Say you have a 1 in by 1 in area of paper and a printing resolution of 300 DPI then there are 300*300 total dots. If the printer is monochrome and each dot is either black or white then there are 2^(300^2) possible symbols.
> If these sets are restricted to be measurable
measurable sets are not Banach-Tarski-able :)
(Anyone know what I'm talking about? I assume it was obsoleted by digital solutions, but I'd like to know if there were any serious flaws in the design).
Ah! Found this: https://www.ripcorddesigns.com/blog-news/who-needs-rastas-an... - there's even a patent! I was searching on space-filling curve, not Hilbert curve.
Perhaps the sense of "I" can be that base, which then allows the expansion of I, I, I (sense of I around I) which expands to in front, behind, etc.
:-D
We can play a lot of games with "what is a symbol", but compactness pervades many of the models that we use to describe reality. The crux of the argument is not necessarily that the symbols themselves are compact as sets, but that the *space of possible descriptions* is compact. In the article, the space of descriptions is (compact) subsets of a (compact) two-dimensional space, which (delightfully) is compact in the appropriate topology.
In your example, the symbols themselves could instead be modeled as a function f:[0,1]^2 -> [0,1] which are "upper semicontinuous", which when appropriately topologized is seen to be compact; in particular, every infinite sequence must have a subsequence that converges to another upper semicontinuous function.
Much of the fun here comes from the Tychonoff theorem, which says that arbitrary products of compact spaces is compact. Since the *measurement space* is compact, the topology of the domain is not as important, as long as the product topology on the function space is the appropriate one. (Mystically, it almost always is.)
My topology is rusty and I don't genuinely doubt the validity of the argument, but I'm having fun trying to poke holes.
<wrong>The idea is that you should measure the amount of black and white ink to change a symbol into another simbol, and if the total amount of ink is less than ε then they indistinguishable. (Where ε is some constant you must choose for the whole system.)</wrong>
I think that every horrible-totally-pathological ink splash has a nice indistinguishable version, but my real analysts is a bit rusty, and there are a few horrible things that I may have forgotten.
Edit: see comment below.
And the article assume that the sets are compact, so they are measurable as you say. Anyway compact sets can be quite pathological (but not as pathological as non measurable sets).
If we're looking for theoretical ways around that constraint, you could allow inks that change over time, so you would have to observe each character for a certain amount of time in order to see what it was representing as its ink changed (or didn't).
https://en.wikipedia.org/wiki/New_riddle_of_induction
The "New Riddle of Induction," proposed by Nelson Goodman in 1955, challenges our understanding of inductive reasoning and the formation of scientific hypotheses. Goodman introduced the concept through his famous "grue" example. Imagine a property "grue," defined as "green up to time t, and blue thereafter." All emeralds examined before time t are both green and grue. The riddle asks: Why is it rational to predict that future emeralds will be green rather than grue? This paradox highlights the problem of choosing between competing hypotheses that equally fit past observations. It questions our basis for preferring "natural" predicates (like green) over "artificial" ones (like grue) in inductive reasoning. Goodman's riddle challenges the idea that induction is based solely on observed regularities. It suggests that our choice of predicates in forming hypotheses is influenced by factors beyond mere observation, such as simplicity, familiarity, or projectibility. The New Riddle of Induction has significant implications for philosophy of science, epistemology, and artificial intelligence. It raises questions about the foundations of scientific prediction, the nature of natural kinds, and the role of language in shaping our understanding of the world.
Related :
https://x.com/eshear/status/1812926436623413285
There are crystal structures that simply won't form anymore, even though they did a few decades ago. Real life Ice-9? Could make an incredible sci-fi book.
The argument from pixels is more natural to us these days, that's why we think it's simpler.
(I had similar reactions to many other older style proofs when I was studying math.)
If we have 300 dpi, we can still code unlimited amounts of information if the color space is continuous, and noise-free.
(The article mentions the human eye limitations, but we could use technological instrumentation to extract info from a print; the human eye doesn't limit what we can code on a BluRay disc, e.g.)
What if you used aperiodic Penrose tiles and could detect intensity? Would it be possible to encode anything in that? There would be no repetition but when would you hit limits of discernability with all the overwriting?
https://www.hypebot.com/hypebot/2020/02/every-possible-melod...
[1] If you want a look at what "Pyramids on Mars"-style musical analysis is like, Erno Lendvai's book about Bartok is pretty awesome. Since Bartok never really wrote or said anything about his compositional methods it's very hard to prove anything conclusively.