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# Questions tagged [set-theory]

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### 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 ...
1 vote
47 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 ...
368 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 ...
71 views

### Given a partition and an element, find the subset that includes this element

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 ...
150 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 ... 165 views

### On the interpretation of coinduction in type theory

The notion of (co)datatype can be modeled satisfactorily in category theory as fixed-points of polynomial functors. Then, (co)induction principles are derived from initial algebras/terminal ...
602 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 ...
240 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 ... 364 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 / ...
416 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 ...
1 vote
474 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 ...
1 vote
84 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 ...
1k 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 ...
209 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 ...
339 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) $\forall S\in \mathcal{F}$, $|S|=a$....
918 views

### Is there a set theoretic way to look at SQL?

I have been learning about SQL and at times it feels like set theory. A statement like SELECT * FROM myTable is like a set $\{ x: x \in \text{myTable} \}$. A ...
980 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. ...
631 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 ...
1 vote
285 views

### How can I formalize key value stores with set theory? [closed]

I'm currently developing a simple key-value NoSQL store and want to build its formal model. I'm interested in knowing if there some work about formalization of key-value stores outside of category ...
1 vote
100 views

### Minimum order of partite in a bipartite graph

I want to create a bipartite graph where the first partite $U$ contains $L$ vertices with degree $k$ and the second partite $V$ contains $N$ other vertices with degree $a$. I need to find the minimum ...
72 views

### Finding exact value with a quotients of products of random values

Sorry for the haphazard title: really not sure what this should be called Suppose we have a set of $z$ random values $S = r_1, \dots, r_z$ drawn from $\mathbb{Z}_N$ (where $N$ is some large prime). ...
74 views

### Explanation of Cantor's diagonal argument? [closed]

I struggled to understand the Cantor's diagonal argument, but I have some problems comprehending the following: By construction, $s$ differs from each $s_n$, since their $n^{th}$ digits differ (...
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 ...
382 views

386 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 ...
778 views

### Which formalism is best suited for automated theorem proving in set theory?

Abbreviations - FOL is first-order logic; NBG is Von Neumann–Bernays–Gödel set theory; SEP is Stanford Encyclopedia of Philosophy; HOL is higher-order logic; ATP is automated theorem proving. Context ...
223 views

### Kruskal-Katona Theorem with Majority?

I am interested in the following problem which seems like an extension of the Kruskal-Katona Theorem. Let $A_k \subseteq \{0,1\}^n$ be a subset of the hypercube such that every element in $A$ has ...