Introduction to Set Theory: Fundamentals, Construction, and Arithmetic

[evinda]

Hello! (Wave)

What is the subject Set Theory about?
What knowledge is required? (Thinking)

That is the Course Content:

Brief report on basic elements (algebra of sets, relations and functions, etc..). Construction of the set of natural numbers. Ordinal numbers and their arithmetic. The axiom of choice. Cardinal numbers and their arithmetic.

 

[mathbalarka]

What knowledge is required? (Thinking)

If I recall correctly, set theory doesn't require any background at all, except basic algebra. It's fundamental enough to understand with as little knowledge of mathematics as possible.

If you are interested in reading about it, I'll refer you to Book of Proof by Hammack.

 

[Evgeny.Makarov]

Brief report on basic elements (algebra of sets, relations and functions, etc..). Construction of the set of natural numbers. Ordinal numbers and their arithmetic. The axiom of choice. Cardinal numbers and their arithmetic.

Set theory is one of the four parts of mathematical logic (the other three are model theory, recursion theory, or theory of computation, and proof theory; see the Contents section of the Handbook of Mathematical Logic). It started in the late 19th century as the study of the foundations of mathematics. Even now it remains one of the most un-applied parts of mathematics because it focuses on infinite sets. Even so, along with other branches of logic, it has some applications to computer science. For example, infinite ordinals (a generalization of natural numbers) are used to characterize functions expressible in lambda-calculus, which is the foundation of functional programming languages. But set theory serves as an important foundation of other parts of mathematics (not didactically, but methodologically).

Set theory is divided into naive and axiomatic. Naive set theory studies algebra of sets, relations and functions and so on from your description. It is absolutely indispensable to any mathematician, and because it is used everywhere, it is sometimes not taught as a separate class, but a student is supposed to absorb it from other subjects. In the US, it is often taught in the discrete mathematics courses.

If one is not careful, using naive set theory one can come up with paradoxes, so axiomatic set theory clarifies rules for forming sets and proceeds to study their deeper properties. Ordinal numbers, the axiom of choice and cardinal numbers are the first topics of this more advanced set theory. Even though it is possible to do math without touching them, at some point it becomes problematic. For example, axiom of choice is used to prove that every vector space has a basis. It is also used to construct the Banach–Tarski paradox.

If I recall correctly, set theory doesn't require any background at all, except basic algebra. It's fundamental enough to understand with as little knowledge of mathematics as possible.

This is true about naive set theory. More advanced parts, like the rest of mathematical logic, do not indeed rely on any particular subject, but they require a certain mathematical maturity. For example, ordinals are sets that contain other sets that in turn contain other sets and so on. Settling this in one's head is not easy.

 

[mathbalarka]

This is true about naive set theory. More advanced parts, like the rest of mathematical logic, do not indeed rely on any particular subject, but they require a certain mathematical maturity. For example, ordinals are sets that contain other sets that in turn contain other sets and so on. Settling this in one's head is not easy.

Fair enough. I have never studied axiomatic set theory except knowing what Zorn's lemma is.

 

[blue_raver22]

Set Theory is a branch of mathematics that deals with the study of collections of objects, known as sets, and the relationships between these sets. It is a fundamental part of mathematics and has applications in various fields such as computer science, physics, and economics.

To understand Set Theory, a strong foundation in basic mathematical concepts such as algebra, relations, and functions is necessary. A solid understanding of logic and proof techniques is also important in order to grasp the fundamental concepts of Set Theory.

The course content typically covers topics such as the algebra of sets, which includes operations such as union, intersection, and complement. It also explores relations and functions, which are important tools for analyzing the connections between sets. The construction of the set of natural numbers, known as the Peano axioms, is another key element in Set Theory.

Ordinal numbers, which are used to describe the order of objects in a set, and their arithmetic are also studied in Set Theory. The axiom of choice, which states that given a collection of non-empty sets, it is possible to choose one element from each set, is an important concept in Set Theory and has implications in other areas of mathematics.

Lastly, cardinal numbers, which are used to describe the size or magnitude of a set, and their arithmetic are also covered in Set Theory. This includes topics such as countable and uncountable sets, the continuum hypothesis, and the concept of infinity.

In summary, Set Theory is a fascinating and essential subject in mathematics that provides a framework for understanding the relationships and properties of collections of objects. It requires a strong foundation in basic mathematical concepts and logic, and its applications can be seen in a wide range of fields.