samspotting said:So for an undergrad talk, I am going to present the proof of dedekind cuts. Besides proving that it makes an ordered field containing the rationals with the least upper bound property, what else could I saw to spice things up a bit?
Dedekind cuts are a mathematical concept created by German mathematician Richard Dedekind. They are a way to represent real numbers by dividing the number line into two sets, with one set containing all numbers less than the chosen number and the other set containing all numbers greater than or equal to the chosen number.
Dedekind cuts are used in mathematics as a way to define real numbers without relying on a specific numerical representation. They are particularly useful in the construction of the real numbers and in rigorous proofs involving real numbers.
Dedekind cuts have several important properties, including the existence of a least upper bound and greatest lower bound for each set, the completeness property (meaning every non-empty set of Dedekind cuts has a least upper bound), and the ability to perform arithmetic operations such as addition and multiplication.
Dedekind cuts are closely related to other mathematical concepts, such as rational numbers, irrational numbers, and the real numbers. They are also used in the construction of other mathematical structures, such as the complex numbers and the p-adic numbers.
Dedekind cuts have applications in various areas of mathematics, including analysis, number theory, and topology. They are also used in computer science and physics, particularly in the study of continuous and infinite structures.