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I'm fairly comfortable dealing with sets of purely Mathematical objects, when I remembered something I heard a long time ago that sets can have elements of any type. Is it possible to define a set with both abstract and physical objects or are the axioms defined only for Mathematical entities? (all our set theory concerns only abstract elements in a set) As this seems something very complicated to visualize and begs a lot of questions with the very different properties of the abstract and the physical, I'm a little bit new to this so I apologise if this is laughably trivial.

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  • It depends on the axioms, but in general, yes. A set is a rational concept. You can put an apple and a circle in a set (e.g. the set of entities that are easy to draw), and that doesn't break any rule. But evidently, a set of geometrical figures cannot include apples.
    – RodolfoAP
    Commented Jan 12, 2022 at 16:53
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    Sure. Take your favorite set of mathematical objects and your favorite set of physical objects, and then take their union.
    – Conifold
    Commented Jan 12, 2022 at 18:48
  • When putting in a physical object does a set work like a set containing numbers where a set with the same elements is 'the same set', when we have an apple as you have, is a set with a different apple in it, a different set? or is 'apple' treated as an object itself, just in the way a number can be thought of as a type in the same way as apple.
    – Confused
    Commented Jan 12, 2022 at 18:51
  • Whichever way you prefer, but the respective sets will be different.
    – Conifold
    Commented Jan 12, 2022 at 18:52
  • the sets with different apples in them will be different? Can I define myself that any set with an apple in it to be the same if all other elements are the same? or do we have to account for all properties when defining a set?
    – Confused
    Commented Jan 12, 2022 at 18:59

2 Answers 2

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Answer

A short, clear answer is yes. You can put pretty much anything you want in a set, including itself. Memberships of sets are arbitrary to the user, and can be defined by set builder notation. For example:

S := {x,y:x is any breed of dog, y is any natural number}

Consequently, given the intensional definition Great Pyrene S, and 5.1 is in S. It should be noted that it is also possible to define sets by explicitly too.

S' := {Great Pyrene, 5.1} would also be acceptable.

Obviously, whether you would want to include physical objects and abstractions is a matter of personal preference, context, and intent.

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The best way to visualize a set is not as a collection, but as the extension of a predicate. A predicate is just a condition or test like "is odd" or "is in Philadelphia". The extension of a predicate is just those things that the predicate is true of--those things that satisfy the predicate. So the extension of "is odd" is all odd numbers, and the extension of "is in Philadelphia" is all things in Philadelphia.

How is the extension different from the predicate itself? It is different because the predicate has more to it than just things it is true of; it has a meaning or significance. For example, the predicate "is a bulldog in that house" is different from the predicate "is a hungry animal in that house". The predicates are different because they have different conditions for being satisfied, but if there are only two animals in the house, say Fido and Ralph, and Fido and Ralph are both bulldogs and are both hungry, then the extensions of the two predicates are the same. Both predicates have the extension {Fido, Ralph}.

This is why sets are said to be "extensional", because there is no such thing as two sets that are different but have the same elements. The set is identified entirely by its elements. Note that the empty set is the extension of every predicate that is true of nothing.

So the question whether you can have a set containing both mathematical objects and physical objects reduces to the question of whether you can have a predicate that is satisfied by one or more mathematical objects and one or more physical objects. There are certainly artificial predicates that answer to that. For example, the predicate "is the Eiffel Tower or is the number two" has the extension {Eiffel Tower, 2}.

Most mathematicians and philosophers would therefore say that a set can have both mathematical and physical objects. However, there are some philosophers who would balk at the artificiality of the predicate and would reject the example.

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  • You're close, but a bit off. Sets are extensional and intentional, true, but you don't quite understand what it means. Read this article. So, when you say, the extension of "is in Philadelphia", you should say the intention of "is in Philadelphia". Predicates that denote membership by rules are intensions if they range across the variable. So, {x:x in 2n-1, n in R} (your odd example) is not an extension clearly, and in fact no extension can be specified because it is infinite. You do write with certainty, I'll give you that.
    – J D
    Commented Mar 2, 2022 at 5:03
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    You are mistaken. Sets are extensional, not intensional. {x:x in 2n-1, n in R} is neither intensional not extensional on its own, but only in how it is understood. If understood as a rule, it is intensional, if understood as the name of a set, the set is extensional. Commented Mar 2, 2022 at 6:34
  • Perhaps you're right. Can you provide an reference so I can straighten myself out?
    – J D
    Commented Mar 2, 2022 at 15:50
  • This one contains a fairly clear explanation: plato.stanford.edu/entries/logic-intensional Commented Mar 3, 2022 at 3:38

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