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Galileo’s Paradox of Infinity—Lectures on Infinity (lecture 6)
Galileo recognized that some infinite collections can be placed into one-to-one correspondence with a proper subcollection. How are we thus to make…
Sep 30
•
Joel David Hamkins
9
2
1
Ordinal definability—how did Gödel do it?
Gödel proposed the concept of ordinal definability well before the reflection theorem of Lévy and Montague existed. How did he do it?
Sep 20
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Joel David Hamkins
16
3
Potential versus Actual Infinity—Lectures on Infinity (lecture 5)
Two radically different conceptions of the infinite—potentialism versus actualism. Disputed for millennia, then a sea change in views. Contemporary…
Sep 9
•
Joel David Hamkins
10
3
Ordinal definability is definable
Why is the notion of ordinal definability definable in set theory, when the underlying notion of definability, we know, is not definable? How do we…
Aug 31
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Joel David Hamkins
15
2
Is Infinity Even or Odd?
The question is naturally taken in a variety of ways—About the ordinals? Specifically ω? But are we dividing into pairs or cutting in half? Standard or…
Aug 24
•
Joel David Hamkins
15
3
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The surreal numbers
Jan 6, 2024
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Joel David Hamkins
40
6
5
Zeno's paradox
Jan 7, 2023
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Joel David Hamkins
35
14
10
Mathematicians disagree on the essential structure of the complex numbers
Nov 10, 2024
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Joel David Hamkins
29
4
The Book of Numbers
Jan 2, 2023
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Joel David Hamkins
44
12
4
Lectures on Infinity
See all
Galileo’s Paradox of Infinity—Lectures on Infinity (lecture 6)
Galileo recognized that some infinite collections can be placed into one-to-one correspondence with a proper subcollection. How are we thus to make…
Sep 30
•
Joel David Hamkins
9
2
1
Potential versus Actual Infinity—Lectures on Infinity (lecture 5)
Two radically different conceptions of the infinite—potentialism versus actualism. Disputed for millennia, then a sea change in views. Contemporary…
Sep 9
•
Joel David Hamkins
10
3
The Largest Tweetable Number—Lectures on Infinity (lecture 4)
The paradox of the largest tweetable number. What is the largest number you can tweet?
Aug 12
•
Joel David Hamkins
35
12
8
The surreal numbers
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Is Infinity Even or Odd?
The question is naturally taken in a variety of ways—About the ordinals? Specifically ω? But are we dividing into pairs or cutting in half? Standard or…
Aug 24
•
Joel David Hamkins
15
3
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
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Joel David Hamkins
9
1
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
14
2
5
The ordinal numbers
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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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Recommendations
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Science with Sabine
Sabine
Mindcreaser - Benjamin Portheault's substack
Benjamin Portheault
The Palindrome
Tivadar Danka
Pershmail
Michael Pershan
Math and Art
Erin Carmody
JDH Links
JDH web page
JDH on Twitter
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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