Set theory Definition and 444 Threads

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. P

    I Construction of sigma-algebras: a counterexample

    Consider a set ##X## and family of sets ##\mathcal E\subset\mathcal P(X)##. Let ##\mathcal E_1=\mathcal{E}\cup\{E^c:E\in\mathcal E\}## and then for ##j>1## define ##\mathcal E_j## to be the collection of all sets that are countable unions of sets in ##\mathcal E_{j-1}## or complements of such...
  2. P

    I Negation of statement involving cardinalities

    In Folland's real analysis book, he defines the following expressions: $$\operatorname{card}(X)\leq\operatorname{card}(Y),\quad \operatorname{card}(X)=\operatorname{card}(Y),\quad \operatorname{card}(X)\geq\operatorname{card}(Y),$$to mean there exists an injection, bijection or surjection from...
  3. B

    Various Intuitions and Conceptualizations of Measurable Cardinals.

    The concept of a "measurable cardinal" is rather difficult for many students of "Intermediate" Set Theory to grasp in terms of more basic set theoretic concepts -- as opposed say to concepts dealing with the relations among various "universes" or "models" etc. In fact, much of the problem may...
  4. elcaro

    Type of system for programming language based on mathematical set theory

    A type in a programming language more or less specifies what values are allowed for a given type. A type has some similarities with a set, but most programming languages lack the operations which a set has. For example, in C one can define a new type based on two other types as a union of those...
  5. Z

    Prove every subset of countable set is either finite or else countable

    There are a lot of steps left out of this proof. Here is my proof with hopefully all the steps. I would like to know if it is correct Let ##A## be a countable set. Then ##A## is either finite or countably infinite. Case 1: ##A## is finite. There is a bijection ##f## from ##A## onto...
  6. john-ice2023

    Are the following statements true? (1) a∈{{a},{a,b}} and (2) b∈{{a},{a,b}} true?

    TL;DR Summary: Look deep into nature, and then you will understand everything better. Albert Einstein. I am new to set theory. I got confused about above questions. For Q(1), I have two solutions, (a) because a is not the element of set {{a},{a,b}}, so a∈{{a},{a,b}} is False. (b) because...
  7. olgerm

    I Question about about halting problem and this particular function

    The most known proof of undecidability of the halting problem is about like that: #assume we have an hypothetical function that can determine whether any program P would halt on input i. def H1(P, i): """ H1 is a hypothetical function that determines whether program P halts on input i...
  8. D

    B I have an issue with Cantor's diagonal argument

    I'm pretty bad at maths, got an A at gcse (uk 16 years old)then never went any further, I've been looking into cantors diagonal argument and I thing I found an issue, given how long its been around I'd imagine I'm not the first but couldn't any real number made using the construct by adding 1 to...
  9. K

    I Is thermal noise a statistical uncertainty?

    Hello! I have a system described by ##y=ax##, where a is the parameter I want to extract and y is the stuff I measure (we can assume that I can measure one instance of y without any uncertainty). x is a parameter I can control experimentally but it has an uncertainty associated to it. In a...
  10. WMDhamnekar

    I How to determine if a set is a semiring or a ring?

    Let E be a finite nonempty set and let ## \Omega := E^{\mathbb{N}}##be the set of all E-valued sequences ##\omega = (\omega_n)_{n\in \mathbb{N}}F##or any ## \omega_1, \dots,\omega_n \in E ## Let ##[\omega_1, \dots,\omega_n]= \{\omega^, \in \Omega : \omega^,_i = \omega_i \forall i =1,\dots,n...
  11. V9999

    I May I use set theory to define the number of solutions of polynomials?

    Let ##Q_{n}(x)## be the inverse of an nth-degree polynomial. Precisely, $$Q_{n}(x)=\displaystyle\frac{1}{P_{n}(x)}$$, It is of my interest to use the set notation to formally define a number, ##J_{n}## that provides the maximum number of solutions of ##Q_{n}(x)^{-1}=0##. Despite not knowing...
  12. H

    Proving that an Integer lies between x and y using Set Theory

    ## y-x \gt 1 \implies y \gt 1+x## Consider the set ##S## which is bounded by an integer ##m##, ## S= \{x+n : n\in N and x+n \lt m\}##. Let's say ##Max {S} = x+n_0##, then we have $$ x+n_0 \leq m \leq x+(n_0 +1)$$ We have, $$ x +n_0 \leq m \leq (x+1) +n_0 \lt y+ n_0 $$ Thus, ##x+n_0 \leq m \lt...
  13. chwala

    How to Efficiently Handle Set Operations Without Inbuilt Functions?

    ok we shall have ##(A-B)∩(A-C)= [1,y]∩[1,2,x,y]=[1,y]## correct?
  14. ShellWillis

    I Exploring the Next Generation of Set Theory: A Discussion on an Intriguing Paper

    https://www.cs.bham.ac.uk/~mhe/papers/omniscient-journal-revised.pdf Might be my favorite article I’ve ever came across I would like to see some interpretations on it to broaden my currently very narrow point of view… Have fun! -oliver
  15. Hallucinogen

    I Common features of set theory and wave functions?

    I would like to know if any of you think there's any sort of connection, analogy, or common features between, sets in set theory and wave functions in QT? Wave functions lack trajectories, so do sets. Wave functions also distribute over areas, as sets can do. To my understanding, wave...
  16. W

    A Range of values for ##2^{\aleph_0}##

    Ok, so assume we have a model for ZFC where CH does not hold. What values may ##2^{\aleph_0}## assume over said models?
  17. M

    I Connection between Set Theory and Navier-Stokes equations?

    Hi, I saw this video by numberphile, and near the end they mention how at the point of a right angle the equation shows infinite velocity for fluids. I'm wondering if this isn't perhaps related to Cantor's solution to Zeno's Paradox of distance (there's always a midpoint). Because I feel like at...
  18. C

    Can you use Taylor Series with mathematical objects other than points?

    I was recently studying the pressure gradient force, and I found it interesting (though this may be incorrect) that you can use a Taylor expansion to pretend that the value of the internal pressure of the fluid does not matter at all, because the internal pressure forces that are a part of the...
  19. S

    MHB Is Theorem 5.2 in SET THEORY AND LOGIC True or False?

    In the book: SET THEORY AND LOGIC By ROBERT S.STOLL in page 19 the following theorem ,No 5.2 in the book ,is given: If,for all A, AUB=A ,then B=0 IS that true or false If false give a counter example If true give a proof
  20. C

    I Question regarding quantifier statement

    Suppose I have the following ( arbitrary ) statement: $$ \forall x\in{S} \ ( P(x) ) $$ Which means: For all x that belongs to S such that P(x). Can I write it as the following so that they are equivalent? ( although it is not conventional ): $$ \forall x\in{S} \land ( P(x) ) $$ Can I write...
  21. D

    B My argument why Hilbert's Hotel is not a veridical Paradox

    Hello there, I had another similar post, where asking for proof for Hilbert’s Hotel. After rethinking this topic, I want to show you a new example. It tries to show why that the sentence, every guest moves into the next room, hides the fact, that we don’t understand what will happen in this...
  22. benorin

    Munkres Topology Ch 1 ex#7) part (c) — basic set theory Q

    Obviously the parenthetical part of the definition of ##F## means ##B\subset C## but we are not allowed to use ##\subset##. I do not know how to express implication with only union, intersection, and set minus without the side relation ##B\cap C = B\Leftrightarrow B\subset C##. This is using the...
  23. F

    MHB Introduction to Set Theory Stream

    Hello everyone. =) In honor of Pi Day I'm going to be explaining the very beginning of set theory (which I consider the beginning of university math) live on Twitch in about two hours (1 PM GMT). For those who do not know Twitch, it's a completely free streaming platform - you can come in and...
  24. S

    I Set Theory - the equivalence relation on elements

    According to https://plato.stanford.edu/entries/zermelo-set-theory/ , Zermelo (translated) said: I don't know if that quote is part of his formal presentation. It does raise the question of whether set theory must formally assume that there exists an equivalence relation on "elements" of...
  25. R

    B Can someone answer this doubt I have on Set theory?

    "The fact that the above eleven properties are satisfied is often expressed by saying that the real numbers form a field with respect to the usual addition and multiplication operations." -what do these lines mean? in particular the line "form a field with respect to"? is it something like...
  26. heff001

    A Higher Set Theory – Cantorian Sets / Large Cardinals in the Infinite

    Zermelo-Fraenkel Axioms - the Axiom of Choice (ZFC), is conceptually incoherent. To me, they stole Cantor’s brilliant work and minimized it. Replies?
  27. S

    On soundness and completeness of ZFC set theory

    Homework Statement: See attached image. Homework Equations: ZFC set theory. Consider the text in the attached image. What is meant with "We require of an axiom system that it be possible to decide whether or not any given formula is an axiom."? Is consistency synonymous with soundness? Is...
  28. A

    I Enumerating a Large Ordinal: Can We Find a Limit to the Continuum?

    The following assertion quoted from the paper below seems as though it couldn’t be true. It is the issue that I would like some help addressing please: “The restriction of ##g(A)## to ##A \cap \omega_1## ensures that ##B## remains countable for this particular ##T## sequence.” ... Define...
  29. C

    MHB Topology Munkres Chapter 1 exercise 2 e- Set theory

    Dear Everyone I am having some difficulties on exercise 2e from Topology 2nd ed by J. Munkres . Here are the directions: determine which of the following states are true for all sets A, B, C, and D. If a double implication fails, determine whether one or the other one of the possible...
  30. C

    MHB Topology Munkres Chapter 1 exercise 2 b and c- Set theory equivalent statements

    Dear Every one, I am having some difficulties on exercise 2b and 2c from Topology 2nd ed by J. Munkres . Here are the directions: determine which of the following states are true for all sets $A$, $B$, $C$, and $D$. If a double implication fails, determine whether one or the other one of the...
  31. J

    A First order logic and set theory: who comes first?

    Goldrei's Propositional and Predicate Calculus states, in page 13: "The countable union of countable sets is countable (...) This result is needed to prove our major result, the completeness theorem in Chapter 5. It depends on a principle called the axiom of choice." In other words: the most...
  32. E

    MHB Isomorphism of logic, arithmetic, and set theory

    Has anybody ever heard of this? I learned about it in a discrete math class in grad school, and I've never heard of it anywhere else !? For example, logical disjunction (OR) and set-theoretic UNION are isomorphic in this sense: 0 OR 0 = 0. {0} UNION {0} = {0}. Similarly, logical AND & set...
  33. S

    Is my Proof Valid for Bijection of Finite Sets?

    <Moderator's note: Moved from a technical forum.> Hi PF, I am learning how to prove things (I have minimal background in math). Would the following proof be considered valid and rigorous? If not any pointers or tips would be much appreciated! Problem: Prove that the notion of number of...
  34. jk22

    B Question about CH (continuum hypothesis)

    Is it possible to calculate this : Suppose the iterative root of ##2^x## : ##\phi(\phi(x))=2^x## (I suppose the Kneser calculation should work, it affirms that there is a real analytic solution) Then how to compute ##\phi(\aleph_0)## ? (We know that ##2^{\aleph_0}=\aleph_1##). Could this be...
  35. nomadreid

    I Cardinality of a set of constant symbols (model theory)

    First, I want to be pedantic here and underline the distinction between a set (in the model, or interpretation) and a sentence (in the theory) which is fulfilled by that set, and also constant symbols (in the theory) versus constants (in the universe of the model) Given that, I would like to...
  36. B

    I Is the Inverse Image of a Computable Function Recursively Enumerable?

    Hello, I am stuck on deciding if given sets are recursive or recursively enumerable and why. Those sets are: set ƒ(A) = {y, ∃ x ∈ A ƒ(x) = y} and the second is set ƒ-1(A) = {x, ƒ(x) ∈ A} where A is a recursive set and ƒ : ℕ → ℕ is a computable function. I am new to computability theory and any...
  37. W

    Set Theory: Power sets of Unions

    Homework Statement I'm having issues understanding a mistake that I'm making, any assistance is appreciated! I know a counterexample but my attempt at proving the proposition is what's troubling me. Prove or disprove $$P(A \cup B) \subseteq P(A) \cup P(B) $$ Homework EquationsThe Attempt at...
  38. danielFiuza

    How to write this in Set Theory notation?

    Hello Everyone, I am trying to write the intersection of a physical problem in the most compact way. I am not really familiar with Set Theory notation, but I think it has the answer. It is about the intersection of two circular areas: - Area 1: A - Area 2: B If I want to write this in Set...
  39. R

    Set theory: Is my proof valid?

    Homework Statement Prove the following for a given universe U A⊆B if and only if A∩(B compliment) = ∅ Homework EquationsThe Attempt at a Solution Assume A,B, (B compliment) are not ∅ if A∩(B compliment) = ∅, x∈A ∨ x∈ (B compliment), but not both If x∈A ∧ x∉(B compliment), then x∈B , because...
  40. T

    Image of a f with a local minima at all points is countable.

    Homework Statement Let ##f:\Bbb{R} \to \Bbb{R}## be a function such that ##f## has a local minimum for all ##x \in \Bbb{R}## (This means that for each ##x \in \Bbb{R}## there is an ##\epsilon \gt 0## where if ##\vert x-t\vert \lt \epsilon## then ##f(x) \leq f(t)##.). Then the image of ##f## is...
  41. ubergewehr273

    Question about a function of sets

    Let a function ##f:X \to X## be defined. Let A and B be sets such that ##A \subseteq X## and ##B \subseteq X##. Then which of the following are correct ? a) ##f(A \cup B) = f(A) \cup f(B)## b) ##f(A \cap B) = f(A) \cap f(B)## c) ##f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)## d) ##f^{-1}(A \cap...
  42. S

    How can I prove that these relations are bijective maps?

    <Moderator's note: Moved from a technical forum and thus no template. Also re-edited: Please use ## instead of $$.> If ##R_{1}## and ##R_{2}## are relations on a set S with ##R_{1};R_{2}=I=R_{2};R_{1}##. Then ##R_{1}## and ##R_{2}## are bijective maps ##R_{1};R_{2}## is a composition of two...
  43. T

    Showing ##\sqrt{2}\in\Bbb{R}## using Dedekind cuts

    1. The problem statement, all variables and given Prove that ##\sqrt{2}\in\Bbb{R}## by showing ##x\cdot x=2## where ##x=A\vert B## is the cut in ##\Bbb{Q}## such that ##A=\{r\in\Bbb{Q}\quad \vert \quad r\leq 0 \quad\lor\quad r^2\lt 2\}##. I believe that I have to show ##A^2=L## however, it...
  44. Wendel

    I How Does the Bernstein-Schröder Theorem Establish Set Equivalence?

    The theorem: Let ##X##, ##Y## be sets. If there exist injections ##X \to Y## and ##Y \to X##, then ##X## and ##Y## are equivalent sets. Proof: Let ##f : X \rightarrow Y## and ##g : Y \rightarrow X## be injections. Each point ##x \in g(Y)⊆X## has a unique preimage ##y\in Y## under g; no ##x \in...
  45. E

    Proof by Induction of shortest suffix of concatenated string

    Homework Statement Wherein α, β are strings, λ = ∅ = empty string, βr is the shortest suffix of the string β, βl is the longest prefix of the string β, and T* is the set of all strings in the Alphabet T, |α| denotes the length of a string α, and the operator ⋅ (dot) denotes concatenation of...
  46. E

    Proof by Induction of String exponentiation? (Algorithms)

    Homework Statement Wherein α is a string, λ = ∅ = the empty string, and T* is the set of all strings in the Alphabet T. Homework Equations (exp-Recursive-Clause 1) : α0 = λ (exp-Recursive-Clause 2) : αn+1 = (αn) ⋅ α The Attempt at a Solution [/B] This one is proving difficult for me. I...
  47. Z

    I Countably Infinite Unions and the Real Numbers: Can They Really Be Uncountable?

    Hello experts, Full disclosure: I am a total layman at math, nothing in my training aside from high school courses and one college calculus class. I'm sure a week doesn't pass without someone posting a question about or challenge to Cantor. I am not here to challenge anything but rather to...
  48. T

    Prove Hausdorff's Maximality Principle by the W.O.P.

    Homework Statement Show Hausdorff's Maximality Principle is true by the Well-Ordering Principle. 2. Relevant propositions/axioms The Attempt at a Solution Case 1: ##\forall x,y\in X## neither ##x\prec y## or ##y\prec x## is true. Hence any singleton subset of ##X## is a maximal linear order...
  49. T

    I Regarding cardinality and mapping between sets.

    why is not always true that if ##\vert A\vert\leq\vert B\vert## then there exist an injection from ##A## to ##B##?
  50. pairofstrings

    I System to represent objects in Mathematics

    Hi. Usually, Computer Programmers use Flow Charts, Algorithms, or UML diagrams to build a great software or system. In the same manner, in Mathematics, what do Mathematicians use to build a great system that they want to build. Category Theory is at the highest level of abstraction; then...
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