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Fundamentals of Mathematics

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Gray code at the pediatrician's office. Gray code at the pediatrician's office Last week we took Iris to the pediatrician for a checkup, during which they weighed, measured, and inoculated her.

Gray code at the pediatrician's office

The measuring device, which I later learned is called a stadiometer, had a bracket on a slider that went up and down on a post. Iris stood against the post and the nurse adjusted the bracket to exactly the top of her head. Then she read off Iris's height from an attached display. How did the bracket know exactly what height to report? (Click to view the other pictures I took of the post.) The pattern is binary numerals. The patterned strip in the left margin of this article is a straightforward translation of binary numerals to black and white boxes, with black representing 1 and white representing 0: If you are paying attention, you will notice that although the strip at left is similar to the pattern in the doctor's office, it is not the same.

Gray codes solve the following problem with raw binary numbers. Tarski's undefinability theorem. Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1936, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics.

Tarski's undefinability theorem

Informally, the theorem states that arithmetical truth cannot be defined in arithmetic. The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system. History[edit] In 1931, Kurt Gödel published his famous incompleteness theorems, which he proved in part by showing how to represent syntax within first-order arithmetic. Each expression of the language of arithmetic is assigned a distinct number. In particular, various sets of expressions are coded as sets of numbers. The undefinability theorem shows that this encoding cannot be done for semantical concepts such as truth.

The undefinability theorem is conventionally attributed to Alfred Tarski. Statement of the theorem[edit] -hard for all k. Arithmetical hierarchy. The arithmetical hierarchy is important in recursion theory, effective descriptive set theory, and the study of formal theories such as Peano arithmetic.

Arithmetical hierarchy

The Tarski-Kuratowski algorithm provides an easy way to get an upper bound on the classifications assigned to a formula and the set it defines. The hyperarithmetical hierarchy and the analytical hierarchy extend the arithmetical hierarchy to classify additional formulas and sets. The arithmetical hierarchy of formulas[edit] The arithmetical hierarchy assigns classifications to the formulas in the language of first-order arithmetic. Morphism. In category theory, morphism is a broadly similar idea, but somewhat more abstract: the mathematical objects involved need not be sets, and the relationship between them may be something more general than a map.

Morphism

The study of morphisms and of the structures (called objects) over which they are defined, is central to category theory. Much of the terminology of morphisms, as well as the intuition underlying them, comes from concrete categories, where the objects are simply sets with some additional structure, and morphisms are structure-preserving functions. In category theory, morphisms are sometimes also called arrows. Definition[edit] There are two operations which are defined on every morphism, the domain (or source) and the codomain (or target).

If a morphism f has domain X and codomain Y, we write f : X → Y. For every three objects X, Y, and Z, there exists a binary operation hom(X, Y) × hom(Y, Z) → hom(X, Z) called composition. Morphisms satisfy two axioms: Some specific morphisms[edit]