Definition and Properties

1. Ordinals as Extensions of Natural Numbers:

   • Ordinals are an extension of natural numbers that allow for describing the position of elements in a well-ordered sequence, including infinite sequences.

   • An ordinal is a type of well-ordered set that is transfinite (i.e., it goes beyond finite numbers).

2. Types of Ordinals:

   • Finite Ordinals: These are simply the natural numbers (0, 1, 2, ...).

   • Transfinite Ordinals: These include ordinals like ω (omega), which is the first infinite ordinal. After ω come ordinals like ω+1, ω+2, ..., and so on.

3. Key Properties:

   • Well-Ordering: Every set of ordinals has a first element, and every non-empty subset has a minimum.

   • Transitivity: An ordinal is a set whose elements are also ordinals, and any element of an ordinal is also a subset of it.

   • Non-linearity: Unlike real numbers, not all ordinals are comparable in terms of magnitude in a continuous sense, but rather in terms of order and position.

4. Operations with Ordinals:

   • Sum of Ordinals: The sum of two ordinals is not commutative. For example, ω + 1 ≠ 1 + ω.

   • Product of Ordinals: Also non-commutative, and used to describe more complex structures.

   • Exponentiation: Defined in a way that extends basic arithmetic operations to ordinals.