# countability

Published articles for countability.

This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles.

## Zero-One Laws for Random Graphs

DevFeed: [Zero-One Laws for Random Graphs](<https://devfeed.tech/articles/zero-one-laws-for-random-graphs-40376.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2015/02/09/zero-one-laws-for-random-graphs/>)

Published: 2015-02-09T09:00:00Z

Content type: article

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Graphs](<https://devfeed.tech/topics/graphs.md>), [Math and Logic](<https://devfeed.tech/topics/math-and-logic.md>)

Tags: [big-o-notation](<https://devfeed.tech/tags/big-o-notation.md>), [connectivity](<https://devfeed.tech/tags/connectivity.md>), [countability](<https://devfeed.tech/tags/countability.md>), [distribution](<https://devfeed.tech/tags/distribution.md>), [erdos-renyi](<https://devfeed.tech/tags/erdos-renyi.md>), [graphs](<https://devfeed.tech/tags/graphs.md>), [logic](<https://devfeed.tech/tags/logic.md>), [logical](<https://devfeed.tech/tags/logical.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [model-theory](<https://devfeed.tech/tags/model-theory.md>), [network-science](<https://devfeed.tech/tags/network-science.md>), [parameter](<https://devfeed.tech/tags/parameter.md>), [random-graph](<https://devfeed.tech/tags/random-graph.md>), [random-graphs](<https://devfeed.tech/tags/random-graphs.md>), [statement](<https://devfeed.tech/tags/statement.md>), [vertex](<https://devfeed.tech/tags/vertex.md>)

### AI overview

This article introduces zero-one laws for Erdős-Rényi random graphs. It explains that many graph properties, including properties expressible in first-order logic, have probabilities that tend toward zero or one as the graph grows, with behavior determined by relevant thresholds or constant edge probabilities.

### Source excerpt

Last time we saw a number of properties of graphs, such as connectivity, where the probability that an Erdős-Rényi random graph $ G(n,p)$ satisfies the property is asymptotically either zero or one. And this zero or one depends on whether the parameter $ p$ is above or below a universal threshold (that depends only on $ n$ and the property in question). To remind the reader, the Erdős-Rényi random "graph" $ G(n,p)$ is a distribution over graphs that you draw from by including each edge independently with probability $ p$.

## Methods of Proof -- Contradiction

DevFeed: [Methods of Proof -- Contradiction](<https://devfeed.tech/articles/methods-of-proof-contradiction-40305.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2013/02/28/methods-of-proof-contradiction/>)

Published: 2013-02-28T11:46:02Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Mathematics](<https://devfeed.tech/topics/mathematics.md>), [Math and Logic](<https://devfeed.tech/topics/math-and-logic.md>), [Computing](<https://devfeed.tech/topics/computing.md>)

Tags: [bijections](<https://devfeed.tech/tags/bijections.md>), [countability](<https://devfeed.tech/tags/countability.md>), [diagonalization](<https://devfeed.tech/tags/diagonalization.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [methods-of-proof](<https://devfeed.tech/tags/methods-of-proof.md>), [proof-by-contradiction](<https://devfeed.tech/tags/proof-by-contradiction.md>), [theory](<https://devfeed.tech/tags/theory.md>)

### AI overview

This tutorial introduces proof by contradiction, explains how it is used to prove impossibility results, and applies the technique to a party-friends problem involving repeated numbers of friends. It also discusses functions on sets and different kinds of infinity.

### Source excerpt

In this post we'll expand our toolbox of proof techniques by adding the proof by contradiction. We'll also expand on our knowledge of functions on sets, and tackle our first nontrivial theorem: that there is more than one kind of infinity. Impossibility and an Example Proof by Contradiction Many of the most impressive results in all of mathematics are proofs of impossibility. We see these in lots of different fields. In number theory, plenty of numbers cannot be expressed as fractions.

## False Proof--The Reals are Countable

DevFeed: [False Proof--The Reals are Countable](<https://devfeed.tech/articles/false-proof-the-reals-are-countable-40228.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2011/07/19/false-proof-the-reals-are-countable/>)

Published: 2011-07-19T17:10:00Z

Content type: article

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [math](<https://devfeed.tech/topics/math.md>)

Tags: [axiom-of-choice](<https://devfeed.tech/tags/axiom-of-choice.md>), [countability](<https://devfeed.tech/tags/countability.md>), [false-proof](<https://devfeed.tech/tags/false-proof.md>), [math](<https://devfeed.tech/tags/math.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [set-theory](<https://devfeed.tech/tags/set-theory.md>), [well-ordering](<https://devfeed.tech/tags/well-ordering.md>)

### AI overview

The article examines a purported proof that the real numbers are countable. It explains that the argument's surjectivity claim is flawed and begins demonstrating the issue using a chosen well-ordering of the integers.

### Source excerpt

It seems that false proofs are quickly becoming some of the most popular posts on Math ∩ Programming. I have been preparing exciting posts on applications of graph coloring, deck stacking, and serial killers. Unfortunately, each requires resources which exist solely on my home desktop, which is currently dismantled in California while I am on vacation in Costa Rica. Until I return from the tropics, I will continue with more of the ever -popular false proofs.

## Set Theory--A Primer

DevFeed: [Set Theory--A Primer](<https://devfeed.tech/articles/set-theory-a-primer-40223.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2011/07/09/set-theory-a-primer/>)

Published: 2011-07-09T18:14:59Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Mathematics](<https://devfeed.tech/topics/mathematics.md>), [Math and Logic](<https://devfeed.tech/topics/math-and-logic.md>)

Tags: [axiom-of-choice](<https://devfeed.tech/tags/axiom-of-choice.md>), [bijections](<https://devfeed.tech/tags/bijections.md>), [cardinality](<https://devfeed.tech/tags/cardinality.md>), [countability](<https://devfeed.tech/tags/countability.md>), [example](<https://devfeed.tech/tags/example.md>), [functions](<https://devfeed.tech/tags/functions.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [numbers](<https://devfeed.tech/tags/numbers.md>), [power-set](<https://devfeed.tech/tags/power-set.md>), [primer](<https://devfeed.tech/tags/primer.md>), [set](<https://devfeed.tech/tags/set.md>), [set-theory](<https://devfeed.tech/tags/set-theory.md>), [symbols](<https://devfeed.tech/tags/symbols.md>), [theory](<https://devfeed.tech/tags/theory.md>), [variable](<https://devfeed.tech/tags/variable.md>)

### AI overview

This primer introduces set theory by defining sets, elements, membership, cardinality, notation, and several ways to construct sets. It uses numerical examples and introduces natural numbers, integers, and rational numbers while noting that unrestricted operations can lead to paradoxes.

### Source excerpt

It's often that a student's first exposure to rigorous mathematics is through set theory, as originally studied by Georg Cantor. This means we will not treat set theory axiomatically (as in ZF set theory), but rather we will take the definition of a set for granted, and allow any operation to be performed on a set. This will be clear when we present examples, and it will be clear why this is a bad idea when we present paradoxes.