What is the purpose of studying set theory?

In summary, the study of set theory has great mathematical utility as it allows for a better understanding of logic and the construction of mathematical objects such as the reals and infinite dimensional vector spaces. It also raises important questions about foundational principles in mathematics.
  • #1
Brunno
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What is the great mathematical utility of set theory?


When you study set theory or when you studied do(did ) you know for what ecxacly you are(were) studying?And what do you know now?
 
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  • #2
Yes, when I studied "set theory" I knew exactly what I was studying. In fact set theory basics (fundamental definitions) are remarkably simple (although like anything, it has some complex ramifications).
 
  • #3
When I studied set-theory, I tended to reject question `for what exactly am I studying this?' - but in those days I was very theory oriented and interested in foundational questions.

After studying it, I knew a lot more set-theory. I also understood a lot more about logic.
I also had some idea of how certain constructions in classical mathematics - the construction of the reals, the existence of bases for infinite dimensional vector spaces - depended on some subtle and contestable basic principles.

But mainly, I knew a lot more set-theory.
 

What is Set Theory?

Set Theory is a branch of mathematics that deals with the study of sets, which are collections of objects. It provides a foundation for all of mathematics and is used in various fields such as computer science, physics, and statistics.

What are the basic concepts in Set Theory?

The basic concepts in Set Theory include sets, elements, subsets, unions, intersections, and complements. A set is a collection of objects, and the objects in a set are called elements. A subset is a set that contains elements from another set. A union is a set that contains all elements from two or more sets. An intersection is a set that contains only elements that are common to two or more sets. A complement is a set that contains all elements not in a given set.

What is the difference between a finite and infinite set?

A finite set has a limited number of elements, while an infinite set has an infinite number of elements. For example, the set of all even numbers is an infinite set, while the set of months in a year is a finite set.

What is the Axiom of Choice in Set Theory?

The Axiom of Choice is a fundamental principle in Set Theory that allows us to choose one element from each set in a collection of sets, even if the sets are infinite. It is often used to prove the existence of mathematical objects that cannot be explicitly constructed.

What is the Continuum Hypothesis in Set Theory?

The Continuum Hypothesis is a statement about the cardinality of sets, which states that there is no set whose cardinality is strictly between the cardinality of the set of natural numbers and the set of real numbers. This statement was proven to be undecidable within the standard axioms of Set Theory, leading to the development of new axioms to address it.

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