况下，它是一个拓扑问题（微不足道与共策）或混合计算拓扑问题（有效微不足道与有效同谋）。