Is there any justification for the existence of sets?

Question

In this Reddit comment I was explaining how natural numbers could be built from the empty set:

A standard set-theoretic way of defining the natural numbers[1] 1,2,3,... is based on the empty set, the set {} that contains zero elements. Suppose it is true that numbers don't exist. Let {} be the set such that every element of {} is a number that exists. Since there does not exist a number, {} contains zero elements. So {} exists. Let 0={} represent the number zero. Now we know 0 exists as a symbol for the set containing zero known elements. Then let 1={{}}={0} represent the set of one element. Now we know 1 exists as a symbol for the set containing one known element. Then 2={{},{{}}}={0,1}, 3={{},{{}},{{},{{}}}}={0,1,2}, etc.

and received the reply:

Why should I accept an ontology of 'sets' at all?

What's the best answer here? Is it just because we assume sets exist? Is there justification for why sets exist?

Answer 1

The first question a mathematician looking at a new concept is not to ask whether it exists but how it proves itself useful. The differential calculus made it easy to write down the equation of motion of Newtonian Physics and Descartes coordinate geometry made it possible to formulate the same equations in a natural way.

This might be called the pragmatic answer, in the sense we think of it as a tool to predict, describe or solve; or a coherent answer, which rejects the (metaphysical) reduction to a foundation, (philosophically this view is called foundatinalism) to a set of beliefs (axioms) that are secure or justified, and is non-inferential ie axioms.

So how do sets prove themselves useful in mathematics. This is connected with Hilberts programme to reduce mathematics to logic, and one is lead to this thought when one considers that sets are aligned with logic, consider Venn Diagrams and observations that all the integers for a set and so on.

A different approach, called Platonism asserts the existence of any set of axioms that is internally consistent. This draws upon Platos theory of Forms which gives existence to pure ideas. This is essentially a metaphysical supposition, and evidence of such a position can only be given in the most indirect kind of way.

Another position that came out of the early work on mathematical logic is Formalism which drops any need to ground a theory in some Platonic world and instead observes that consistency is all that is important when it comes to manipulating axioms and theorems. This is actually grounded in logic, and observes that a mathematical theory can be seen as simply as a set of strings that are rewritten according to some set of rules.

Finally, the analytic or positivist approach that Rostomyan mentions with regards to Carnap, and which came out of a philosophy of science in Vienna, also dispenses with metaphysics by rendering them 'meaningless' - the metaphysical voice is silenced.

A standard set-theoretic way of defining the natural numbers[1] 1,2,3,... is based on the empty set

Note that we can start from any other set, the set of cups for example: Define 0:=Cups, 1:={0}, 2:={1} etc. Of course the void set was chosen for two reasons, it is a given, and also it is the minimal set that is included in any other set. Another point to notice is that we can say 2 is a member of 3 which doesn't make much sense. Finally note we can just define the natural numbers as a ring with certain arithmetic properties or via the Peano Axioms.

As one notes from this, there are quite a few distinct positions and which position one adopts is most likely a function of ones own inclination as the schools of thought one is aligned with.

Answer 2

We may have two different point of view on this issue.

1) The mathematical one.

From Joseph Shoenfield, Mathematical Logic (1967), page 238 :

a set A is formed by gathering together certain objects to form a single object, which is the set A. Thus before the set A is formed, we must have available all of the objects which are to be members of A. [...]

We are thus led to the following description of the construction of sets. We start with certain objects which are not sets and do not involve sets in their construction. We call these objects urelements. We then form sets in successive stages. At each stage we have available the urelements and the sets formed at earlier stages; and we form into sets all collections of these objects.

A collection is to be a set only if it is formed at some stage in this construction [emphasis added].

We can carry out this construction with any collection of urelements. If we carry it out with no urelements, the sets which we obtain are called pure sets. It turns out that these are sufficient for mathematical purposes; and they are also sufficient to illustrate all the problems which arise in the general case.

[Thus, in the "standard" treatment of set theory, like ZF] we shall therefore restrict ourselves to this case, and henceforth take set or class to mean pure set.

Thus, if our choice is to have urelements, we can start with a collection of objects whatever: physical or abstract ones.

But if we start with physical objects, we are not licensed to assume the existence of infinitely many of them, while some sort of "axiom of infinity" is necessary for the development of "current" mathematics.

In conclusion, the mathematical theory of sets is a basic one, because with a very "limited" amount of initial ideas can build up an unified "vision" of (quite) all current mathematics.

Having said that, a "formalist" mathematician can says that he is "playing the game" of sets without committment about the existence of abstract objects called "sets" [as we can do with numbers and any other kind of abstract objects], but it seems to me a very unsatisfactory view about the "meaning" of mathematical practice.

2) The logico-philosophical one.

According to the logicist tradition of Frege and Russell, concepts are primitive and classes are the extension of concepts (thus sets or classes have a sort of "secondary" nature).

But the well known difficulties in which the logicist program incurred has blocked (up to now ?) the development of a consistent and shared theory of concepts.

Answer 3

The mathematical concept of a set has a close connection to the notion of a well order, and a weaker connection to the notion of a quotient object. Cantor developed set theory and the theory of well orders together. Rigorous introductions to set theory still treat well orders as an integral part of set theory. Even most simplified introductions to set theory include well orders.

Note that even programming languages make use of (well) order to implement efficient set data structures. It's true that set data structures in programming languages can also cope with well-orders on equivalence classes of elements, but often they implicitly create an order among the elements of an equivalence class.

The equivalence classes just mentioned can help explain how most axiomatic set theories can avoid a prefered global well order, while still being able to well order (any set of) their universe. There exists a prefered well order on equivalence classes of sets, like von Neuman's cumulative hierarchy or Gödel's constructible hierarchy.

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There is nothing wrong with well orders. But if we imagine a set, most of us will imagine it together with some sort of implicit well order, and this well order makes the concept of a set so appealing to our intuition. Without some sort of well order, sets are much harder to grasp, and no longer intuitive.

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What did I mean by the weaker connection to the notion of a quotient object? We might look at sets as functions from a universe of objects to the set {0,1}. The functions can be defined in many different ways. Two functions are intensionally equal, if their definitions coincide. (Even if you define a function by a list of all the objects that it maps to 1, two such functions will only be intensionally equal if the objects are listed in the same order.) Two functions are extensionally equal, if their mapping behavior of objects from the universe is identical. Two intensionally equal functions are automatically also extensionally equal, but the converse is not always true. A set in this sense is a quotient object of extensionally equal functions.

The existence of a well order makes it easy to work with quotient objects, because it enables us to pick a prefered representant from each quotient object. But at least in theory, it should also be possible to work with quotient objects directly. In that case however, one might have to worry about some intiutionistic distinctions. We might have a system of quotient objects, where we can always find out when two elements belong to different quotient objects (if we only search long enough), but where we sometimes can't prove that two objects really belong to the same quotient object.

Answer 4

What is the meaning of "to exist"? We can take the position that all material objects exist and that all immaterial objects (usually addressed as ideas) exist as far as they correspond to an activity in our brains. In general it is simple to define: An object exists if two persons can be sure to be talking about one and the same object when they mention a name, for instance the number pi.

In the gist of this view we can take Cantor's definition of set: "every collection of defined well-distinguished objects of our visualization or thinking." Then the set {1, pi, my father, sun} exists.

A statement by Jan Mycielski supports this point:

David Hilbert in 1904 [...] wrote that sets are thought-objects which can be imagined prior to their elements. At request of the referee who asked what is a thought-object let me add: I understand it to be a thought about an object which may exist or not. Thus it is an electrochemical event in the brain or/and its record in the memory. In particular it is a physical thing in space time. Of course it is difficult to characterise any physical phenomena. But we have the ability to recognize thoughts as identical or different, just as we have the ability to recognize a silent lightning from a thunderous one. Hence I understand Hilbert's words as follows: mathematicians imagine sets which do not exist, but their thoughts about sets do exist and they can arise prior to the thoughts of most elements in those sets.

[J. Mycielski: "Russell's paradox and Hilbert's (much forgotten) view of set theory" in G. Link (ed.): "One hundred years of Russell's paradox: mathematics, logic, philosophy", de Gruyter, Berlin (2004) p. 534]

So we can state that infinite sets like all real numbers exist, because we can talk about them and we all understand the same. But we can also be sure that not all elements of this set exist, because there are only countably many possible definitions but, allegedly, more real numbers which, according to this view, do not exist.

Answer 5

Is there any justification for why sets exist? No. But then there is no justification for any position. Arguments make assumptions to try to show all of those assumptions true would lead to a new argument with a new set of assumptions that have to be shown to be true and so on: infinite regress. The question you should ask is what is the value of arguments like the one that models the properties of natural numbers in terms of sets. The answer might be something like it's a good idea to try to model natural numbers in terms of set theory because it ties natural numbers and set theory together. If this works you have discovered a deep connection between these two areas of mathematics, if it doesn't you have discovered a flaw in our understanding of maths that should be fixed.

For more on non-justificationist philosophy of maths you might want to read "Realism and the Aim of Science" by Karl Popper, "Proofs and Refutations" by Imre Lakatos and "The Fabric of Reality" and "The Beginning of Infinity" by David Deutsch.