Hyperreal §

Group: 4 #group-4

Relations §

• Ordered Field: The hyperreal field is an ordered field, meaning it has a total order relation that is compatible with the field operations.
• Simulacra and Simulation: The book explores the concept of the hyperreal, where simulations become more real than reality itself.
• Surreal Numbers: The hyperreal numbers are a proper extension of the surreal numbers, which also contain infinitely large and infinitely small numbers.
• Infinitesimal: Hyperreal numbers contain infinitesimals, which are numbers smaller than any positive real number.
• Nonstandard Models: The hyperreal field is a nonstandard model of the real numbers, meaning it extends the real number system with additional elements.
• Infinitely Small: Hyperreal numbers include infinitely small numbers, which are smaller than any positive real number.
• Non-Archimedean Field: The hyperreal field is a non-Archimedean field, meaning it contains infinitely large and infinitely small numbers.
• Mathematical Logic: The study of hyperreal numbers and nonstandard analysis involves concepts from mathematical logic, such as model theory and nonstandard models.
• Nonstandard Analysis: Hyperreal numbers are used in nonstandard analysis, which is a branch of mathematical analysis that deals with infinitesimals and infinitely large numbers.
• Ultrafilter: The construction of the hyperreal field involves the use of ultrafilters, which are mathematical objects used in model theory.
• Model Theory: The construction of the hyperreal field is based on model-theoretic techniques, such as the use of ultrafilters and nonstandard models.
• Calculus: Hyperreal numbers can be used to provide a rigorous foundation for calculus and to simplify certain calculations and proofs.
• Hyperreal Field: The hyperreal field is the field of hyperreal numbers, which extends the real number system with infinitesimals and infinitely large numbers.
• Infinitely Large: Hyperreal numbers include infinitely large numbers, which are larger than any real number.
• Algebraic Structure: The hyperreal field is an algebraic structure that satisfies the axioms of a field and has additional properties related to infinitesimals and infinitely large numbers.
• Transfer Principle: The transfer principle is a fundamental tool in nonstandard analysis that allows transferring properties and theorems from the real numbers to the hyperreal numbers.