In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, particularly in set theory

Set theory

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

, Fodor's lemma states the following:

If is a regular

Regular cardinal

In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. So, crudely speaking, a regular cardinal is one which cannot be broken into a smaller collection of smaller parts....

, uncountable cardinal

Cardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...

, is a stationary subset

Stationary set

In mathematics, particularly in set theory and model theory, there are at least three notions of stationary set:-Classical notion:If \kappa \, is a cardinal of uncountable cofinality, S \subseteq \kappa \,, and S \, intersects every club set in \kappa \,, then S \, is called a stationary set....

 of , and is regressive (that is, for any , ) then there is some and some stationary such that for any . In modern parlance, the nonstationary ideal is normal.

Proof

We can assume that (by removing 0, if necessary).
If Fodor's lemma is false, for every there is some club set

Club set

In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal which is closed under the order topology, and is unbounded relative to the limit ordinal...

  such that . Let . The club sets are closed under diagonal intersection

Diagonal intersection

Diagonal intersection is a term used in mathematics, especially in set theory.If \displaystyle\delta is an ordinal number and \displaystyle\langle X_\alpha \mid \alphaDiagonal intersection is a term used in mathematics, especially in set theory....

, so is also club and therefore there is some . Then for each , and so there can be no such that , so , a contradiction

Contradiction

In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other...

.

The lemma was first proved by the Hungarian set theorist, Géza Fodor

Géza Fodor (mathematician)

Géza Fodor was a Hungarian mathematician, working in set theory. He proved Fodor's lemma on stationary sets, one of the most important, and most used results in set theory. He was a professor at the Bolyai Institute of Mathematics at the Szeged University. He was vice-president, then president of...

 in 1956. It is sometimes also called "The Pressing Down Lemma".

Fodor's lemma also holds for Thomas Jech

Thomas Jech

Thomas J. Jech is a mathematician specializing in set theory who was at Penn State for more than 25 years. He was educated at Charles University and is now at the of the Academy of Sciences of the Czech Republic....

's notion of stationary sets as well as for the general notion of stationary set.