mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...

, the Paris–Harrington theorem states that a certain

combinatorial principle In proving results in combinatorics several useful combinatorial rules or combinatorial principles are commonly recognized and used. The rule of sum, rule of product, and inclusion–exclusion principle are often used for enumerative purposes. Bij ...

Ramsey theory Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in Ramsey theory typically ask ...

, namely the strengthened finite Ramsey theorem, is true, but not provable in

Peano arithmetic In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearl ...

. This has been described by some (such as the editor of the ''Handbook of Mathematical Logic'' in the references below) as the first "natural" example of a true statement about the integers that could be stated in the language of arithmetic, but not proved in Peano arithmetic; it was already known that such statements existed by Gödel's first incompleteness theorem.

Strengthened finite Ramsey theorem

The strengthened finite Ramsey theorem is a statement about colorings and natural numbers and states that: : For any positive integers ''n'', ''k'', ''m'', such that ''m ≥ n'', one can find ''N'' with the following property: if we color each of the ''n''-element subsets of ''S'' = with one of ''k'' colors, then we can find a subset ''Y'' of ''S'' with at least ''m'' elements, such that all ''n''-element subsets of ''Y'' have the same color, and the number of elements of ''Y'' is at least the smallest element of ''Y''. Without the condition that the number of elements of ''Y'' is at least the smallest element of ''Y'', this is a corollary of the finite Ramsey theorem in

K_

, with ''N'' given by: :

\binom = , \mathcal_n(S),  \ge R(\,\underbrace_k\,).

Moreover, the strengthened finite Ramsey theorem can be deduced from the infinite Ramsey theorem in almost exactly the same way that the finite Ramsey theorem can be deduced from it, using a compactness argument (see the article on

Ramsey's theorem In combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. To demonstrate the theorem for two colours (say ...

for details). This proof can be carried out in

second-order arithmetic In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics. A precur ...

. The Paris–Harrington theorem states that the strengthened finite Ramsey theorem is not provable in

Paris–Harrington theorem

Roughly speaking, Jeff Paris and Leo Harrington (1977) showed that the strengthened finite Ramsey theorem is unprovable in Peano arithmetic by showing that in Peano arithmetic it implies the consistency of Peano arithmetic itself. Since Peano arithmetic cannot prove its own consistency by Gödel's second incompleteness theorem, this shows that Peano arithmetic cannot prove the strengthened finite Ramsey theorem. The smallest number ''N'' that satisfies the strengthened finite Ramsey theorem is a

computable function Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable if there exists an algorithm that can do ...

of ''n'', ''m'', ''k'', but grows extremely fast. In particular it is not primitive recursive, but it is also far larger than standard examples of non-primitive recursive functions such as the

Ackermann function In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. All primitive recursive functions are total ...

. Its growth is so large that Peano arithmetic cannot prove it is defined everywhere, although Peano arithmetic easily proves that the Ackermann function is well defined.

References

*
mathworld entry
*

External links

''A brief introduction to unprovability''
(contains a proof of the Paris–Harrington theorem) b
Andrey Bovykin
{{DEFAULTSORT:Paris-Harrington Theorem Independence results Theorems in the foundations of mathematics

Strengthened finite Ramsey theorem

Paris–Harrington theorem

See also

References

External links