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If the empty set is unique, how can it occur twice in a construction like for example the von Neumann ordinal for 2: {{},{{}}}.

Or if the number 1 is unique, how is it possible you can add it to an identical copy of itself to give 2?

If something is unique, an "identical copy" is a contradiction in terms, which is why you can't have a set {{},{}}.

Is this problem discussed anywhere in the philosophy of set-theory, Ontology or maybe somewhere else?

I guess you could maybe argue that there is a "concept" of a 1, which is unique, but many "instances" of 1 of which there can be many. But this seems contrived and non-parsimonious. Also it would mean you can add the concept of 1 to the concept of 2 to get the concept of 3, but you can't add the concept of 1 to itself to get the concept of 2, which makes no sense.

Stefan
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  • The word "object" is used ambiguously in mathematics. 1 in a fixed structure is the name of an object type with unlimited supply of instances, but this object type is itself an "object", which is unique. Then you have an unlimited supply of isomorphic instances of positive integers (von Neumann's, Zermelo's, etc.), all of which are instances of the same type of structure, which is also a unique "object", positive integers. Natural language functions the same way, "man" is a type with many instances, but you can use the noun in a sentence as if it was an object. – Conifold Sep 19 '20 at 20:12
  • Closely related is Benacerraf's identification problem, see [SEP, What Numbers Could Not Be](https://plato.stanford.edu/entries/philosophy-mathematics/#WhaNumCouNot). – Conifold Sep 19 '20 at 21:22
  • ' but this object type is itself an "object", which is unique ' - but that's what I am saying, if you have a unique 1-type, like maybe the set of all sets which only contain one element, then the problem repeats on the type level: How come you can define a plus operation and add two 1-types to get a 2-type, if the 1-type is supposedly unique? – Stefan Sep 19 '20 at 21:30
  • @Stefan Welcome to Stack Exchange, my friend. I hope you find what you are looking for here. :) ^^ :D – Donate to the Edhi Foundation Sep 19 '20 at 21:33
  • Because you are not adding types, only their instances. When you want to "add types" you are passing to a higher type they are instances of, and they are no longer unique. "Unique" is relative, just like "object". – Conifold Sep 19 '20 at 21:40
  • Benacerraf's identification problem is about the fact that you can define numbers using sets in several different ways, each being equally valid. The problem I am struggling with appears even if Benacerraf's identification problem didn't exist and there was just one unique way to define numbers using sets, because even in this unique definition of numbers, you would want to add 1 to 1 to get two, so you need two copies of a single thing...? – Stefan Sep 19 '20 at 21:45
  • 1 is not a thing, it is a label, and since existence in mathematics is cheap you can have an unlimited supply of things it labels. Or, if you prefer platonism, you can think of it as a form under which multiple particulars fall, some call them ["abstract particulars"](https://en.wikipedia.org/wiki/Abstract_particulars). Plato even attached form-copies to each at one point. – Conifold Sep 19 '20 at 21:47
  • Googled "abstract particulars" and "Trope theory", which does indeed seem to touch on the problem - thanks for that pointer Conifold. But the resolution presented seems to be just postulating: Copies of the same thing (label, trope,...) can exist many times in the platonic realm, full stop - But this still leaves me with the niggle: If you put "two" of the same into a set, the set only contains one thing, i.e. {1,1}={1}, so sometimes you CANNOT have copies and sometimes like in the set {1,{1,2}} you CAN have copies - which one is it? How can it be both? – Stefan Sep 19 '20 at 22:58
  • One way to handle it is to distinguish [qualitative and numerical identity](https://plato.stanford.edu/entries/identity/#1). Copies are qualitatively but not numerically identical, when in a set, numerically only distinct things are taken as a single element. If you *really* want copies in a set you can index them to make pairs (including the index) qualitatively distinct. – Conifold Sep 19 '20 at 23:37
  • You speak of *concepts*: maybe this is the right track. See Frege: number are "abstract" objects derived from number-concepts. The number 2 is a different concept from the number 1: the concept is unique but we have many concepts that falls under it: all the concepts that are instantiated by two only objects. – Mauro ALLEGRANZA Sep 21 '20 at 08:11

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