# Questions tagged [set-theory]

50 questions
Filter by
Sorted by
Tagged with
4k views

### Which interesting theorems in TCS rely on the Axiom of Choice? (Or alternatively, the Axiom of Determinacy?)

Mathematicians sometimes worry about the Axiom of Choice (AC) and Axiom of Determinancy (AD). Axiom of Choice: Given any collection ${\cal C}$ of nonempty sets, there is a function $f$ that, given a ...
• 27.5k
1k views

### Results in Theoretical CS independent of ZFC

I'm going to ask a quite vague question, since the borderline between theoretical computer science and math is not always easy to distinguish. QUESTION: Are you aware of any interesting result in CS ...
• 501
694 views

### Applications for set theory, ordinal theory, infinite combinatorics and general topology in computer science?

I am a mathematician interested in set theory, ordinal theory, infinite combinatorics and general topology. Are there any applications for these subjects in computer science? I have looked a bit, and ...
• 263
793 views

### Forcing method used in Baker-Gill-Solovay Relativization paper and Cohen's Proof of Continuum Hypothesis Independence

I am generally interested in the forcing method used by Baker-Gill-Solovay and Cohen. I am looking for as many sources as I can get my hands on concerning either the technique itself or its use. Does ...
• 533
2k views

### Type system based on naive set theory

As I understand, in computer science data types are not based on set theory because of things like Russell's paradox, but as in real world programming languages we can't express such complex data ...
• 381
471 views

### The state of art for sunflower system

I am interesting in the sunflower system and its applications in computer science. Given a Universe $U$ and a collection of $k$ sets $A_i$ is called a k-sunflower system if $A_i \cap A_j = Y$ for ...
• 329
1k views

### Universal and existential types

I'm trying to wrap my head around the concepts of universal and existential types but everywhere I look, I see either logical or operational intuitions (or implementations) (e.g. TAPL book by B. ...
• 121
489 views

### What are the issues with a set-like interpretation of quantifiers in type theory?

In his answer to a question that tries to treat universal and existential quantifiers as intersections and unions of sets, Andrej Bauer says: Forget the intersections and unions. People get this idea ...
404 views

### Cantor's theorem in type theory

Cantor's theorem states that For any set A, the set of all subsets of A has a strictly greater cardinality than A itself. Is it possible to encode something like this using only types / ...
• 193
2k views

### Formal Definition/counter part in mathematics for “Objects” of Object Oriented Models

This is a question I asked in mathematics SE forum, and I was referred here. So here is the question- I'm a newbie in both formal mathematics and theoretical computer science, so please bear with me ...
• 193
2k views

### Are there presentations of set theory in terms of lambda-calculus?

I am planning to implement in software a set theory language, based on a binary function, which in set theory is the so called adjunction operation: $f(x, y) = x \cup$ {y}. Therefore, a presentation ...
• 81
224 views

### Is there a known automatic proof of the independence of the continuum hypothesis?

In 2002, L.C. Paulson gave a mechanized proof of the consistency of the axiom of choice by formalizing $V=L$ and its consistency. We could ask whether there is a formalized proof of the independence ...
• 181
2k views

### In the context of regular languages, must the alphabet be finite?

In The Theory of Parsing, Translation, and Compiling, Volume I, Section 0.2.1 (p.15 / 1972), Aho and Ullman casually write that "[a]n alphabet need not be finite or even countable, but for all ...
• 71
172 views

### Trying understand a move in Cohen's proof of the independence of the continuum hypothesis

I've read a few different presentations of Cohen's proof. All of them (that I've seen) eventually make a move where a Cartesian product (call it CP) between the (M-form of) $\aleph_2$ and $\aleph_0$ ...
• 533
180 views

### Order notation quirk

Is it true that $$O(n) = \bigcap \{ O(g) \mid g \in \omega(n) \}?$$ This appears to be a straighforward question about sets of functions, but on closer examination leads to some murky waters. I would ...
351 views

### A Combinatorial Problem on Extremal Set Theory

Given a ground set $[n]$, under what condition of parameters $a,b,c$ does a family of subsets $\mathcal{F}\subseteq 2^{[n]}$ with the following property exist? (i) $\forall S\in \mathcal{F}$, $|S|=a$....
• 437
453 views

### Do we know a specific $L_{ZFC}$ such that $K(s) \ge L_{ZFC}$ is unprovable in ZFC for all strings $s$?

Chaitin's incompleteness theorem states for any formal system $F$ (which satisfies various criteria), there is a $L$ such that for any $s$ the statement $$K(s) \ge L_F$$ is unprovable in that formal ...
143 views

### What's complexity of this set problem which looks like "Linear Programming"?

I came up with a problem below, which looks like a linear programming problem: Given $n$ sets $S_{1}, S_{2},..., S_{n}$, with constraints of :  \forall i=1, 2, 3,...,n\space\space \left | S_{i} ...
• 439
654 views

### Data structure to determine if sets are disjoint in o(n) time

My initial question was exactly the title of this post, but after feedback from commenters I have formulated a more precise version of the question that attempts to capture its essence. Does there ...
• 159
447 views

• 41
161 views

### Set-theoretic encoding of functions in type theory

Functions usually get encoded in set theory as follows. A function $A\to B$ is a subset $f\subset A\times B$ such that $\pi_1:f\to A$ is a bijection. In type theory to give a function $A\to B$ is to ...
251 views

### Optimal partition according to partition cardinality

Given $N$ sets of integers $S_1, \ldots,S_N$ with $|S_i| \le K$. We want to partition those sets such that the union of all sets in any given partition doesn't contain more than $K$ elements. Can ...
390 views

### Theorem prover fails to find simple set theory proof?

I am trying to use an automated theorem prover (SNARK) to prove a theorem in first-order logic. Tarski claims in his "a work on mereology" that the goal is provable from assertions 1-3 but he does ...
• 279
1 vote
3k views

### What's the relation between the dominating set and vertex cover?

I wonder if the minimal dominating set is always a subset of the minimal vertex cover in any graph. If so, what's the proof?
• 121
1 vote
246 views

• 439
1 vote
53 views

### Better approximation of the subset in the membership oracle

A standard tool for estimating the size of a subset via membership oracle queries is given below. Lemma 2.8: . Consider two (finite) sets $B ⊆ U$, where $n = |U|$. Let $ε ∈ (0, 1)$ and $γ ∈ (0, 1/2)$ ...
• 41
1 vote
57 views

### Is this a variant of the set cover problem?

$\textbf{Decision Problem:}$ Given a finite set of elements $E$ and a collection $C$ of non empty sets, $C=\{E_1,...,E_n\}$, such that each $E_i$ covers at least one element of $E$. The goal is to ...
1 vote
483 views

### Equality Constraints over Sets with Tree Automata

Tree Automata can be used to model sets of values of a Herbrand Universe, for example, to model possible values in a functional program. Systems of subtype constraints over set expressions have ...
• 2,832
1 vote
85 views

### Equality Theorems with Type Theoretic Proof

I am investigating how I might be able to translate even commonplace equalities/ inequalities via the so-called Curry-Howard Correspondance - from a generic, set theoretic plus AOC foundation - into a ...
1 vote
94 views

### Is there a name for this property in set-valued analysis or combinatorics?

I asked this question a few days ago on MO, but I haven't received an answer. So I thought I would ask here. I have also added a relaxed version of the question here. Let $F$ be a set-valued, finite-...
• 779
I am interested in the following simple problem: Let $X$ be a set and $X_1\cup X_2\cup\cdots\cup X_k$ be a finite partition of $X$. Given $x\in X$, find the subset $X_i$ for which $x\in X_i$. I am ...