How to construct the natural numbers with hypersets

[Stoney Pete]

I'm not a logician or mathematician but a philosopher (with dyscalculia) so please forgive me for skipping the technicalities...

My question is this: Is there in the theory of non-well-founded sets (hypersets) something analogous to the set-theoretic construction of the natural numbers in ZF? How does this work? What are the axioms used? And where can I read more about this?

Thanks in advance!

 

[TeethWhitener]

Why would there be any difference? The von Neumann construction of the ordinals starts with equating zero with the empty set and constructing each successor as the set of its predecessors. So 0=∅, 1={0}, 2={0,1}, 3={0,1,2}, etc. Non-well-foundedness simply means suspending the axiom of foundation (there are a few different ways to do this consistently). The axiom of foundation says that no set is a member of itself. Suspending this means that some sets are members of themselves. If a set is a member of itself, it is called non-well-founded.

But note, negating the axiom of foundation doesn't imply that well-founded sets don't exist; it simply allows non-well-founded sets to exist (alongside well-founded sets). (E.g., if you allowed only non-well-founded sets, you wouldn't even be able to define the empty set, because the empty set is well-founded.) But all the sets from standard set theory are well-founded, so I'm not aware of any reason why we'd have to give them up simply because we've allowed non-well-founded sets to exist as well. Do you have a particular reason in mind for why allowing non-well-founded sets would be a problem for the von Neumann construction?

 

[Stoney Pete]

Well, I am interested in the relation between set theory, self-consciousness and mathematics. I read a paper by cognitive scientist Kenneth Williford who uses hypersets to model self-consciousness and its self-referential structure (a la Douglas Hofstadter). This made me think: perhaps we can see mathematics (insofar as it can be constructed from sets) as deriving from the structure of self-consciousness (insofar as it can be modeled by hypersets)? I guess this idea belongs to the philosophy of mathematics rather than mathematics proper. I am interested in the question: what is the reality of mathematics? Is it just a bunch of formal symbolic systems? Is it a platonic realm? And I guess my idea (that self-consciousness is the 'origin' of mathematics) comes close to intuitionism about mathematics, though I don't think my idea implies finitism. I know it's pretty vague. Hence my interest in hypersets, to make this more precise.

So my basic idea is this: If we model self-consciousness using hypersets, we can say that self-consciousness is like a set S = {S}. Self-consciousness is like this because I am not just aware of myself, but I am also aware that I am aware of myself, and I am aware that I am aware that I am aware of myself. So you get a nested hierarchy similar to S = {S} = {{S}} = {{{...}}}. If we then use something like Von Neumann's definition of an ordinal as a set that contains its predecessor, we could -- in a sense -- say that S = ω because S contains a predecessor which contains a predecessor etc.

But this way of constructing the natural numbers won't do, because here we start the construction with ω and we never get to 0...

So perhaps we should select another hyperset as the appropriate model for self-consciousness which can be seen as generating the natural number system. Perhaps S = {∅, S} where ∅ and S form an ordered pair. In the model presented by Kenneth Williford S={∅, S} would model something like: I am conscious of ∅ and of myself as having this consciousness. So in psychological terms S={∅, S} would be a possible model. Now when we let S unfold its cycles we get: S = {∅, {∅, {∅, ... }}}. The first element of S is ∅ and this is naturally interpreted as 0. After that, however, the analogy with the natural number system breaks down...

So, in short, this is my question: Is it possible to design one hyperset that in its cyclical (= recursive?) structure can be seen as constructing or otherwise mirroring the natural number system?

Hope this is not too vague for you...

 

[Stoney Pete]

But perhaps S={S} might work after all... I have been trying to find a circular graph and the above is the best one I could find. So let's rename our hyperset and call it "v" to match the picture above. Let's take v0 as the 'mother node'. And let's assume for the sake of the argument that the graph contains infinitely many nodes (ω nodes). If we take "→" to mean "contains as element" then we can write: v0→v1→v2→v3...etc. In a sense this means that v0=ω because v0 has infinitely many predecessors. But at the same time the graph comes full circle in v0 itself so that we would have to say that v0 is also ∅ or 0, the first predecessor! And looked at it this way, v1= -1, v2= -2 etc.

Does this make sense?

 

[TeethWhitener]

A lot of this is philosophy, which, per forum rules, doesn't really belong in these forums. But I'll try to just address the mathematical content of your post.

If we then use something like Von Neumann's definition of an ordinal as a set that contains its predecessor, we could -- in a sense -- say that S = ω because S contains a predecessor which contains a predecessor etc.

First, von Neumann's construction of the ordinals is that a number is the set that contains all of its predecessors, not just its immediate predecessor. This is a very important point, because the criterion "is a member of" gives a convenient and simple well-ordering. Thus, we can say that 2={0,1} is less than 3={0,1,2} because 2∈3. So the ordinal numbers are ordered (hence the name) by the inclusion relation. This wouldn't work if you defined, e.g., 1={0}, 2={1}={{0}}, etc., because now 0∉2, for example. You could potentially come up with another criterion for ordering, however.

If we take "→" to mean "contains as element" then we can write: v0→v1→v2→v3...etc. In a sense this means that v0=ω because v0 has infinitely many predecessors. But at the same time the graph comes full circle in v0 itself so that we would have to say that v0 is also ∅ or 0, the first predecessor!

v0 can't be the empty set, because it's not empty. The point of equating zero with the empty set is that by construction, it doesn't have a predecessor, so therefore it's the first member in the well-ordered set. A well-ordering is only possible if you single out a member that is the first in the set and say "this is the smallest," or "this is the Xest" for whatever property X you want to order.

So, in short, this is my question: Is it possible to design one hyperset that in its cyclical (= recursive?) structure can be seen as constructing or otherwise mirroring the natural number system?

I don't know for sure, but I doubt it. Non-well-founded sets (what you call hypersets) are infinitely recursive, so they have no bottom (or no first element, so to speak). Without a first element, you can't get a well-ordering. Since we generally think of well-ordering as a crucial property of natural numbers, I seriously doubt any system like this would work.

 

[Stoney Pete]

That's a clear answer, thank you!