Infinitely More

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

7

4

4

Counting to Epsilon Naught

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

Mar 4 • Joel David Hamkins

14

2

3

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

6

3

1

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

9

8

3

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

15

3

4

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

13

4

2

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

10

5

2

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

18

7

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

10

3

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

10

6

Ultrafinitism

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

Dec 12, 2025 • Joel David Hamkins

25

11

12

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

11

2

4