This is an intensional definition of a set, a definition by specifying a common property of the set's elements. An extensional definition of a set would be a definition by listing all the particular elements. — litewave
What about these two: the property of redness, and the property of being an instance of redness (or the property of having the property of redness). Both properties seem to be instantiated in all instances of redness, so the instances form one and the same set. — litewave
However, if "being an instance of redness" is referring to several things, as in "being an exemplification of redness", then it means the same as "redness". — RussellA
A set is a collection of objects. An average person surely knows what a collection is. Not so surely a universal. — litewave
The mainstream view among mathematicians is that sets are abstract objects. You can see them with the mind's eye, but not physical eyes. — frank
Would that mean that "being in that collection of objects [or individuals, per Banno]" is a shared property? Can an object "wander in," so to speak, and partake of that property? This may not be a question about your definition so much as an expression of uncertainty about "property". — J
Yes. So what, if anything, would we want to say about identifying such a set with some property? I take it you don't want "being in set X" to count as a property -- nor could it, on the OP's proposal. — J
I'd put it to you that the collection of individuals is an abstract object. To use your cell phone example -- we can think about the cell phone as a collection of particular objects and then name this in accord with set-theory. I.e. we can make sets which refer to concrete individuals, but to treat something as a set is still an abstraction. — Moliere
I'm not sure what you mean by "abstract" or "abstraction" here. Is the phone a concrete or an abstract object? Is it a collection of other objects or not? — litewave
What I mean by "abstraction" is that you can treat the phone in either way without changing anything real. — Moliere
A collection consisting of a phone located in my house and another phone located in my friend's house would be a concrete object too, although some might resist that because the two phones are separated "too much". A collection consisting of my phone and of another phone in a different universe that is in a different spacetime might be regarded as a concrete object because its elements are located in a spacetime but then again, they are in different spacetimes, so this collection transcends a single spacetime. — litewave
And then there is the general property/universal "phone" (or "phoneness") - that which all particular phones have in common - and I guess this would be regarded as an abstract object by almost anyone because unless we identify it with the set of all phones, it seems to transcend spacetime or be located in spacetime in an especially weird way. — litewave
heh, well, once you flip your opinion I'll return -- but my work is done :D — Moliere
But, no, they're different -- a set is its' elements, rather than a property which all the elements share. — Moliere
But I still don't think that a set is identical to its elements because a single object cannot be identical to multiple of object. So a set is another object, additional to its elements. — litewave
To go back to the subset relation: if any element is a member of B, and B is a subset of A, then any element of B is an element of A. — Moliere
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