Saturday, March 21, 2015

To Infinity... and Beyond (English)

См. также:
О бесконечном
Dangerous Knowledge
Прикоснуться к бесконечности за неполных два часа приводится вся наивная теория множеств со всеми идеями доказательств.
A brief intro to ordinal numbers

Disclaimer: Эта подборка видео не моя.
Ниже есть продолжение.

Part 1 of a general-audience talk about various notions of infinity in mathematics. In this part I talk about ordinal and cardinal counting, to set the stage for two ways of talking about different kinds of infinity.

In this part I start talking about infinite sets and comparing them using cardinality. In particular I show one version of Cantor's famous diagonalization method.

In this part I talk about how we can be fooled by our intuitions about the size of various infinite sets.

Про множество Кантора см. у меня тут.

In this part I show a more general version of the diagonalization argument, which proves that there are an infinite number of different infinities. I briefly discuss some famous questions that come out of that.

Про континуум-гипотезу см. у меня тут. Про аксиому выбора см. тут.

Here I start talking about a seemingly unrelated topic, namely how to use recursion to define various arithmetic operations, from the ordinary to the extraordinarily large. We'll see later how this leads us (in a reasonably concrete way) to the idea of ordinal infinities.

I show how a version of the diagonalization trick creates a super-fast-growing function which deserves an infinite ordinal label, and then how we can go further, which gives a hint of the usefulness of infinite ordinals.

In this part I talk in general terms about why the average working mathematician would need to know about the different kinds of infinity.

Про теорию меры см. у меня тут.

Про аксиому выбора см. тут.

In this last part, I bring in a tiny bit of history and conclude with Russell's Paradox, which is a sort of ultimate use of the diagonal trick.

Про бесконечно-малые величины см. тут.

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