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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
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
14
3
4
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
12
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
1
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
6
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Zeno's paradox
Jan 7, 2023
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Joel David Hamkins
35
14
9
The surreal numbers
Jan 6, 2024
•
Joel David Hamkins
39
6
5
Mathematicians disagree on the essential structure of the complex numbers
Nov 10, 2024
•
Joel David Hamkins
27
4
The Book of Numbers
Jan 2, 2023
•
Joel David Hamkins
44
12
4
The surreal numbers
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The surreal line is topologically compact—or is it?
Shocking instances of compactness in the surreal line
Nov 28, 2025
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Joel David Hamkins
11
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, 2025
•
Joel David Hamkins
12
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, 2025
•
Joel David Hamkins
11
5
4
The ordinal numbers
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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
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
14
3
4
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
12
4
2
Ultrafinitism
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The Book of Infinity
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Infinite Games
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A Panorama of Logic
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Proof and the Art
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Philosophy of Mathematics
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The mathematics and philosophy of the infinite
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Pershmail
Michael Pershan
The Palindrome
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Science without the gobbledygook
Sabine
The Nature-Nurture-Nietzsche Newsletter
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TrueSciPhi.AI
Kelly Truelove
JDH Links
JDH web page
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JDH on Notes
JDH on MathOverflow
JDH on YouTube
JDH on Google Scholar
JDH at MIT Press
My Books
Lectures on the Philosophy of Mathematics, MIT Press 2021
Proof and the Art of Mathematics, MIT Press 2020
Proof and the Art of Mathematics: Examples and Extensions, MIT Press, 2021
A Mathematician's Year in Japan, Kindle KDP, 2015
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