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The ordinal numbers
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Regrettable Failures in the Natural Ring of Ordinals
The natural ring of ordinals has unique prime factorization, but other natural features go wrong—the concept of even goes awry, greatest common divisors…
May 24
13
9
4
The Natural Ring of Ordinals Has Prime Factorization
The natural ring of ordinals is a unique factorization domain—every number factors uniquely as a finite product of primes.
May 14
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Joel David Hamkins
16
4
6
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…
May 4
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Joel David Hamkins
17
2
4
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
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Joel David Hamkins
10
6
2
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
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Joel David Hamkins
11
4
4
Counting to Epsilon Naught
Let us aspire to count much higher in the ordinals. How high can you count?
Mar 4
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Joel David Hamkins
14
5
3
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
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Joel David Hamkins
10
8
3
Indecomposable Ordinals
Which ordinals are closed under addition? Which are closed under multiplication? Let us try to identify them exactly.
Feb 1
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Joel David Hamkins
17
3
5
Ordinal arithmetic
Let's review the basics of ordinal arithmetic, addition, multiplication, and exponentiation, providing both the order-theoretic semantic definitions as…
Jan 22
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Joel David Hamkins
13
4
2
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, 2025
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Joel David Hamkins
9
9
2
The infinite subway paradox
The first in a series of essays on the infinite subway paradox
Sep 3, 2025
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Joel David Hamkins
13
9
3
The surreal numbers
All the numbers great and small. The surreal numbers, generated in a recursive process of completion, unify the real numbers, the ordinals, and the…
Jan 6, 2024
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Joel David Hamkins
39
6
5
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