"Extension by new constant and function names" usually refers to a concept in formal logic and model theory, particularly in the context of extending a theory by adding new symbols for constants and functions. In formal logic, a theory can be thought of as a set of sentences in a formal language. Sometimes, one needs to expand or extend the language of the theory to include additional elements. Here's how this works in practice: 1. **New Constants**: You can introduce new constant symbols into the language.

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.

The Cut-Elimination Theorem is a fundamental result in proof theory, particularly in the context of sequent calculus and formal systems. It asserts that any proof in a certain logical system that includes the use of "cut" inference rules can be transformed into a proof that does not use these cut rules, thus ensuring that the proof is "cut-free.

Craig's theorem is a result in the field of mathematical logic, particularly in model theory. It is named after William Craig, who formulated it in the context of first-order logic. The theorem states that if a set of first-order statements (a theory) has a model, then it has a countable model.

The concept of completeness in the context of atomic initial sequents is primarily discussed in the realm of formal logic and proof theory, particularly in relation to sequent calculi, which are systems used for representing logical deductions. **Atomic Initial Sequents** refer specifically to sequents that consist of atomic formulas only. A sequent generally has the form \( A_1, A_2, ..., A_n \vdash B \), where the formulas \( A_1, A_2, ...

Codd's theorem is a fundamental result in the field of relational databases, formulated by Edgar F. Codd, who is also credited with developing the relational model for database management systems. The theorem essentially states that a relational database can be fully understood and manipulated using only a set of operations, specifically based on the relational algebra, without needing to rely on the underlying implementation details.

The Bourbaki–Witt theorem is a result in the field of mathematics, specifically in the area of linear algebra and the theory of groups and fields. It establishes a connection between vector spaces over division rings and certain algebraic structures related to linear transformations. In its most common formulation, the Bourbaki–Witt theorem provides a characterization of the structure of finite-dimensional vector spaces.

The Borel determinacy theorem is a significant result in set theory, particularly in the context of descriptive set theory. It concerns games played on sets of natural numbers and specifically establishes that certain types of games are determined.

The Barwise Compactness Theorem is a result in model theory, specifically concerning first-order logic and structures. It extends the concept of compactness, which states that if every finite subset of a set of first-order sentences has a model, then the entire set has a model. The Barwise Compactness Theorem applies this idea to certain kinds of structures known as "partial structures.

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.

The Von Staudt–Clausen theorem is a result in the field of number theory, particularly concerning the theory of continued fractions and the approximation of numbers. The theorem provides a way to express a specific class of numbers, notably the values of certain mathematical constants, as a sum involving continued fractions.

The Turán–Kubilius inequality is a result in number theory and probabilistic number theory, often related to the distribution of prime numbers. It provides a bound on the probability that certain events, often concerning the sums of random variables, will occur.

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.

Sophie Germain's theorem is a result in number theory concerning prime numbers. It states that if \( p \) is a prime number, and \( 2p + 1 \) is also prime, then \( p \) is called a Sophie Germain prime, and \( 2p + 1 \) is called a safe prime.

The Skolem–Mahler–Lech theorem is a result in number theory and in the study of sequences which concerns the behavior of integer sequences defined by linear recurrence relations. More specifically, it deals with the properties of the zeros of such sequences.

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.

Serre's modularity conjecture, proposed by Jean-Pierre Serre in the 1980s, is a deep and influential hypothesis in the field of number theory, particularly concerning the relationship between modular forms and elliptic curves.

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.