Monday, March 16, 2015

A brief intro to ordinal numbers (English)

См. также:
О бесконечном
Dangerous Knowledge
Прикоснуться к бесконечности за неполных два часа приводится вся наивная теория множеств со всеми идеями доказательств.
To Infinity... and Beyond

Disclaimer: Эта подборка видео не моя.

Here I give von Neumann's elegant construction of the natural numbers from (quite literally) nothing.

There is more below.
Ниже есть продолжение.

Here I introduce the axiom of infinity and the first two infinite ordinals, omega and its successor.

In this part I introduce the mathematical definition of an ordering on a set, and in particular a well-ordering, which is fundamental to the definition of an ordinal number.

Here I use the definition of a general ordinal number as a well-ordered set to introduce a few more explicit infinite ordinals. (Note: the technical definition of an ordinal is more restrictive, but this definition captures the idea.)

In this last part I do a little ordinal arithmetic, including some of the surprising facts, such as how addition and multiplication are not commutative. I close with some hand-waving remarks about how ordinals are best seen as representing patterns, in particular, the kinds of patterns that show up in the study of recursion. You'll want to watch my "Ridiculously Huge Numbers" and "To Infinity... and Beyond" video series to make full sense of what I'm saying here near the end. (Note that the former is at a higher level than the latter, especially in its later parts.)

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