Paul Cohen

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Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician and logician, best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was awarded a Fields Medal.


  • 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.
  • 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.
    • p. 2408 of (2005). "Skolem and pessimism about proof in mathematics". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 363 (1835): 2407–2418. ISSN 1364-503X. DOI:10.1098/rsta.2005.1661.

Set Theory and the Continuum Hypothesis (1966)

  • 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 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".
  • When the Löwenheim-Skolem theorem is applied to particular formal systems, we obtain as special cases: Every group, field, ordered field, etc., has a countable subsystem of the same type. A more spectacular result follows from applying the theorem to set theory (a system which we shall later formalize): There is a countable collection of sets, such that if restrict the membership relation to these sets alone, they form a model for set theory (more precisely all the true statements of set theory are true in this model). In particular, within this model which we may denote by M, there must be an uncountable set. This paradox, that a countable model can contain an uncountable set, is explained by noting that to say a set is uncountable merely asserts the nonexistence of a one-one mapping of the set with the set of integers. The "uncountable" set in M set actually has only countably many members in M, but there is no one-one correspondence M of this set with the set of integers.

Quotes about Paul Cohen

  • In 1963 P. J. Cohen completed Gödel's linguistic attack on set theory by introducing the immensely valuable, syntactic notion of forcing, and by using it to demonstrate that the axioms of set theory were not powerful enough to prove Cantor's continuum hypothesis. Thus the presently existing axioms of set theory leave the most celebrated hypothesis of set theory shrouded in uncertainty.
  • To him mathematics was a unified subject that one could master broadly. He had a deep understanding of most areas, and he taught advanced courses in logic, analysis, differential equations, algebra, topology, Lie theory, and number theory on a regular basis. He felt that good mathematics should be easy to understand and that it is always based on simple ideas once you get to the bottom of the issue. This attitude extended to a strong belief that the well-recognized unsolved problems in mathematics are the heart of the subject and have clear and transparent solutions once the right new ideas and viewpoints are found. This belief gave him the courage to work on notoriously difficult problems throughout his career.
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