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of the real numbers, are a special case of this. Some striking applications of ultraproducts include very elegant proofs of the compactness theorem and...

of ultraproducts Fundamental theorem of vector analysis Carl Friedrich Gauss referred to the law of quadratic reciprocity as the "fundamental theorem" of...

Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability...

Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories...

logic, a theorem is a statement that has been proven, or can be proven. The proof of a theorem is a logical argument that uses the inference rules of a deductive...

I. An ultraproduct of copies of the same structure is known as an ultrapower. The key to using ultraproducts in model theory is Łoś's theorem: Let M...

undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations of mathematics...

Cantor's theorem is a fundamental result which states that, for any set A {\displaystyle A} , the set of all subsets of A , {\displaystyle A,} known as the power...

mathematical theorems by computer programs. Automated reasoning over mathematical proof was a major impetus for the development of computer science. While the roots...

theorem Five color theorem Five lemma Fundamental theorem of arithmetic Gauss–Markov theorem (brief pointer to proof) Gödel's incompleteness theorem Gödel's...

and Ultraproducts. North Holland Publishing Company. Chapter 5, Theorem 4.3, page 103. Blass, Andreas (1984). "Existence of bases implies the Axiom of Choice"...

a number of theoretical limitations": ...the magnitudes involved should lead one to suspect that theorems and arguments based chiefly on the mere finiteness...

then X is a dense subspace of βXPages displaying wikidata descriptions as a fallback The fundamental theorem of ultraproducts – Mathematical constructionPages...

then X is a dense subspace of βXPages displaying wikidata descriptions as a fallback The fundamental theorem of ultraproducts – Mathematical constructionPages...

for Aristotle and Euclid. The ancient Greeks considered geometry as just one of several sciences, and held the theorems of geometry on par with scientific...

also known as a "helping theorem" or an "auxiliary theorem". In many cases, a lemma derives its importance from the theorem it aims to prove; however...

g(x)\in [x].} Then the following three connected theorems hold: ~ partitions A into equivalence classes. (This is the Fundamental Theorem of Equivalence Relations...

concepts of theorems, proofs, algorithms, etc. This may also include the philosophical study of the relation of this framework with reality. The term "foundations...

originated in the 1930s, with the work of Kurt Gödel, Alonzo Church, Rózsa Péter, Alan Turing, Stephen Kleene, and Emil Post. The fundamental results the researchers...

}}_{p}\cong \mathbf {C} .} The Ax–Kochen theorem mentioned above also follows from this and an isomorphism of the ultraproducts (in both cases over all primes...