MHB Introduction to Set Theory: Fundamentals, Construction, and Arithmetic

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Set theory is a fundamental area of mathematics that explores the nature of sets, including their construction and arithmetic, such as natural numbers, ordinal numbers, and cardinal numbers. It requires minimal background knowledge, primarily basic algebra, making it accessible to beginners. The field is divided into naive set theory, which can lead to paradoxes, and axiomatic set theory, which provides a structured framework to avoid these issues. Set theory serves as a foundational component of mathematical logic and has applications in computer science, particularly in functional programming. Understanding both naive and axiomatic set theory is essential for deeper mathematical study, though advanced concepts require a higher level of mathematical maturity.
evinda
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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.
 
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evinda said:
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.
 
evinda said:
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.

mathbalarka said:
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.
 
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.
 
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
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