On Set Theory (1950-1974)

"Every mathematician agrees that every mathematician must know some set theory; the disagreement begins in trying to decide how much is some. [...] The student's task in learning set theory is to steep himself in unfamiliar but essentially shallow generalities till they become so familiar that they can be used with almost no conscious effort. In other words, general set theory is pretty trivial stuff really, but, if you want to be a mathematician, you need some, and here it is; read it, absorb it, and forget it [...] the language and notation are those of ordinary informal mathematics. A more important way in which the naive point of view predominates is that set theory is regarded as a body of facts, of which the axioms are a brief and convenient summary; in the orthodox axiomatic view the logical relations among various axioms are the central objects of study." (Paul R Halmos, "Naive Set Theory", 1960)

"Categorical algebra has developed in recent years as an effective method of organizing parts of mathematics. Typically, this sort of organization uses notions such as that of the category G of all groups. [...] This raises the problem of finding some axiomatization of set theory - or of some foundational discipline like set theory - which will be adequate and appropriate to realizing this intent. This problem may turn out to have revolutionary implications vis-`a-vis the accepted views of the role of set theory." (Saunders Mac Lane, "Categorical algebra and set-theoretic foundations", 1967)

"Many examples occur in the theory of sets in which the definition of the set is self-contradictory. The study of the question of the conditions under which this takes place leads to deep questions of logic. Consideration of these questions has completely changed the face of the subject." (Naum Ya. Vilenkin, "Stories about Sets", 1968)

"Set theory is concerned with abstract objects and their relation to various collections which contain them. We do not define what a set is but accept it as a primitive notion. We gain an intuitive feeling for the meaning of sets and, consequently, an idea of their usage from merely listing some of the synonyms: class, collection, conglomeration, bunch, aggregate. Similarly, the notion of an object is primitive, with synonyms element and point. Finally, the relation between elements and sets, the idea of an element being in a set, is primitive." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"To the average mathematician who merely wants to know his work is securely based, the most appealing choice is to avoid difficulties by means of Hilbert's program. Here one regards mathematics as a formal game and one is only concerned with the question of consistency [...] The Realist position is probably the one which most mathematicians would prefer to take. It is not until he becomes aware of some of the difficulties in set theory that he would even begin to question it. If these difficulties particularly upset him, he will rush to the shelter of Formalism, while his normal position will be somewhere between the two, trying to enjoy the best of two worlds." (Paul J Cohen, "Axiomatic Set Theory", 1971)