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Whovian

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**Parody math paper - defining "properties"**

And no, I don't mean properties in the mathematical sense, but rather in the "everyday" sense.

In a parody mathematics paper I'm writing, I'm trying to define the "properties" of an object as a stepping stone to defining similarity. If we let P(o) be the set of all properties of o, similarity, in this case, is a function ##S## which takes two objects as an input and gives an element of (0,1] as an output, such that ##P\left(o\right)\cap P\left(o'\right)\subset P\left(o''\right)\cap P\left(o'\right)\implies S\left(o,o'\right)<S\left(o,o''\right)##, among other things. Clearly, if we just let properties be arbitrary elements of the power set of the set of objects (in which the objects having said property correspond to the elements of the set,) this axiom is basically moot, since for any o≠o'', the "if" bit of that statement is always false.

I'm thinking maybe just a topology over the set of objects, where properties are open sets in this topology? Does anyone have any ideas for a better or other definition?

(If this is a bad place for this post, can this be locked, deleted, or moved to the appropriate forum?)