Mathematics \ Mathematical Logic

Mathematical logic is a subfield of mathematics that explores the applications and foundations of logic within mathematical contexts. The main goal of mathematical logic is to formalize the intuitions and principles underlying mathematical reasoning, and to study the structure and implications of these formal systems. This area is vital not only for theoretical purposes but also for practical applications in computer science, linguistics, and philosophy.

Key Concepts and Areas of Study:

1. Propositional Logic (or Sentential Logic):
   • Deals with propositions which can be either true or false.
   • Studied through logical connectives like AND (∧), OR (∨), NOT (¬), IMPLICATION (→), and BICONDITIONAL (↔︎).
   • An essential aspect is the formation of truth tables that show the truth value of logical expressions depending on the truth values of their components.
2. First-Order Logic (Predicate Logic):
   • Extends propositional logic by including quantifiers and predicates, thereby increasing its expressive power.
   • Syntax:
     • Variables (e.g., \( x, y, z \))
     • Constants (e.g., \( a, b, c \))
     • Predicates (e.g., \( P(x), Q(x, y) \))
     • Quantifiers: Universal (\( \forall \)) and Existential (\( \exists \))
   • Semantics:
     • Interpretation for the domain of discourse, assigning meanings to symbols.
     • An example formula in first-order logic is \( \forall x \ (P(x) \to \exists y \ Q(x, y)) \), interpreted as “for all \( x \), if \( P(x) \) is true, then there exists a \( y \) such that \( Q(x, y) \) is true.”
3. Model Theory:
   • Studies the relationship between formal languages (like those of propositional and first-order logic) and their interpretations or models.
   • A key focus is on the concept of satisfaction, where a model satisfies a sentence if the sentence is true in that model.
4. Proof Theory:
   • Concerns the nature of mathematical proofs.
   • It formalizes proofs as mathematical objects, often using formal systems such as Hilbert systems, natural deduction, and sequent calculus.
   • Gödel’s incompleteness theorems are a major result, showing inherent limitations in formal systems for arithmetic.
5. Set Theory:
   • Provides the foundational language for virtually all of mathematics.
   • Studies sets, which are collections of objects, and the relations between them.
   • Axiomatic set theory, such as Zermelo-Fraenkel set theory (ZF), includes axioms intended to avoid paradoxes like Russell’s paradox.
6. Recursion Theory (Computability Theory):
   • Investigates what problems can be solved by algorithms and which mathematical functions are computable.
   • Central concepts include Turing machines, the halting problem, and decidability.

Mathematical Expressions:

One widely used system in proof theory and first-order logic is the sequent calculus, where a sequent is an expression of the form:

\[ \Gamma \vdash \Delta \]

Here, \( \Gamma \) and \( \Delta \) are, respectively, sets (or multisets) of formulas, indicating that the conjunction of formulas in \( \Gamma \) implies the disjunction of formulas in \( \Delta \).

In model theory, we often formalize the satisfaction relation with the notation:

\[ \mathcal{M} \models \phi \]

Where \( \mathcal{M} \) is a model and \( \phi \) is a formula, indicating that \( \phi \) is true in \( \mathcal{M} \).

Mathematical logic combines deep theoretical insights with rigorous formal systems, providing the backbone for much of modern mathematics and theoretical computer science. It seeks to understand the nature of logic itself, formalize reasoning, and rigorously establish the foundations upon which mathematical theories are built.