# axiom of choice

Published articles for axiom of choice.

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## 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.

## Well Orderings and Search

DevFeed: [Well Orderings and Search](<https://devfeed.tech/articles/well-orderings-and-search-40202.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2011/06/14/well-orderings-and-search/>)

Published: 2011-06-14T11:18:04Z

Content type: tutorial

Language: en

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

Topics: [Algorithm](<https://devfeed.tech/topics/algorithm.md>), [Sorting](<https://devfeed.tech/topics/sorting.md>), [ordering](<https://devfeed.tech/topics/ordering.md>), [implementation](<https://devfeed.tech/topics/implementation.md>)

Tags: [algorithm](<https://devfeed.tech/tags/algorithm.md>), [axiom-of-choice](<https://devfeed.tech/tags/axiom-of-choice.md>), [binary-search](<https://devfeed.tech/tags/binary-search.md>), [mathematica](<https://devfeed.tech/tags/mathematica.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [optimization](<https://devfeed.tech/tags/optimization.md>), [ordering](<https://devfeed.tech/tags/ordering.md>), [pseudocode](<https://devfeed.tech/tags/pseudocode.md>), [recursion](<https://devfeed.tech/tags/recursion.md>), [sorting](<https://devfeed.tech/tags/sorting.md>), [well-ordering](<https://devfeed.tech/tags/well-ordering.md>)

### AI overview

This tutorial explains binary search on sorted lists, including its recursive structure and O(log n) runtime. It then introduces strict total and well orders to explain why sorting and comparison work.

### Source excerpt

Binary Search Binary search is perhaps the first and most basic nontrivial algorithm a student learns. For the mathematicians out there, binary search is a fast procedure to determine whether a sorted list contains a particular element. Here is a pseudocode implementation: # Binary Search: # Given a list L, sorted via the total order <, and a sought # element x, return true iff L contains x. function binarySearch(L, x, <): # base case if(length(L) == 1): return L[0] == x middleIndex = floor(length(L) / 2) if (L[middleIndex] == x): return true # inductive step, with ellipsis notation meaning slices of L # from the beginning and to the end, respectively if (x < L[middleIndex]): return binarySort(L[.