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In mathematics we deal with 'sets' they are abstract as the objects in them are abstract, they have no tempo-spatial location. How about standard 'collections' we would encounter in real life, if I put three objects next to each other, how does this differ from a 'set'?

For example a singleton 'set' is always considered different from the object inside of it, this suggests they are 'abstract' in nature, but my 'collection' physically exists, is this collection different from a set due to this distinction?

Abstract objects can be described as sets, but could they be defined as a 'collection' in a similar manner, their entire make-up can be defined as a grouping of ideas, but not as a 'set' with the formality of using the idea of a 'set'? I have my doubts on this, as we somewhat have to 'repackage' our ideas to form new ones from others.

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  • When you talk mathematics, you talk metaphysics. So, there's essentially no difference: both are ideals. Your shopping list is metaphysical (ideas written in a paper are just a set or a collection of ideas). But your actual shopping bag can be considered a physical object, a whole, a set, not really a collection. See the link.
    – RodolfoAP
    Commented Mar 8, 2023 at 11:23

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There are theories of physical sets, e.g. the causal set research program or the somewhat older idea of quasi-sets (with intended quantum-physics applications). See also Augenstein[96] for rampant speculation about the intersection of the physics and set-theory domains of discourse.

As for theories of non-physical sets, these often reach a conceptual plateau whereupon we speak of "proper classes," which have elements under the basic elementhood relation but aren't themselves elements—except this is moreso modulo the basic elementhood relation in that the issue of "hyperclasses" emerges:

... we may distinguish the mathematico-set theoretic part of the realm, the sets, from the classes. As there are no ’super-parts’ a hierarchy over and above the universe V does not threaten.

The reference is in part(!) to Maddy[88]:

In other words, we assume that our theory of sets is the universal theory of collections, and hence that it applies to these new objects. This gesture produces lots and lots of these class-like entities, lots of ordinal-like objects greater than ORD, lots of stages of construction after V; and they all obey the axioms of set theory.

This treatment is neat, so neat that we begin to wonder if these new layers really consist of entities of a new and different type; perhaps we just forgot to finish the iterative hierarchy in the first place.

In a book about category theory, they say:

The basic concepts that we need are those of “sets” and “classes”. On a few occasions we will need to go beyond these and also use “conglomerates”. ...The concept of “conglomerate” has been created to deal with “collections of classes”. ... Since our main interest lies with such categories as the category of all sets, the category of all vector spaces, the category of all topological spaces, the category of all automata, and possible “extensions” of these, no need arises to consider any “collections” beyond the level of conglomerates, such as the entity of “all conglomerates”. ... [There is a problem that] could be resolved, e.g., by introducing in addition to “sets”, “classes”, and “conglomerates”, one higher level of entities, say, “collections”. Then the collection of all conglomerates and all functions between them would form a rather well-behaved “quasi-quasicategory”. Since there are only a few occasions in this text where the use of something like this would be advantageous, we have refrained from complicating the foundations by introducing it.

The upshot is that if we think of multiple elementhood relations, we end up with an isomorphic system wherein the things we weren't calling sets "because they're not elements of other things" now can become proper sets again, just under further elementhood relations (types of these I suppose would be easiest to say). That is, a set is an element0 of a proper class, a proper class is an element1 of a conglomerate, a conglomerate is an element2 of a collection, etc.

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A collection is not a set and a set should not be thought of as a virtual bag of things. A set should be thought of as the extension of a property or concept. A property could be, for example, "is an odd number" or "is an apple on that tree". Any objects of which the property is true is said to be in the extension of the property. A set then, is all of the objects in the extension of a property, viewed as a whole.

So if you make a pile of apples, there is a property associated with that pile, namely, "is an apple in that pile", and every apple in the pile is in the extension of that property, so there is a set associated with the pile. But it is a mistake to think of the set as being the pile. There is nothing in the notion of a set about things being brought together, just a notion of the elements having something in common like being in a pile together, or being an odd number, or being red. Of course, what they have in common could be completely artificial: "is an apple in that pile or is an odd number", for example.

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  • good point, is the idea of a collection valid for abstract objects, or is it simply that they can be ;associated; with other ideas we feel as falling under them, is this idea of a 'collection' just a set due to a lack of spatial and temporal location?
    – Confused
    Commented Oct 8, 2022 at 15:32
  • " A set should be thought of as the extension of a predicate. " How about if I give you the predicate x ∉ x ? Does that give us a set? No, that's Russell's paradox. A set is NOT the extension of a predicate. Surely you know that.
    – user4894
    Commented Oct 8, 2022 at 21:06
  • @user4894, I did not say that every predicate has an extension. Actually, the word "predicate" was wrong; I should have said property or concept. Commented Oct 8, 2022 at 23:41
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    @DavidGudeman Then I'd give x ∉ x as a property or concept. Any such formulation fails. You wrote, "A set should be thought of as the extension of a property or concept" and I can only respond to what you wrote. OP is asking about the nature of sets, and you gave them an inaccurate idea. Your use of "should" is wrong. Extensions of predicates don't always give sets. They almost never do, unless you are qualifying over an already existing set. As I believe you already know. Don't know why you're defending this point. If you'd said "can informally be" instead of "should" it would have been ok.
    – user4894
    Commented Oct 9, 2022 at 2:00
  • @user4894, my switch from predicate to property was not motivated by your criticism; it was motivated by my lack of sleep and my sudden feeling that "predicate" meant "proposition". It was some sort of temporary delusion. In any case, that change was not a response to your criticism. My response to your criticism was given in the first sentence of my reply. Since you don't seem to have understood, let me elaborate: for my claim to be true it is not necessary that all predicates or properties have extensions, it is only necessary that all extension have predicates or properties. Commented Oct 9, 2022 at 16:09

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