The Szpilrajn extension theorem, also known as the Szpilrajn-Sierpiński extension theorem, is a result in order theory, specifically within the area concerning partially ordered sets (posets). The theorem provides a method for extending a given partial order to a total order.

Lusin's separation theorem is an important result in the field of measure theory and topology, particularly in the context of Borel sets and measurable functions. The theorem deals with the separation of measurable sets by continuous functions.

The Kanamori–McAloon theorem is a result in the field of combinatorial optimization and discrete mathematics, particularly related to the study of perfect matchings in bipartite graphs. It is named after researchers Yoshihiro Kanamori and Jim McAloon. While the specific theorem may not be universally recognized or widely published under that name, it typically pertains to conditions under which certain structured forms of bipartite graphs possess perfect matchings.

Herbrand's theorem is an important result in mathematical logic, particularly in the field of model theory and proof theory. It connects syntactic properties of first-order logic formulas to semantic properties of their models. There are several formulations of Herbrand's theorem, but one of the most common versions concerns the existence of models for a set of first-order logic sentences. ### Herbrand's Theorem (Informal Statement) 1.

In set theory, the term "lemma" generally refers to a proven statement or proposition that is used as a stepping stone to prove other statements or theorems. In mathematical writing, authors often introduce lemmas to break down complex proofs into smaller, more manageable pieces. A lemma may not be of primary interest in itself, but it helps to establish the truth of more significant results.

Zeckendorf's theorem states that every positive integer can be uniquely represented as a sum of one or more distinct non-consecutive Fibonacci numbers.

A semicircle is a shape that represents half of a circle. It is formed by cutting a circle along a diameter. The key characteristics of a semicircle are: 1. **Definition**: A semicircle consists of the arc of a circle that spans 180 degrees and its endpoints, which are the endpoints of the diameter. 2. **Diameter**: The line segment joining the endpoints of the arc is called the diameter of the semicircle.

The Subspace Theorem is a significant result in Diophantine approximation and algebraic geometry, primarily associated with the work of mathematician W. Michael M. Schmidt. It provides a strong criterion for understanding when certain types of linear forms in algebraic numbers can approximate other algebraic numbers closely.

The Six Exponentials Theorem is a result in complex analysis and differential equations that deals with the solutions of certain classes of linear differential equations. It establishes conditions under which specific linear combinations of exponential functions can represent the solutions to these equations.

Roth's theorem, established by mathematician Klaus Roth in 1951, is a significant result in the field of number theory, particularly in the study of arithmetic progressions and additive combinatorics. The theorem specifically deals with the distribution of rational approximations to irrational numbers. In its classical form, Roth's theorem states that if \(\alpha\) is an irrational number, then it cannot be well-approximated by rational numbers in a very precise way.

Lindström's theorem is a significant result in model theory, a branch of mathematical logic that deals with the relationships between formal languages and their interpretations, or models. Formulated by Per Lindström in the 1960s, the theorem characterizes the logical systems that enjoy certain completeness and categoricity properties, specifically those known as the "Lindström properties.

Ribet's theorem is a fundamental result in number theory related to the Taniyama-Shimura-Weil conjecture, which is a key element in the proof of Fermat's Last Theorem. The theorem, proved by Ken Ribet in 1986, establishes a crucial connection between elliptic curves and modular forms.

The Nagell–Lutz theorem is a result in the theory of Diophantine equations, specifically concerning the representation of integers as sums of powers of natural numbers. It states that if a prime \( p \) can be expressed as a sum of two square numbers, i.e.

The Modularity Theorem, which is a significant result in number theory, asserts a deep connection between elliptic curves and modular forms. Specifically, it states that every rational elliptic curve over the field of rational numbers is modular.

Robinson's joint consistency theorem is a result in the field of decision theory and economics related to the consistency of preferences and the representation of preferences by a utility function. The theorem addresses the question of how to represent preferences over a set of choices that may vary according to certain parameters. Specifically, it deals with the conditions under which a joint distribution of choices can be consistent with the preferences of agents when making those choices.

Frege's theorem is a significant result in the foundations of mathematics and logic, attributed to the German mathematician and philosopher Gottlob Frege. It establishes the connection between logic and mathematics, specifically concerning the foundations of arithmetic. At its core, Frege's theorem asserts that the basic propositions of arithmetic can be derived from purely logical axioms and definitions. More specifically, it shows that the arithmetic of natural numbers can be defined in terms of logic through the formalization of the concept of number.

The Deduction Theorem is a fundamental principle in propositional logic and mathematical logic. It establishes a relationship between syntactic proofs and semantic entailment. The theorem can be stated as follows: If a formula \( B \) can be derived from a set of premises \( \Gamma \) along with an additional assumption \( A \), then it is possible to infer that the implication \( A \rightarrow B \) can be derived from the premises \( \Gamma \) alone.

Faltings's theorem, proven by Gerd Faltings in 1983, is a significant result in number theory and algebraic geometry. The theorem states that: **For a given algebraic curve defined over the rationals (or more generally, over any number field), there are only finitely many rational points on the curve, provided the genus of the curve is greater than or equal to 2.

Behrend's theorem is a result in the field of combinatorial number theory, particularly concerning the distribution of numbers that are free of a specific type of arithmetic progression.

The Thom conjecture, proposed by mathematician René Thom in the 1950s, relates to topology and singularity theory. Specifically, it concerns the structure of non-singular mappings between manifolds and the conditions under which certain types of singularities can occur. The conjecture asserts that every real-valued function defined on a manifold can be approximated by a function that has a certain type of "generic" singularity.