What is Set theory: Definition and 442 Discussions

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole.
The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of naive set theory. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and Burali-Forti paradox) various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied.
Set theory is commonly employed as a foundational system for the whole of mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beside its foundational role, set theory also provides the framework to develop a mathematical theory of infinity, and has various applications in computer science, philosophy and formal semantics. Its foundational appeal, together with its paradoxes, its implications for the concept of infinity and its multiple applications, have made set theory an area of major interest for logicians and philosophers of mathematics. Contemporary research into set theory covers a vast array of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.

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  1. G

    Can Set Difference and Union Operations Prove Subset Relations in Set Theory?

    Homework Statement Suppose B is a set and suppose \mathcal{F} is a family of sets. Prove that \cup {A\setminus B|A \in \mathcal{F}}\subseteq \cup(\mathcal{F}\setminus \mathcal{P}(B)) For want of a better way I'm denoting powerset of B as \mathcal{P}(B)) Homework Equations The...
  2. A

    Symmetric/Antisymmetric Relations, Set Theory Problem

    Homework Statement Prove that if R is a symmetric relation on A, and Dom(R) = A, then R = the identity relation. 2. The attempt at a solution My problem is... I don't believe the claim. At all. If A = {1, 2, 3} and R = {(1, 2), (2, 1), (3, 1), (1, 3)}, that satisfies the antecedent, and...
  3. F

    Probability - Set Theory Question

    So I'm taking a probability class right now. We are going over elementary set theory, and the professor brought something up which seems non-intuitive to me. He said that a set must have distinct objects, so... A = \{ 1, \,\, 1, \,\, 2, \,\, 3 \} is not properly defined, because the 1 is...
  4. G

    Classical First-Order Logic, Axiomatic Set Theory, and Undecidable Propositions

    It has been known for some time that the Axiom of Choice (if you treat it as a proposition to be proved rather than an axiom) and the Continuum Hypothesis are independent of Zermelo-Fraenkel set theory (ZF). These and other statements (Suslin's Problem, Whitehead's Problem, the existence of...
  5. G

    Quantifier equivalence in set theory

    Homework Statement I have been asked to show that \exists xAP(x)\vee\exists xBP(x) is equivalent to \exists x(A\cup B)P(x) Homework Equations 1) P\rightarrow Q \equiv \neg P \vee Q 2) \neg(P\vee Q)\equiv \neg P \wedge \neg Q 3) P \vee (Q\vee R) \equiv (P\vee Q) \vee R \equiv P...
  6. Oxymoron

    Elementary Set Theory Problem Checking

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  7. J

    Set Theory Q: Is |P(A)| = |P(B)| iff A=B? Hints Needed

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  8. M

    Filling in the blanks proof, having some issues Set Theory Unions/Subsets

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  9. M

    Set Theory, Can you check to see if this is right, Is AoA3 a parition of Z?

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  10. M

    Can Nested Subsets Prove Equality in Limits?

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  11. phoenixthoth

    What is the structure of the Ultra Power Space?

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  12. J

    Math Preparing for Grad School: Set Theory & Logician Career Prospects

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  13. Oxymoron

    What is the Relationship Between Union and Power Sets?

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  14. S

    Proving A=B When A U B is a Subset of A Intersect B: Set Theory Explained

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  15. D

    Set Theory vs Logic: Which Should Come First?

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  16. U

    Basic Set Theory: Understanding Problems

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  17. MathematicalPhysicist

    Some quetions about set theory.

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  18. J

    Current Set Theory Research: Where to Find Programs in CA

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  19. JasonJo

    Nightmares with formal proofs in set theory

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  20. MathematicalPhysicist

    Equivalence of Functions and Power Sets in Set Theory

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  21. P

    Real Analysis (Set Theory) Proof

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  22. E

    Understanding Set Theory: Equivalence Relations and Partitions Explained

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  23. E

    Set Theory: How Many Elements are in the Cartesian Product of Two Sets?

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  24. MathematicalPhysicist

    Set Theory Questions: Equivalence Rules and Tautology Proof

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  25. agro

    Naive Set Theory by Paul R. Halmos

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  26. P

    Set theory proves that god does NOT exists Or DOES exist

    As I have not received an answer for the post I submitted under the thread "Does the set theory prove that there is no God?" I decided to submit it here. Without having much knowledge of set theory, How exactly does the Set theory prove that there is no god? Thank you in advance for your kind...
  27. G

    What is Set Theory and How is it Used in Science?

    Can someone introduce me to what the basics of set theory are, mean, are use in, have been created by, and it's branches?
  28. C

    Does the set theory prove that there is no God?

    One of the axioms of the axiomatic set theory is that there are no universal sets. However, God is omnipresent, so he would have to be this universal set. Thus, God would immediately lead to a contradiction within the set theory. Does this prove that there is no God?
  29. A

    When did set theory prevail?

    Dear members, it has recently been put forth to me the question as to when and by which means did it become clear to the majority of logicians and mathematicians that set theory prevailed over Russell's theory of types (as in Principia)? (By set theory I mean as in the axioms of...
  30. K

    Infinite set theory has any place in finding limits

    Hi...I'm wondering if infinite set theory has any place in finding limits. Is there any way that tabling elements of sets can find you the answer to a limit question?
  31. C

    Set Theory Proof: A vs. B-C vs. C

    Hi all (i) A- (B-C) = (A-B) U C (ii) A - (B U C) = (A-B) - C Which one is always right and which is sometimes wrong? My solution If x is an element of A - (B-C), then x is not contained in B-C. If y is an element of (A-B) U C, then y is at least one of A-B or C. (i) is sometimes...
  32. C

    How do I calculate the number of elements in set Z using set theory?

    Hello all Set X has x elements and Sset Y has y elements and Set Z consists of all elements are are in either set X or set Y with the exception of the k common elements. I know that the answer is x+ y - 2k, however how would I get this? Should I just use a practical example?
  33. E

    Russell's type theory, ZF set theory

    What are the weaknesses and strengths of Russell's type theory? I'm having trouble understanding exactly what makes it weaker than ZF. I've been told its too restrictive, but what exactly makes it more restrictive than ZF? Are there any objections to ZF?
  34. J

    Cantor's diagonal argument (I guess technically this comes under set theory)

    I have a question about the potentially self-referential nature of cantor's diagonal argument (putting this under set theory because of how it relates to the axiom of choice). If we go along the denumerably infinite list of real numbers which theoretically exists for the sake of the example...
  35. T

    Some elementary set theory questions

    I am currently reviewing for an upcoming test over sets. What the instructor did was to give out the test he gave out for last semester for us to study from. I can answer most of these questions but there are a few that I am a little bit unsure of. Some of the questions are complete the...
  36. T

    Elementary set theory problem

    Let C be a circle and let D be the set of all diameters of C. What is \capD? I think it is the center of the circle since that would be the only point of intersection of all the diameters of the circle. Could someone let me know if I am correct? Regards Jeremy
  37. S

    Finding the Intersection of Infinite Sets: A Basic Set Theory Question

    ok, find the intersection of i=1 to infinitie of A(i). A(i) = [0, to 1/i]. I don't understand what it is asking. what do they mean to find the intersection of that? and what if A(i) = [0, 1/n)
  38. K

    Is x not in A or B AND not in C?

    I'm trying to prove something small with set theory and since I'm new to it, I've run into a problem. I can't understand what the following means exactly and how to proceed further. Or where the mistake is, if there is one. I think there is, because it seems... freaky...
  39. E

    Set Theory Proofs: f:X->Y Function and Subset B of Y

    Let f:X->Y be a function 1) Given any subset B of Y, prove that f(f^-1(B)) is a subset of B 2) Prove that f(f^-1(B))=B for all subsets B of Y if and only if f is surjective Help anybody?
  40. E

    Some basic definition questions of set theory

    I have to prove the following theorem, 1) If f:A->B is a surjection, and g:B->C is a surjection then g dot f:a->C is a surjection Well this makes sense and I am not sure how to PROVE it Is it sufficient to say the following if for every element b of B, there exists a element a of A...
  41. Y

    Is the Image of an Open Set Under a Continuous Function Always Open?

    Let f be a real-valued function defined and continuous on the set of real numbers R. Which of the following must be true of the set S = {f(c): 0<c<1}? I. S is a connected subset of R II. S is an open subset of R III. S is a bounded subset of R The answer is I and III only. I understand...
  42. MathematicalPhysicist

    When category theory and set theory meet

    was there an attempt to unite between those two fields?
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