# well-ordering

Published articles for well-ordering.

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## False Proof--All Numbers are Describable in at Most Twenty Words

DevFeed: [False Proof--All Numbers are Describable in at Most Twenty Words](<https://devfeed.tech/articles/false-proof-all-numbers-are-describable-in-at-most-twenty-words-40233.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2011/07/28/false-proof-twenty-word/>)

Published: 2011-07-28T16:03:27Z

Content type: article

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>), [Statement](<https://devfeed.tech/topics/statement.md>)

Tags: [computer](<https://devfeed.tech/tags/computer.md>), [false-proof](<https://devfeed.tech/tags/false-proof.md>), [kolmogorov-complexity](<https://devfeed.tech/tags/kolmogorov-complexity.md>), [language](<https://devfeed.tech/tags/language.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [program](<https://devfeed.tech/tags/program.md>), [set-theory](<https://devfeed.tech/tags/set-theory.md>), [turing-machine](<https://devfeed.tech/tags/turing-machine.md>), [well-ordering](<https://devfeed.tech/tags/well-ordering.md>)

### AI overview

The article examines a false proof claiming that every natural number can be described in fewer than twenty words. It explains that the contradiction arises from imprecise notions of description and set construction, connecting the issue to Russell's paradox and the Richard-Berry paradox.

### Source excerpt

Problem: Show that every natural number can be unambiguously described in fewer than twenty words. "Solution": Suppose to the contrary that not every natural number can be so described. Let $ S$ be the set of all natural numbers which are describable in fewer than twenty words. Consider $ R = \mathbb{N}-S$, the set of all words which cannot be described in fewer than twenty words. Since $ R$ is a subset of the natural numbers, which is well-ordered, it has a unique smallest element which we call $ r$.

## 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.

## 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[.