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The lattice of sets of natural numbers is rich (2021)
- flobosg 2mo ago(2021)
- munchler 2mo agoWhat a beautiful illustration. It makes intuitive the very abstract concepts discussed in the text. It’s fun to zoom in and browse around the structure.
- michael0church 2mo ago[dead]
- voidmain 2mo agoThe visualization is of the power set, which is uncountable.
- michael0church 2mo ago[dead]
- zaebal 2mo agoTREE(3) is unimaginably small, compared to ω
- zygentoma 2mo agoWell, any natural number is unimaginably small, compared to ω …
- tromp 2mo agoTREE(3) is also unimaginably tiny compared to the normal form size of (λa.aaa(λbλcλdλe.ebbbcde)aaaa)(λfλx.f(fx)) [1]. [1] https://wiki.bbchallenge.org/wiki/Lambda_Calculus#Champions https://wiki.bbchallenge.org/wiki/Lambda_Calculus#Champions
- aeneasmackenzie 2mo agoAll describable or recognizable complexity is part of the subcountable set of computable subsets of N. Higher infinities thus mostly contain fake elements about which nothing can be said, so they don’t feel any bigger.
- __MatrixMan__ 2mo ago"Fake elements," feels right to me. They're allegedly in there but we can't find any of them. It's funny that these elements comprise the majority of the "real" numbers.
- gregw2 2mo agoWhat a great visualization! Now can your favorite LLM make me a similar one for the Real #s?
- stavros 2mo agoWhy can't yours?
- gregw2 2mo agoIt was a joke/tease question. I have yet to be able to create diagrams or visualizations I am happy with with LLMs. I can get a diagram, but tweaking it via prompts is extremely painful. I do think it's a interesting moderately straightforward test case for an LLM that is beyond today's frontier.
- MarkusQ 2mo agoNope.
- deleted 2mo ago[deleted]
- scythmic_waves 2mo ago> The power set lattice (P(N)) of all sets of natural numbers, not to scale, some sets omitted...
- nphardon 2mo agoIf the universe contains a finite amount of information, would that disprove the existence of an infinite set? I.e. if the representation of a number contained more information than the amount of information available in the entire universe.
- benmandrew 2mo agoIt's a very interesting idea; if you want to learn more about it, look up "ultrafinitism".
- amavect 2mo agoNot really. Math uses no physical observation, only axioms. Nothing can "prove" or "disprove" axioms. However, if observation supports the axiomatic theory, then we use the theory for physical prediction. If observation doesn't, then we don't use the theory. Does that count as "disproof"? In practice, infinite sets never exist as enumerations of every element, but as ways to generate more elements along with descriptions for which elements to include. Infinite set theories allow for equivocating a finite description with the infinite enumeration. In contrast, programming languages usually make a distinction between data (always finite) and data generation (possibly infinite). I would think that counts as a "disproof" in a way.
- d4ng 2mo agoThere exist subsets of the natural numbers which are infinite, but which are not finitely definable in first-order arithmetic.
- amavect 2mo agoFor example? And does "first-order arithmetic" mean ZFC?
- d4ng 2mo agoI got this example from an LLM: 1. Fix a formal system S. In the LLM example, it uses first-order arithmetic, but I don't see why we wouldn't be able to use ZFC. 2. Let D be the set of subsets of the natural numbers N which are definable by a finite formula in S. 3. There are countably many finite formulas, so |D| <= |N|. 4. Cantor's theorem says that the size of the power set of N is greater than |N|. 5. Therefore there must be subsets of N which are not definable by a finite formula in S. If you disagree with this, I would be interested to know.
- gregfjohnson 2mo agoJoel Hamkins is my new favorite mathematician :-) There is a beautiful application of the power set of the Naturals to denotational semantics. I assume Prof. Hamkins's book will cover that topic, but it is not mentioned in the linked article. Dana Scott (and apparently Gordon Plotkin independently) came up with a clever way to create a model of the lambda calculus that employs the power set of the Naturals. The problem is that in lambda calculus, the formal language permits every expression to appear in the left-hand slot of the "Function_Application" operator. I.e., every term is simultaneously permitted to be given as an argument to a function, and also to be used as a function. So we have the conundrum of finding some set "S" where every element of S is a function (not a problem so far), BUT, those functions all take elements of S as inputs and produce elements of S as outputs. So we need a set S that is isomorphic to the set of functions "S -> S". Cardinality arguments show that this is not possible: the function space for any non-trivial set has greater cardinality than the set itself. So, Scott and Plotkin devised a "computationally sensible" way to interpret an arbitrary set of integers as a function over sets of integers. By standard encodings, interpret any integer "n" as an ordered pair "<M,u>". Now, again via standard encoding, interpret M as a finite set of integers M_set. The "function" defined by a singleton set { n } applied to some other set Q is: {u} if M_set is a subset of Q, Empty_set otherwise. Then for a set of more than one element, take the union of the outputs of each of the elements applied to Q as above. One can define a topology over the power set of the Naturals, and the above functions turn out to be exactly the continuous functions relative to that topology. The continuous functions so defined have the same cardinality as the power set of the Naturals. An interesting historical side-note: historically, mathematical structures were the starting point, and axiometized formal languages a la Frege, Russell & Whitehead, etc. were built later. In the case of lambda calculus, it was the other way around: the formal language came first, and it was a multi-decade riddle what actual mathematical structure (if any) this formal language actually described.