# Set theory Definition and 74 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. ### I Next generation set theory

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
2. ### 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?
3. ### I 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...
4. ### 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...
5. ### 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...
6. ### 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...
7. ### 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...
8. ### I Enumerating a Large Ordinal

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...
9. ### Tips for Bijection Proof

<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...
10. ### 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...
11. ### 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...
12. ### I Decide if a set is recursive

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...
13. ### 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...
14. ### 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...
15. ### 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...
16. ### 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...
17. ### 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...
18. ### I Bernstein-Schröder Theorem

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...
19. ### 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...
20. ### 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...
21. ### I Confusion over countability

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...
22. ### 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...
23. ### 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##?
24. ### 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...
25. ### B A Rational Game

This post is to set forth a little game that attempts to demonstrate something that I find to be intriguing about the real numbers. The game is one that takes place in a theoretical sense only. It starts by assuming we have two pieces of paper. On each is a line segment of length two: [0,2]...
26. ### I Countability of ℚ

I know there are many proofs of this I can google, but I am interested in a particular one my book proposed. Also, by countable, I mean that there is a bijection from A to ℕ (*), since this is the definition my book decided to stick to. The reasoning is as follows: ℤ is countable, and so iz ℤxℤ...
27. ### B Empty domains and the vacuous truth

So, here's my question. I read somewhere that all universal truths on empty domains are vacuously true, whereas all existential are false. However, if all statements of the form (∀x∈A)(P(x)) , where A is an empty set, are vacuously true, then the statement (∃x∈A)(P(x)) should also be true...
28. ### Prove that a set is an event

Homework Statement Suppose that the sample space is the set of all real numbers and that every interval of the form (-infinity, b] for any real number b is an event. Show that for any real number b (-infinity, b) must also be an event. The Attempt at a Solution use the 3 conditions required...
29. ### Prove Zorn's Lemma is equivalent to the following statement

Homework Statement From Introduction to Set Theory Chapter 8.1 exercise 1.4 Prove that Zorn's Lemma is equivalent to the following statement: For all ##(A,\leq)##, the set of all chains of ##(A,\leq)## has an ##\subseteq##-maximal element.[/B] Homework Equations N/A The Attempt at a...
30. ### Help me Prove that such a set does NOT exist

Let Q denote the theory of Robinson Arithmetic. A theory T is nice iff T is consistent, is p.r. adequate and extends Q. The fixed-point lemma states that for all nice theories T, for any formula φ, there is a sentence σ such that T ⊢σ↔φ("σ")...
31. ### Relation closures proof

Homework Statement Suppose R1 and R2 are relations on A and R1 ⊆ R2. Let S1 and S2 be the transitive closures of R1 and R2 respectively. Prove that S1 ⊆ S2. Please check my proof and please explain my mistakes. thank you for taking the time to help. Homework Equations N/A The Attempt at a...
32. ### Topology by Simmons Problem 1.3.3

Homework Statement Let ##X## and ## Y## be non-empty sets, ##i## be the identity mapping, and ##f## a mapping of ##X## into ##Y##. Show the following a) ##f## is one-to-one ##~\Leftrightarrow~## there exists a mapping ##g## of ##Y## into ##X## such that ##gf=i_X## b) ##f## is onto...
33. ### Topology by Simmons Problem 1.2.1

Homework Statement If ##\bf{A}## ##= \{A_i\}## and ##\bf{B}## ##= \{B_j\}## are two classes of sets such that ##\bf{A} \subseteq \bf{B}##, show that ##\cap_j B_j \subseteq \cap_i A_i## and ##\cup_i A_i \subseteq \cup_j B_j## Homework Equations The Attempt at a Solution Since ##\bf{A}...
34. ### I Sets, Subsets, Possible Relations

Given a set, there are subsets and possible relations between those arbitrary subsets. For a given example set, the possible relation between the subsets of the example set will narrow down to the "true" possible relations between those subsets. a) {1} Number of Subsets: ##2^1 = 2## (∅, {1})...

Sets which doesn't contain themselves are called normal sets while sets that contain themselves are called abnormal. Let ##N## be a set of all normal sets. Prove that ##N## is normal if and only if ##N## is abnormal. Proof. ##~~\rightarrow ~~ ## Suppose ##N## is normal such that ##N \not\in N##...
36. ### I Existential Import paper by Corcoran and Massoud

In the paper John Corcoran & Hassan Masoud (2014): Existential Import Today: New Metatheorems; Historical, Philosophical, and Pedagogical Misconceptions, History and Philosophy of Logic, http://dx.doi.org/10.1080/01445340.2014.952947, already in the introduction it says, as self-evident, that...
37. ### I Can an ordered pair have identical elements?

Hi guys, Here is a wacky question for you: Suppose you have a simple recursive function f(x)=x. Given the fact that a function f(x)=y can be rewritten as a set of ordered pairs (x, y) with x from the domain of f and y from the range of f, it would seem that the function f(x)=x can be written...
38. ### Difficulty proving a relation is an equivalence relation

Homework Statement Homework Equations I don't think there are any in this case The Attempt at a Solution I know that in order to prove R is an equivalence relation, I'd have to show that it is Reflexive, Symmetric, and Transitive. I'm not sure why, but I'm finding this a bit difficult in...
39. ### B What are surreal numbers?

Hey guys! I have heard of this concept in various places and sort of understands what it attempts to do. Can anybody please explain it to me in more detail like how it works, how to notate it, and how to expand it to infinities and infinitesimals. Thanks in advance! Aakash Lakshmanan xphysx.com...
40. ### Set Theory Question

Homework Statement ##C \subseteq A \cap B \implies A \cap B \cap C = C## Homework Equations How do I get rid of the "belongs to" term on the right hand side? I know I need to prove either the left hand or the right hand side of the "or" term is correct, I'm just not sure how to get there. The...
41. ### Interval Proof

Homework Statement Let ##A_n = (n − 1, n + 1)##, for all natural numbers n. Find, with proof, ##∪_{n≥1}A_n## Homework Equations What does that last statement mean? Union for n greater than or equal to one times the interval? The Attempt at a Solution I can't understand the question.
42. ### A Question about axiom of regularity

It has been stated that in axiom of regularity , a set cannot be an element of itself and there is a proof for which S={S} . I can understand his proof since S is the only element and hence its method of proof is viable here . But , what if I change the question to S= {S,b} ( it is a set which...
43. ### Basic probability with set theory

Homework Statement [/B] P(A | \overline{B}) = ? Homework Equations Multiplicative rule: \begin{equation} P(A | B) = \frac{P(A \cap B)}{P(B)} \end{equation} Additive rule: \begin{equation} P(A \cup B) = P(A) + P(B) - P(A \cap B) \end{equation} Difference: \begin{equation} A \backslash B = A...
44. ### I Basic question about set theory

Hi, Is the following notation correct? X = {xt> max(xt-k,...,xt-1,xt+1,...,xt+k)|(t-k,...,t+k) ∈ T2k+1∧ k ∈ ℕ\{0}} where T = [1,n]∩ℕ denotes the time periods over which x runs. I basically want to say that X is the set of points that are local maxima in a neighbourhood of k points to the right...
45. ### I Logically equivalent

From the text it says (P -> Q) or (P -> R) is equivalent to P -> (Q or R) I tried to see if this is true so I tried (P \to Q) \lor (P \to R) \\ (P \lor \neg Q) \lor (P \lor \neg R) \\ P \lor \neg Q \lor \neg R \\ P \lor \neg(Q \land R) \\ P \to (Q \land R) and P \to (Q \lor R) \\ P \lor...
46. ### I Logic and distributive laws

Im just reading this one example and i am stumped at this one step. (R\to C) \land (S \to C) \\ (\neg R\lor C) \land (\neg S \lor C) \ \ \ \ \ \textrm{by conditional law}\\ (\neg R\land \neg S) \lor C \ \ \ \ \textrm{by distributive law} I don't understand how it went from the second step to...
47. ### Overcoming abstraction

Hi As I am venturing in graduate level mathematics, I am having a recurring problem; I keep getting stuck in the abstraction of it. Usually it involved set theory; I never get "fluent" in it. However, the main problem is abstraction. For instance, this semester I had topology, and the...
48. ### Markov chains: period of a state

Hello, I am trying to understand the intuition of the definition of the period of a state in a Markov chain. Say for example we can go from state i to state i in 3,5,7,9,11.... and so on steps. The gcd here is one. So is this aperiodic state or one with periodicity of 2. Thanks
49. ### I Are These Rules new Conjectures in Set Theory?

We can easily find out below rules in set theory: 1. Let consider set “A” as follows: A = {a1, a2, a3, a4… an} and also power set of A is set C: C = {{}, {a1}, {a2}, {a3}, {a4}, {a1, a2}, {a1, a3},….{an}} Rule 1: To find the number of subsets with precise members number, we can use Binomial...
50. ### Looking for book on ZFC axiomatic set theory (I think)

I am looking for a book that starts at the standard ZFC axioms and progresses to the point where some recognisable non-trivial mathematical statement is proved. By recognisable I mean something that you may encounter in school/early university level and is not purely set-theoretical (e.g...