Infinitely More

The surreal line is topologically compact—or is it?

Shocking instances of compactness in the surreal line

Nov 28 • 

Joel David Hamkins

10

2

4

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 • 

Joel David Hamkins

10

8

4

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 • 

Joel David Hamkins

10

5

4

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 • 

Joel David Hamkins

13

4

5

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 • 

Joel David Hamkins

9

13

5

September 2025

The uncountable transfinite subway

We explore a more sophisticated version of the infinite subway paradox, with stations all the way to the uncountable ordinals and beyond

Sep 28 • 

Joel David Hamkins

9

20

1

The infinite subway paradox—extending into the transfinite

We extend the infinite subway paradox into the countable ordinals. Can you meet the infinite subway challenge?

Sep 19 • 

Joel David Hamkins

8

9

2

The infinite subway—a full range of paradox

The second in a series of essays on the infinite subway paradox, in which we explore the full range of paradoxical behavior

Sep 11 • 

Joel David Hamkins

9

1

2

The infinite subway paradox

The first in a series of essays on the infinite subway paradox

Sep 3 • 

Joel David Hamkins

10

9

2

August 2025

The Hilbert program

An excerpt from Lectures on the Philosophy of Mathematics

Aug 25 • 

Joel David Hamkins

26

4

Tactics versus strategies—the case of chess

Does chess admit of winning or drawing tactics? Which information exactly do we need to include as part of the board position?

Aug 17 • 

Joel David Hamkins

9

6

2

The tactical variation of the fundamental theorem

We prove the tactical variation of the fundamental theorem of finite games—for finite games with sufficiently rich board positions, one of the players…

Aug 10 • 

Joel David Hamkins

6

2

2