Talk:Property (mathematics)

Rewrite?

I'm proposing the article be replaced with something like this:

In mathematics, a property P is any way of defining, for each mathematical object x, whether P(x) does or does not hold. Formally, in mathematical logic, it is a unary predicate.

The class $\{x:P(x)\}$, using set-builder notation, is called the extension of P. Properties with identical extensions are considered identical, and, since every class corresponds to a property, P is often identified with its extension.

Very often, properties are defined to hold only for some members of a set X, in which case the extension of the property is a subset A of X, and we can define the indicator function $\mathbf {1} _{A}$ on X.

Examples of properties include:

• the property of being an even natural number, the extension of which is the set of even natural numbers
• the property that holds for all sets, the extension of which is the class of all sets
• the property that holds for no object, the extension of which is the empty set

The current text is neither rigorous (it assumes properties whose extensions are sets) nor intuitive (surely someone who doesn't have an understanding of what a property is will not be helped by defining it as a "characteristic").

I'm posting this on the talk page mostly in the hope that someone comes up with a much better text :-)

IpseCustos (talk) 12:51, 6 June 2022 (UTC)