Infinitely More

Indecomposable Ordinals

Which ordinals are closed under addition? Which are closed under multiplication? Let us try to identify them exactly.

Feb 1 • Joel David Hamkins

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January 2026

Ordinal arithmetic

Let's review the basics of ordinal arithmetic, addition, multiplication, and exponentiation, providing both the order-theoretic semantic definitions as…

Jan 22 • Joel David Hamkins

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Ultrafinitism as arithmetic potentialism

We may fruitfully view the philosophy of ultrafinitism in a potentialist light, helping to illuminate its philosophical commitments.

Jan 12 • Joel David Hamkins

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Anthropomorphizing the Russell paradox

Anthropormorphization in mathematics—an excerpt from my podcast with Lex Fridman, a sweeping conversation on infinity, philosophy, and mathematics.

Jan 5 • Joel David Hamkins

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December 2025

Two visions of ultrafinitism intricately intertwined

These two concepts of ultrafinitism—one positing a largest number, another asserting totality for addition and multiplication but not exponentiation—are…

Dec 29, 2025 • Joel David Hamkins

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Ultrafinitism with a largest number

According to the theory of finite arithmetic (FA), there is a largest natural number, and consequently addition and multiplication are only partially…

Dec 19, 2025 • Joel David Hamkins

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Ultrafinitism

Ultrafinitism is the philosophical view that only comparatively small or accessible numbers exist.

Dec 12, 2025 • Joel David Hamkins

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November 2025

The surreal line is topologically compact—or is it?

Shocking instances of compactness in the surreal line

Nov 28, 2025 • Joel David Hamkins

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The surreal line is topologically disconnected—or is it?

The surreal line is topologically disconnected according to a natural conception of connectedness. Nevertheless, on another conception—attending to…

Nov 15, 2025 • Joel David Hamkins

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The omnific integers are strange

We shall explore several surprising failures of the analogy between the omnific integers and the integers. It turns out that Oz is not so very like ℤ…

Nov 4, 2025 • Joel David Hamkins

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October 2025

The omnific integers are an integer part of the surreal numbers

Can we find a surreal-numbers analogue of the integers? An integer part of the surreal numbers, a discretely ordered subring, for which every surreal…

Oct 23, 2025 • Joel David Hamkins

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The surreal ω × ω chessboard is bigger than you think

What is the nature of the surreal ω × ω chessboard? How many squares are there? How many chess pieces shall we require to set up the board?

Oct 13, 2025 • Joel David Hamkins

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