Tagged Questions

74 views

What background is needed to start Set Theory?

So I'm doing calculus at this moment in my first year, but I think I'm lacking in terms of mathematical notation, logic, etc. After some research I found that Set Theory is sort of mandatory for any ...
29 views

Good, memorable explanations for relationship between functions and set operations

Recently I was in a position to explain basic relationships between functions and set operations (preimages commute with all set operations, images commute with unions, but they fail to commute with ...
83 views

De Morgan's laws in logic and set theory

In logic De Morgan's law means $\lnot (A \land B) \Leftrightarrow \lnot A \lor \lnot B$ In set theory De Morgan's law means $(A \cap B)^C = A^C \cup B^C$ I'm surprised that the same idea is true in ...
53 views

Is the collection of all functions between two sets a set?

Can we say "the set of all functions between two sets" as easily as we could say "the set of all real numbers", for example?
59 views

Is there any interesting interpretation of the set of all functions between two sets?

Is there any way to interpret the set of all functions from a set $X$ to a set $Y$? There is an interpretation of it as the cartesian product of $X$-many copies of $Y$, but I am asking for a more ...
171 views

Natural uses for the co-product of sets?

I had come across countless uses of the (Cartesian) product of sets long before I first ever met the concept of a "co-product"1 of sets. In fact, anyone who has learned basic analytic geometry in ...
25 views

Elementary set theory problem - lack of working memory

Here's an example to illustrate the problem: I have this problem from a book about set theory: $\wp(\cup_{i \in I}A_i)\nsubseteq \cup_{i \in I} \wp(A_i)$ I'm supposed to translate it to pure ...
233 views

What is the right interpretation of the axiom of extensionality

A set $a$ can be called extensional if it has the following propery: $$\forall b\left[\forall x\left[x\in b\iff x\in a\right]\Rightarrow a=b\right]$$ Based on this the axiom of extensionality can be ...
135 views

Recommendation on a rigorous and deep introductory logic textbook

In this post, I don't mean any word by its somewhat "mathematical or logical" meaning but just "literally". It's been three years since I started "formal" mathematics, and now I'm familiar with set ...
69 views

Generalization of principle of inclusion and exclusion (PIE)

The PIE can be stated as $$|\cup_{i=1}^n Y_i| = \sum_{J\subset[n], J\neq \emptyset} (-1)^{|J|-1} |Y_J|$$ where $[n]=\{1,2,...,n\}$ and $Y_J=\cap_{i \in J} Y_i$. Problems using it are usually reduced ...
96 views

Largest infinite cardinal used in a proof

I've heard before that Knuth holds the record for the largest constant used in a mathematical proof. I was wondering what is the largest cardinal ever explicitly considered in set theory. I presume ...
115 views

Correct formal interval notation

I can't find any definitive answer on this topic, maybe that's because there isn't one, but I figured if there was a place to ask then SE was it! To describe a set in which $x$ and $y$ are in the ...
342 views

What is the purpose of sets? Why do we use them?

All is in the title. Why sets? Why do we need them and where are they important?
1k views

Refuting the Anti-Cantor Cranks

I occasionally have the opportunity to argue with anti-Cantor cranks, people who for some reason or the other attack the validity of Cantor's diagonalization proof of the uncountability of the real ...
29 views

Enumerations in totally ordered sets

Suppose that $A$ is a subset of the positive integers and you want to enumerate all the elements of $A$ in a sequence $a_1, a_2, \ldots$ such that $a_n < a_{n+1}$. What is the most concise and ...
56 views

Proving Equality of 2 Functions

I have a general question, illustrated by a specific example. The general question is how to methodically prove that two functions are equal. Much like trying to prove an "if-and-only-if" statement ...
57 views

Analogue of the term 'summand' for unions and intersections.

If we have a sum $\sum_{i=1}^na_i$, we call the terms $a_i$ summands. In fact, in the cases of addition, subtraction, multiplication, and division, we have a large vocabulary to describe the various ...
175 views

Is the intersection of every non-empty family of inductive sets equal to the intersection of every inductive set?

In Halmos Naive set theory, there is the following passage (excuse my french) in his section introducing natural numbers : In this language the axiom of infinity simply says that there exists a ...
200 views

What's correspondence between the model theoric and the set theoric kernel of homomorphism?

A kernel of a mapping $h$ from $\mathfrak{A}$ to $\mathfrak{B}$, generally, is an equivalence relation $\{(a,a') \in \mathfrak{A} \times \mathfrak{A} \mid h(a)=h(a')\}$. However, in model theory, ...
121 views

Can we use proper classes in this way, to define a new infinity larger than |Ord|?

I believe there is a way to do this that makes sense, and I explain it below. I would like to know if I did some obvious mistake, or if the idea doesn't make sense for some reason I didn't figure it ...
567 views

What does it mean for a set to exist?

Is there a precise meaning of the word 'exist', what does it mean for a set to exist? And what does it mean for a set to 'not exist' ? And what is a set, what is the precise definition of a set?
236 views

Examples of “transfer via bijection”

On some occasions I have seen the following situation: We want find out whether a set of a given cardinality $\varkappa$ has some property P. If this property is invariant under bijective maps, then ...
125 views

Examples of dictionaries between two distinct fields of mathematics (or between “differents” structures of math).

I'd like to meet explicit examples of dictionaries between two distinct fields of Mathematics (or between two "different" structures of Mathematics). I'm not interested in the usual sense dictionary ...
481 views

The way into set theory

Given that I am going through Munkres's book on topology , I had to give a glance at the topics included in the first chapter like that of Axiom of choice, The maximum principle, the equivalence of ...
130 views

What does means the $\frown$ in sequence notation?

In the theorem 3.6 of Juhász's Cardinal Functions in General Topology appears the following symbol about sequence: $\frown$ The role context of it's appearance is the following: Theorem. Let X be an ...
173 views

What sets satisfy $V = V^V$?

I'm not sure if said set exist or whether it is unique, but what name could I use to find more about it and what kind of interesting properties does it have? Clarification edit: I meant a set $V$ ...
61 views

Convention on comparing cardinality

When showing that two sets $A$ and $B$ have the same (finite or infinite) cardinality, it is usually done by constructing a function and showing that it is a bijection. However, in some cases, ...
283 views

$\wedge,\cap$ and $\vee,\cup$ between Logic and Set Theory always interchangeable?

In "$\wedge,\cap,\times$ and $\vee,\cup,+$ are always interchangeable?" It has been shown that arithmetic shouldn't be included. So the new modified question is: The analogy of $\wedge,\cap$ and ...
191 views

$\wedge,\cap,\times$ and $\vee,\cup,+$ are always interchangeable?

Update : Should have left the Arithmetic out of this question, the new modified question is posted here : $\wedge,\cap$ and $\vee,\cup$ between Logic and Set Theory always interchangeable? ...
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What would be the consequence of restricting multiplication by Zero to only Finite Cardinals? [closed]

What would be the consequence of restricting multiplication by Zero to only Finite Cardinals? Would this lead to contradictions? How could it be achieved?
143 views

Heuristics suggesting a unit interval is uncountable

This is maybe a soft question, I am not sure yet. Anyway, I am delivering a 8 (+ 4 supervisions) hour course on 'basic set theory' for undergraduates : set notation, bijections, functions, ...
912 views

Inherently discrete concepts

Are there any concepts which are naturally defined only for the integers and so far has resisted any attempts at extension to other fields such as rationals or reals? Does not meet criteria: ...
335 views

What does “a set of things” mean?

Suppose we defined some mathematical object $P$, where $P$ is natural number, polynomial, endofunction, geometric figure, etc. What does the expression “$A$ is a set of $P$s” mean: Set inclusion) ...
142 views

What is a statement?

This perhaps is not a math problem. I donot know if it fits in here. But it confuses me a lot these two days. I have wiki it. But it does not work for me. I will give some easy "proofs" to explain ...
I have defined this simple sheaf. Take $E:set, B:set, p:E\to B$. Let $P(B)$ be a set of subsets of $B$. Let $S$ be the set of sections of $p$. Let $F$ be a contravariant functor from $P(B)$ as a poset ...