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Invariance
Invariance
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As Conifold and Michael Carey mentioned in the comments, the Zermelo ordinals do not extend to infinite sets. I want to emphasize this because it's very important to the way ordinal numbers are used. Very often (perhaps nearly always) when one is using the von Neumann ordinals, the infinite versions of the ordinals come into play.

For example, we typically use the ordinals to define cardinality in that |X| is the least ordinal number bijective to X, so we'd need to find some other way to formalize size. With the von Neumann ordinals, we get a series of infinite sets of increasing cardinality (starting with ω₀, ω₁, ω₂, ω₃, and continuing on for every ordinal subscript) and we know that every infinite set is bijective to ωₙ for some ordinal n (assuming the axiom of choice).

There are some other convenient things about the von Neumann ordinals, like how A ≤ B just means A ⊆ B, or how |X| = X for finite X, but infinity is a big one.

As Conifold and Michael Carey mentioned in the comments, the Zermelo ordinals do not extend to infinite sets. I want to emphasize this because it's very important to the way ordinal numbers are used. Very often (perhaps nearly always) when one is using the von Neumann ordinals, the infinite versions of the ordinals come into play.

For example, we typically use the ordinals to define cardinality in that |X| is the least ordinal number bijective to X, so we'd need to find some other way to formalize size. With the von Neumann ordinals, we get a series of infinite sets of increasing cardinality (starting with ω₀, ω₁, ω₂, ω₃, and continuing on for every ordinal) and we know that every infinite set is bijective to ωₙ for some ordinal n (assuming the axiom of choice).

There are some other convenient things about the von Neumann ordinals, like how A ≤ B just means A ⊆ B, or how |X| = X for finite X, but infinity is a big one.

As Conifold and Michael Carey mentioned in the comments, the Zermelo ordinals do not extend to infinite sets. I want to emphasize this because it's very important to the way ordinal numbers are used. Very often (perhaps nearly always) when one is using the von Neumann ordinals, the infinite versions of the ordinals come into play.

For example, we typically use the ordinals to define cardinality in that |X| is the least ordinal number bijective to X, so we'd need to find some other way to formalize size. With the von Neumann ordinals, we get a series of infinite sets of increasing cardinality (starting with ω₀, ω₁, ω₂, ω₃, and continuing on for every ordinal subscript) and we know that every infinite set is bijective to ωₙ for some ordinal n (assuming the axiom of choice).

There are some other convenient things about the von Neumann ordinals, like how A ≤ B just means A ⊆ B, or how |X| = X for finite X, but infinity is a big one.

Source Link
Invariance
Invariance
  • 119
  • 2

As Conifold and Michael Carey mentioned in the comments, the Zermelo ordinals do not extend to infinite sets. I want to emphasize this because it's very important to the way ordinal numbers are used. Very often (perhaps nearly always) when one is using the von Neumann ordinals, the infinite versions of the ordinals come into play.

For example, we typically use the ordinals to define cardinality in that |X| is the least ordinal number bijective to X, so we'd need to find some other way to formalize size. With the von Neumann ordinals, we get a series of infinite sets of increasing cardinality (starting with ω₀, ω₁, ω₂, ω₃, and continuing on for every ordinal) and we know that every infinite set is bijective to ωₙ for some ordinal n (assuming the axiom of choice).

There are some other convenient things about the von Neumann ordinals, like how A ≤ B just means A ⊆ B, or how |X| = X for finite X, but infinity is a big one.