Foundations of mathematics

Foundations of mathematics is a branch of mathematical logic that seeks to understand the fundamental concepts and principles that underpin mathematics as a whole. It explores the nature of mathematical objects, the validity of mathematical reasoning, and the scope and limitations of mathematical systems. The field addresses several key areas, including: 1. **Set Theory**: This is the study of sets, which are collections of objects.

Theorems in the foundations of mathematics are statements or propositions that have been rigorously proven based on a set of axioms and previously established theorems. The field of foundations of mathematics investigates the nature, structure, and implications of mathematical reasoning and its underlying principles.

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.

The Condensation Lemma is a result in the context of automata theory and formal languages, particularly concerning context-free grammars and their equivalence. It mainly states conditions under which certain types of grammars can be simplified without losing their generative power. In a broader sense, the lemma is often framed as follows: 1. **Grammar Definitions**: Consider a context-free grammar (CFG) that generates a language.

Fodor's lemma is a result in set theory that is often used in the context of infinite combinatorics and descriptive set theory. It is named after the Hungarian mathematician Géza Fodor, who introduced it. **Statement of Fodor's Lemma:** Let \( X \) be a set and let \( \kappa \) be an infinite cardinal.

The Moschovakis coding lemma is a result in mathematical logic, particularly in the area of recursion theory and effective descriptive set theory. It is named after Yiannis N. Moschovakis, who made significant contributions to these fields. The lemma is concerned with the concept of **coding sets of natural numbers** using **recursive (or computable) functions**.

The Mostowski Collapse Lemma is a result in set theory, particularly in the context of the theory of well-founded relations. It is used primarily to show that any well-founded relation can be transformed into a linear order (or a total order) and that any set can be "collapsed" to obtain a well-founded set.

The Rasiowa–Sikorski lemma is a result in the field of mathematical logic, particularly in set theory and model theory. It provides a criterion for determining whether a certain kind of subset exists in a model of set theory. The lemma is named after the mathematicians Helena Rasiowa and Andrzej Sikorski, who contributed to the field of logic in the mid-20th century.

Zorn's Lemma is a principle in set theory that is often used in the context of proving the existence of certain types of mathematical objects. It is equivalent to the Axiom of Choice and the Well-Ordering Theorem, meaning that if one of these statements is assumed to be true, then the others can be proven true as well.

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.

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 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.

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 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, ...

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 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.

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.

"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.

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.

Gödel's speed-up theorem is a result in the field of mathematical logic, particularly in the study of formal systems and computability. It essentially states that for certain mathematical statements that can be proven in a relatively weak formal system, there exist stronger systems in which those statements can be proven more efficiently—specifically, in what is known as "faster" or more succinct proofs.

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.

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.

Kleene's recursion theorem, named after mathematician Stephen Cole Kleene, is a fundamental result in the field of computability theory. It addresses the existence of computable functions that can be defined recursively. The theorem states that for any total computable function \( f \), there exists a program (or particular index in the sense of the arithmetical hierarchy) that produces itself as an output when given its own index (or code) as input.

The Knaster-Tarski theorem is a fundamental result in the field of fixed-point theory, particularly in the context of partially ordered sets (posets).

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.

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.

Löb's theorem is a result in mathematical logic, particularly in the area concerning formal systems and provability. It deals with self-referential statements in formal systems and is often discussed in the context of Gödel's incompleteness theorems.

The Paris–Harrington theorem is a result in the field of mathematical logic and combinatorics, specifically in the area of set theory and the study of large cardinals. It is a form of combinatorial principle that exemplifies the limits of certain deductive systems, particularly in relation to the axioms of Peano arithmetic and other standard set theories.

Post's theorem, named after Emil Post, is a result in the field of mathematical logic and computability theory. It specifically deals with the properties of recursively enumerable sets, particularly in the context of formal languages and decision problems. The theorem states that: **"For any countable set of recursive (or computable) functions, there exists a recursively enumerable set that captures all the functions from the set.

Rice's theorem is a fundamental result in computability theory that addresses the limits of what can be determined about the behavior of Turing machines and languages recognized by them. Specifically, the theorem states that any non-trivial property of the languages recognized by Turing machines is undecidable.

The Rice–Shapiro theorem, often referred to as Rice's theorem in the context of computability theory, is a fundamental result concerning the properties of recursively enumerable (r.e.) sets and the functions computable by Turing machines. In its standard form, Rice's theorem states that any non-trivial property of the languages recognized by Turing machines is undecidable.

Richardson's theorem is a result in the field of mathematical logic, specifically in the area of computability theory. The theorem states that if \( A \) is a recursively enumerable (r.e.) set, then the set of its recursive subsets is r.e. This theorem has significant implications for understanding the structure of recursively enumerable sets and their relationships to recursive sets. In more technical terms, the theorem provides a comprehensive characterization of the recursive subsets of a recursively enumerable set in terms of effective enumerability.

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.

The Schröder–Bernstein theorem is a fundamental result in set theory that provides a criterion for the existence of a bijection (one-to-one and onto correspondence) between two sets, given certain conditions about the existence of injections (one-to-one functions) between those sets. In the context of measurable spaces, the theorem can be reformulated to pertain to the measurability of the functions involved.

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.

Tarski's undefinability theorem is a result in mathematical logic that deals with the concept of truth within formal languages. Named after the logician Alfred Tarski, the theorem asserts that the notion of truth cannot be defined within a sufficiently expressive formal language that can express arithmetic truths about itself.