41–50 of 74 posts
> I would like to understand why numbers looks like leaves 1st of all they don't. The graph doesn't look like pinnatids or palmatids. There is some resemblance of an alternating disposition of leaves, but that's not the shape of the leaf itself but the distribution of them, and it's a stretch. Secondly, I'll take the generous interpretation of the question which is, why the graph looks mathematically like leaves, and…
Earlier quoted context omitted.
It always felt arbitrary to me : next(x)={x} would give 1={0} 2={{0}} 3={{{0}}} Kuratowski's encoding gives : 0=Ø=() 1={Ø}=(0) 2={Ø,{Ø}}=(0,1) 3={Ø,{Ø},{Ø,{Ø}}}=(0,1,2) The cardinal of N is n and every element in N are the predecessors of n. Von Neuman's encoding gives : 0=Ø 1=0U{0}={Ø,{Ø}} 2=1U{1}={Ø,{Ø},{Ø,{Ø}}} Now the cardinal of N is n+1, and n is the maximum of the set N defining n. Both Von Neuman's and Kurato…
For Von Neuman: 1 = Ø U {Ø} = {Ø} 2 = 1 U {1} = {Ø,{Ø}}
I love the imagination here and imagination in general but the framing really stretches it... I think this whole article would be more substantive if it was a little more grounded in the concept of induction. Well and if it would admit very openly that in the sentence "numbers are leaves" the word "are" is about the existence of an isomorphism between the natural numbers and a series of nested sets... but that it's f…
Author here. Wasn't expecting to see this on the front page! I'm really very far from a mathematician and this was a write up of a fun side project. I think the title would be unforgivably misleading in a formal context (if this was a paper claiming any new insights) but really it was a fun side project I wanted to right about. Maybe you read this and learned a little bit about set theory if you had no idea what it w…
You could think about organizing a neural network with layers or nodes that are indexed by Von Neumann ordinals, where the structure of the network follows the natural progression of ordinals. For example:
Each layer or node in the neural network could correspond to a finite ordinal (such as 0, 1, 2, etc.) or transfinite ordinal (like ωω, ω+1ω+1, etc.). The way the network expands and evolves could follow the ordering and progression inherent in the Von Neumann ordinal system.
This could lead to an architecture where early layers (low ordinals) represent simpler, more basic computations (e.g., feature extraction or basic transformations). Later layers (higher ordinals) could correspond to more complex, abstract processing or deeper, more abstract representations.
But I'm afraid there is no hardware substrate upon which to build such a thing.
I do take some issue with intepreting set theory's membership relation in terms of the tree child relation, though.
First, the child relation is presumably transitive, whilst set membership is not. (The subset relation is transitive. Presumably it is 'direct child' relation we have in mind here.)
Second, as seen in the third diagram, nodes don't map well to set entities, because the same entity can be a member of distinct sets, but these would count as distinct nodes on some trees. E.g., in the diagram both leaf nodes are distinct, but they both represent the empty set, and hence should be identical. So the identity of sets is not preserved in the tree encoding.
But this is picky — a lovely read. Thanks author!
I think people will like the following tangent. https://en.m.wikipedia.org/wiki/Benacerraf%27s_identificatio... In the philosophy of mathematics, Benacerraf's identification problem is a philosophical argument developed by Paul Benacerraf against set-theoretic Platonism and published in 1965 in an article entitled "What Numbers Could Not Be". Historically, the work became a significant catalyst in motivating the deve…
The important part about the construction of the natural numbers from axiomatic set theory is that it can be done, not that it brings us closer to the Platonic idea of numbers. It can of course be done in many ways (OP's post lists just two). There's no reason to believe any specific representation within set theory is the true order of the universe, but it is extremely useful and we should be glad it works so well.
Despite several negative comments, I thought the author did a great job of explaining and playing with set theory (which, as can be seen by the response, is good fun.) I do take some issue with intepreting set theory's membership relation in terms of the tree child relation, though. First, the child relation is presumably transitive, whilst set membership is not. (The subset relation is transitive. Presumably it is '…
Honestly sets as trees isn't original. While I was learning about ZFC I came across some lectures[0] by Richard Borcherds which was the seed of insipiration for this project.
In mathematical tree nomenclature, the leaf nodes of the trees in the article are all 0 (represented by the empty set). So only the number zero is truly a leaf. ;)