Infinitely More

The Natural Ring of Ordinals

The natural ring of ordinals is the discretely ordered ring generated by the ordinals in the natural arithmetic. The ring exhibits many attractive…

6 hrs ago • Joel David Hamkins

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April 2026

The big bang of numbers

On the big bang of numbers, the surreal genesis—an excerpt from my podcast with Lex Fridman, a sweeping conversation on infinity, philosophy, and…

Apr 23 • Joel David Hamkins

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The natural product of ordinals

Five different self-standing but equivalent accounts of the natural product of ordinals, reflecting five different philosophical perspectives on this…

Apr 12 • Joel David Hamkins

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March 2026

The Book of Infinity—pre-orders are open

Order now at your favorite bookseller

Mar 28 • Joel David Hamkins

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Natural Ordinal Addition

Five different self-standing accounts of natural addition in the ordinals, reflecting five different philosophical perspectives on how we should best…

Mar 15 • Joel David Hamkins

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Counting to Epsilon Naught

Let us aspire to count much higher in the ordinals. How high can you count?

Mar 4 • Joel David Hamkins

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February 2026

On the greats and mathematical style

Lex Fridman and I discuss who is the greatest mathematician in history, and what are the different mathematical styles of undertaking mathematical…

Feb 22 • Joel David Hamkins

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Cantor Normal Form

Cantor proved a remarkable fact about ordinals, providing an ordinal notation system in which every ordinal admits a unique canonical representation by…

Feb 12 • Joel David Hamkins

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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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