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ordinal arithmetic
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The Natural Field of Ordinals
Which numbers are transcendental over the ordinals? Which are irrational? Let us introduce the natural field of ordinals and consider the status of √2…
Jun 25
•
Joel David Hamkins
9
2
6
Fermat’s last theorem in the natural ring of ordinals
Are there any nontrivial solutions of the famous Fermat equation in the natural ring of ordinals?
Jun 13
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Joel David Hamkins
11
3
5
Regrettable Failures in the Natural Ring of Ordinals
The natural ring of ordinals has unique prime factorization, but other natural features go awry—there are consecutive odd numbers, entire intervals of…
May 24
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Joel David Hamkins
13
9
5
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
18
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
5
The natural product of ordinals
Five self-standing equivalent accounts of the natural product of ordinals, reflecting five different philosophical perspectives on this beautiful, core…
Apr 12
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Joel David Hamkins
10
10
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
•
Joel David Hamkins
11
6
4
Counting to Epsilon Naught
Shall we count higher in the ordinals? Let us strive for the ordinal epsilon naught, the first exponential fixed point—we shall completely grasp all the…
Mar 4
•
Joel David Hamkins
15
5
3
Cantor Normal Form
A remarkable ordinal notation system, as useful as the decimal number system, but for arbitrary ordinals and grounded in base ω rather than base ten.
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
18
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
14
4
2
How to count
Shall we count together in the ordinals? Let us venture into that transfinite realm beyond infinity. Is infinity even or odd? Can we count to an…
Feb 22, 2023
•
Joel David Hamkins
22
12
1
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