cantors-attic

Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.

View the Project on GitHub neugierde/cantors-attic

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The upper attic
The middle attic
The lower attic
The parlour
The playroom
The library
The cellar

Sources
Cantor's Attic (original site)
Joel David Hamkins blog post about the Attic
Latest working snapshot at the wayback machine

Monotone

Monotonicity is a property of functions.

A function $f$ is monotone if and only if when $x \le y \implies f(x) \le f(y)$, for all $x$ and $y$ in the domain of $f$.

A function $f$ is strictly monotone if and only if, for all $x$ and $y$ in the domain of $f$, $x \lt y \implies f(x) \lt f(y)$. All strictly monotone functions are monotone, but not vice versa.

A function $f$ is called a strictly monotone ordinal function if and only if it is strictly monotone, its domain is an ordinal number, and its range is a subset of the ordinals.

Properties

All strictly monotone functions are injective.

If $f$ is a strictly monotone ordinal function, then $x \le f(x)$ for any $x$ in the domain of $f$.

If $f$ provides an order-isomorphism between an ordinal and a subset of the ordinals, then $f$ is strictly monotone.

Examples of Monotone functions

The identity function is an example of a monotone function that is not strictly monotone.

The aleph function is a strictly monotone ordinal function.