The Universe of Discourse (Mark Dominus Blog)
Previously: Ordinal numbers and basic set theory Ordinals as nim-heaps Nim always ends, even with infinite ordinals Infinite Nim as a coin-moving game Coin-moving games without the coins Previously we saw how to interpret the difficult-seeming ordinal as a particular ordering of the set of finite sequences of numbers, revealing what seemed like a scary monster as gentle and straightforward. Now…
In mathematical study there are two kinds of theorems, which serve very different purposes. Math instruction follows the same pattern. Students are often very puzzled by this, and rightly so, because it's never explained, or at least I've never seen it explained. There is this crucial, critical piece of mathematical methodology which is never made explicit, students just have to figure it out on…
Previously: Ordinal numbers and basic set theory Ordinals as nim-heaps Nim always ends, even with infinite ordinals Infinite Nim as a coin-moving game In the previous article we saw how to interpret Nim heaps of up to beans as coins on a quarter-infinite array: The coin here represents a heap of beans. The heap can be reduced to any smaller number of beans. In the coin version of the game, this…
The bookshelf by my elbow, the one that I can reach without getting up, has seven books on it, not necessarily the ones I look in the most, but the ones whose emanations I most hope will infuse me as I write. Roget's Thesaurus (4th edition) The one I actually refer to most often is the Harper and Row Roget's Thesaurus . I thought I had acquired this in my teens, but the note on the flyleaf says…
Previously: Ordinal numbers and basic set theory Ordinals as nim-heaps Nim always ends, even with infinite ordinals In the previous articles I talked about the game of Nim, a very simple game for two players: There are some piles of beans Players alternate turns A legal move is to take any number of beans from one pile Whoever takes the last bean wins I wrote about how Nim could be extended to…
( Previously ) Flukes and Breakthroughs In the NBA 2015–16 season, Steph Curry set the all-time single-season record for three-point field goals, 402, completely crushing the old record of 286. Curry's record still stands. The New York Times was rather breathless about this : The record is an outlier that defies most comparisons, but here is one: It is the equivalent of hitting 103 home runs in a…
Previously: Ordinal numbers and basic set theory Ordinals as nim-heaps [Yesterday][prev-2] I talked about the game of Nim, which involves two players taking beans from several piles, and an extension that includes green tokens that behave a bit like infinite piles: When there's a pile with one or more green tokens, it's legal for a player to remove any or all of them, and then to add any number of…
Previously: Ordinal numbers and basic set theory We're going to get to in a long and roundabout way. First I want to talk about the game of Nim. Nim Nim is a very simple game for two players. There are some piles of beans, which are called nim-heaps . When it's your turn, you are allowed to remove as many beans as you like, as long as they are all in the same pile. Whoever takes the last bean…
This post is going to be about what infinite ordinal numbers are, and about is in particular. I had a brainwave a while back (18 months now, wow, I have definitely not been blogging enough) and suddenly understood much better than I did before. I have several related ideas here and I am going to try to write one blog post about each of them, instead of one gigantic blog post about all of them…
This is my great-grandfather, born Dominusz Andor in Szeged, Hungary in 1886. In the picture he is in Brooklyn, New York, probably sometime in the early 1950's. By 1911 Andor had moved from Hungary to Vienna and had changed the spelling of his name to “Dominus” to save confusion. He worked as a goldsmith, and owned his own jewelry shop, so he must have been doing OK. There's a family legend about…
Making the rounds last week was this magnificent article on the complications of Arabic typesetting, An interactive introduction to the terrific experience of rendering Arabic typography and its technical debt . The author, Saleh, promises: The reply took and the closure of the ticket took half an hour or so. The reasons behind it took five hundred years to pile up, and they involve a…