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A formal theory is said to be incomplete if it contains fewer theorems than would be possible while still retaining consistency.

Gödel's first incompleteness theorem states that all consistent axiomatic formulations of number theory which include Peano arithmetic include undecidable propositions ...

A theorem sometimes called "Euclid's first theorem" or Euclid's principle states that if p is a prime and p|ab, then p|a or p|b (where | means divides). A corollary is that ...

Gödel's second incompleteness theorem states no consistent axiomatic system which includes Peano arithmetic can prove its own consistency. Stated more colloquially, any ...

Let s_i be the orders of singular points on a curve (Coolidge 1959, p. 56). Harnack's first theorem states that a real irreducible curve of order n cannot have more than ...

There are three theorems related to pedal circles that go under the collective title of the Fontené theorems. The first Fontené theorem lets DeltaABC be a triangle and P an ...

The fundamental theorem(s) of calculus relate derivatives and integrals with one another. These relationships are both important theoretical achievements and pactical tools ...

Let f(theta) be Lebesgue integrable and let f(r,theta)=1/(2pi)int_(-pi)^pif(t)(1-r^2)/(1-2rcos(t-theta)+r^2)dt (1) be the corresponding Poisson integral. Then almost ...

Let (Omega)_(ij) be the resistance distance matrix of a connected graph G on n nodes. Then Foster's theorems state that sum_((i,j) in E(G)))Omega_(ij)=n-1, where E(g) is the ...

Let p be a prime number, G a finite group, and |G| the order of G. 1. If p divides |G|, then G has a Sylow p-subgroup. 2. In a finite group, all the Sylow p-subgroups are ...