The ordinal numbers

Counting to Epsilon Naught

Let us aspire to count much higher in the ordinals. How high can you count?

Mar 4 • Joel David Hamkins

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

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

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

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

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 • Joel David Hamkins

The infinite subway paradox

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

Sep 3, 2025 • Joel David Hamkins

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 • Joel David Hamkins

Well orders and the ordinal numbers

A well order is a linear order with the further property that every nonempty subset of the domain has a least element. If the well order has any…

Dec 15, 2023 • Joel David Hamkins

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

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