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.
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
## 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...
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
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...
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...
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...
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
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...
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...
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...
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...
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...
"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...
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...
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...
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...
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...
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...
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...
<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...
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...
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...
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...
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...
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...
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...
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...
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...
<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...
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...
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...
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...
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...
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...
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...
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...