"The object of mathematics is to discover "true" theorems. We shall use the term "valid" to describe statements formed according to certain rules and then shall discuss how this notion compares with the intuitive idea of "true."

“To the average mathematician who merely wants to know that 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.”

“However, in all honesty, I must say that one must essentially forget that all proofs are transcribed in this formal language. In order to think productively, one must use all the intuitive and informal methods of reasoning at one's disposal. At the very end one must check that no errors have been committed; but in practice set theory is treated as any other branch of mathematics. The reason that we can do this is that we will never speak about proofs but only about models.”

“It is now known that the truth or falsity of the continuum hypothesis and other related conjectures cannot be determined by set theory as we know it today. This state of affairs regarding a classical and presumably well-posed problem must certainly appear rather unsatisfactory to the average mathematician. One is tempted to look more closely and perhaps more critically at the foundations of mathematics. Although our present "Cantorian" mathematics is highly successful in its treatment of abstractions, one must not overlook the fact that from the very beginning the use of infinite processes was regarded with suspicion by many people.”

“The theorem of Löwenheim–Skolem was the first truly important discovery about formal systems in general, and it remains probably the most basic. It is not a negative result at all, but plays an important role in many situations. For example, in Gödel's proof of the consistency of the Continuum Hypothesis, the fact that the hypothesis holds in the universe of constructible sets is essentially an application of the theorem.”