How to understand Tetration in a common-sense intuitive way that isn't just symbol manipulation?For example, if I have [math]2[/math] pairs of pants and [math]5[/math] shirts, then:* I have [math]2+5=7[/math] total articles of clothing.* I have [math]2*5=10[/math] possible outfits I can wear.* I have [math]2^5 = 32[/math] possible fashion senses of which shirts go best with black pants or khaki pants.* However, what possible physical interpretation would the number [math]^{5}2 = 2^{2^{2^{2^2}}} = 2^{2^{2^{4}}} = 2^{2^{16}} = 2^{65536}[/math] have in this situation?Physical interpretations for inverse functions like super roots or super logarithms or other fast-growing discrete finite functions like described in https://neugierde.github.io/cantors-attic/Parlour are also fair game.Bonus points if the interpretations can fit into the Curry-Howard Correspondence for sum types ("tagged unions"), product types ("pairs"), and exponent types ("functions") generalizing to tetration somehow.
Nobody?Bumping with some physical interpretations of ordinal infinities (those described in the lower attic).https://youtu.be/ZYj4NkeGPdM?t=2265https://youtu.be/CQ4Ap5itTX4?t=62As a refresher, the difference is ordinal infinites are non-commutative, unlike cardinal infinities. I.E.* [math]1 + \infty = \infty = \infty + 1[/math]* [math]1 + \omega = \omega \neq \omega + 1[/math]My personal physical interpretation is deadlines. If you ask me how long until commercial fusion power is available, I could tell a ridiculous answer today e.g. in 100 years; or I could tell you to ask me tomorrow and then give a ridiculous answer tomorrow anyway.
>>16892129Ordinal infinities don't require infinite nesting. Technically, every number is an ordinal infinity, and Shannon entropy kicks in within actually reality (quantum foam, information theory of 6bit beyond 4^3 etc.Same thing in genetics past 4^3 splitting of DNA down to 64codons. Beyond that, noise to signal ratio and fragmentation dominates
>>16890579Why would there be a physical interpretation?
>>16890579maybe its the 6 am but I dont get your example for the 2^5.
>>16892186Abbreviate a person's "Preference" for pants as [math]\mathcal{P}_\text{P}[/math]. Some people might universally "Prefer" black pants ([math]\mathcal{P}_\text{P}=\text{Black}[/math]) and others the opposite ([math]\mathcal{P_\text{P}}=\text{Khaki}[/math]). However a person with a more developed fashion sense might vary their "Preference" depending on what goes best with each shirt, giving a Preference "function" e.g.:[math]\begin{align*}\mathcal{P_\text{P}}(\text{white}) &= \text{Black} \\\mathcal{P_\text{P}}(\text{light grey}) &= \text{Black} \\\mathcal{P_\text{P}}(\text{maroon}) &= \text{Black} \\\mathcal{P_\text{P}}(\text{olive green}) &= \text{Khaki} \\\mathcal{P_\text{P}}(\text{brown}) &= \text{Khaki} \\\end{align*}[/math]For the scenario with 2 pants (disallowing ties) and 5 shirts there are [math]2^5=32[/math] possibilities for [math]\mathcal{P_\text{P}}[/math]. Pic related has a tie where [math]\mathcal{P_\text{P}}(\text{white}) = \text{Black} \cup \text{Khaki}[/math]; in this case allowing ties behaves as a 3rd "virtual pants" option, giving [math]3^5=243[/math] possibilities for [math]\mathcal{P_\text{P}}[/math].Note that exponentation is the first non-commutative operation here. [math]\mathcal{P_\text{P}} \neq \mathcal{P_\text{C}}[/math] There are less ways to prefer among 5 shirts given 2 pants [math]\mathcal{P_\text{C}} = 5^2 = 25[/math].Note also that there are more kinds of possible "ties". If you only allow a single overall tie it behaves as a 6th "virtual shirt" option, giving [math]6^2=36[/math] possibilities for [math]\mathcal{P_\text{C}}[/math]. However if you allow for "sub-ties" e.g. [math]\text{W} \succ \text{LG} \approx \text{M} \succ \text{OG} \approx \text{B}[/math] then https://en.wikipedia.org/wiki/Ordered_Bell_number gives that there are 541 "virtual shirt" options giving [math]541^2=292,681[/math] possibilities for [math]\mathcal{P_\text{C}}[/math].
>>16890579https://en.wikipedia.org/wiki/Von_Neumann_universe#Finite_and_low_cardinality_stages_of_the_hierarchy
>>16892748That's interesting that repeatedly taking the power set of something resembles tetrating by 2's. But how can I generalize that into an interpretation of tetrating by e.g. 3's or 5's?
tetration sounds like a mathematican interacted with a chemist, couldn't recall that titrations were discussed, then assigned his wrong memory to his schizophrenic numerology
>>16892148https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_SciencesI also think it interesting that the Dunbar numbers resemble the Reynolds numbers.
>>16893364Instead of sets, you could use multisets with a limit on the number of times an element can occur.
>>16893365That's funny
>>16890579tl;dr you cannot.You morons are so tiresome with your physical interpretations. Not every piece of math needs to have connection to the physical world.Proper ordinal theory requires the full Axiom of Choice so that there exists a well-order isomorphism between a set and some ordinal. The Axiom of Choice automatically implies there exists subsets of the reals with no Lebesgue measure on them (eg Vitali sets). This leads to the Banach-Tarski “paradox”, which is called such because it’s extremely unphysical. So any mathematical theory that assumes the Axiom of Choice is intrinsically unphysical and it is vanity of all vanities to try and get “physical intuiton” from it. Use your brain instead.Btw the Axiom of Dependent Choice is enough to develop non-pathological real analysis (see Solovay’s model of ZF+DC). But Dependent Choice is not enough to establish a proper well-order isomorphism between every set and the ordinals. In practice this means you never encounter transfinite arithmetic in physics.
>>16895979>tl;dr you cannot.Prove it. Or are you not a mathematician?>You moronsIntroduce me to them; then I'll stop bothering you as I'll have someone more interesting to talk to.>you never encounter transfinite arithmetic in physicsThis is retarded reasoning, on par with "you can't have sqrt(-1) apples therefore you never encounter imaginary numbers in physics". I don't know jack about physics and even I know that capacitors and inductors are modeled with "impedance", the generalization of resistance in Ohms to the complex plane.>Axiom of Choice is intrinsically unphysicalBertrand Russell stated that without AC you can choose one shoe from every pair, but not one sock from every pair.Choice is inherently physical, we make choices every day, and mathematics models choices in Game Theory. I've already linked videos modeling infinite choices earlier in the thread, so here's one on modeling AC called the Determinancy Game: https://youtu.be/Kj5RCs1FHcc?t=648As for the physical interpretation of Banach-Tarski specifically, we already have one: voting. (As an aside, the correspondence between Cantor's Theorem and Arrow's Theorem is yet another fascinating one I wanted to get to in this thread.)Just as in 1D the rearrangement theorem proves you can sum grandi's series to any number you want, in 2D you can jerrymander the districts of a voting population to ensure victory for any party you want.The only limit, of course, is that the smallest "quantum" of a district is a single person, which means the resolution of Banach-Tarski parallels in physics the resolution of Zeno's motion or the Rayleigh–Jeans ultraviolet catastrophe.
>>16897155>I don’t know jack about physics>proceeds to post some pop-sci mumbo-jumbo to “prove” me wrong
>>16897155>Banach-Tarskihttps://en.wikipedia.org/wiki/Cayley_graph
>>16890579There are algorithms where the complexity in worse case is expressed in tetrations (e.g. cut elimination theorems).As for the "axiom of choice" or any other controversial piece of mathematical axiomatization since; every quantity above 10^(10^(10^(10^50))) cannot be realized as a measurement or even a parameter in any conceivable physical experiment, the disagreements about whether AC (or CH; or DC; or AD; or excluded middle; or if "I have the right to have sigma^0_n but not pi^0_{n+1} sentences as axioms"; or "the nonlinear tautology (A -> A -> B) -> (A -> B) makes copies and allow Gödel-type incompleteness hence I'm not allowed to use it") is real consititutes a 100% byzantine debate identical to "angels wear trousers instead of dresses" and by the way the overwhelming of what so-called constructive mathematics call "concrete" isn't for the same reason. The most important feature of maths is consistency (only warranted by empirism because of Gödel) and how rge vast majority of mathematical ideas "eliminate themselves" (through cut-elimination a.k.a. program execution like a C++ program full of templates which ultimately work as organizational ideas but at the very end the program only shifts data made of bits on a ram range) when deriving a concrete statement (of bounded arithmetic).
>>16897257>The most important feature of maths is consistency (only warranted by empirism because of Gödel) Math doesn’t care about empricism. It is a priori by construction. What you likely mean is that axiomatic systems should be “sieved out” via agreement with experiment. But empirical science is inductive by nature and no experiment can yield truth in finite number of measurements.
Think of an x by x grid of points. x^x is the number of paths from left to right through this grid. We will call this the path set.Now, for even height tetrations, we can think of the number of ways of extending each path through more x by x grids.For example, for x^^4 = x^(x^(x^x)), we want to extend each path through another x by x grid. This is not the same as number of paths through an x by 2x grid. Our paths do lie in this grid, but we want to assign to each path through the first grid a path in the second grid. Now, x^^2k is thought of the same way, just by extending through k grids.Now for odds, say 2k + 1, we can think of first doing x^^2k to get our number of ways of going through k grids, and then think of the number of ways of selecting x paths. (Repeats allowed) thus counting the number of “ribbons” of width x.
You should be able to copy [math]\LaTeX[/math] commands from 4chan posts by right-clicking the LaTeX -> "Show Math As" -> "TeX Commands". When posting there should be a TeX preview button in the top left to the left of the "Name" and "Options" fields. You'll have to bracket the commands in a post e.g. [math]\texttt{[math]\LaTeX[/math]}[/math] Extra challenge: How do I make this a proper quine?>>16897517>we want to extend each path through another x by x grid>x^^2kThank you for your idea. However it does not appear to describe [math]^{(2k)}x[/math] but rather [math]^{k}(x^2)[/math]. E.g. for [math]k=2[/math] it does not appear to describe [math]x^{(x^{(x^x)})}[/math] but rather [math](x^x)^{(x^x)} = x^{(x \cdot (x^x))} = x^{(x^{x+1})}[/math]. Likewise for [math]k=3[/math] it does not appear to describe [math]x^{(x^{(x^{(x^{(x^x)})})})}[/math] but rather [math](x^x)^{((x^x)^{(x^x)})} = (x^x)^{(x^{(x \cdot (x^x))})} = (x^x)^{(x^{(x^{x+1})})} = x^{(x \cdot (x^{(x^{x+1})}))} = x^{(x^{((x^{x+1}) + 1)})}[/math]. So it is making some kind of tetration tower with a physical interpretation and I am grateful for that, although it is a complicated one of height [math]k+1[/math] rather than a simple one of height [math]2k[/math].I would be interested in focusing the path metaphor away from homogenous grids onto something more hierarchical. For example I might transition from long-distance freeways to short-distance local roads as I get closer to a destination. I expect neuron connections in the brain follow a similar hierarchy.
>>16897344In some sense math axiom systems are already "sieved out by experiment"(unless the Church Turing thesis is wrong, which would be an extremely disturbing discovery, nothing that cannot rephrased as the halting of a Turing machine is experimentally testable and we can even go lower than that by bounding the number of execution steps). The empirism claim was about non contradiction ("so far", like any other empirism claim), not about what various math axiomatization say.
>>16898050>unless the Church Turing thesis is wrongI thought this paper was interesting: https://arxiv.org/abs/2206.06473 (free, rip SciHub and Anna's Archive)>A Dilemma for Solomonoff Prediction>The framework of Solomonoff prediction assigns prior probability to hypotheses inversely proportional to their Kolmogorov complexity.>There are two well-known problems.>First, the Solomonoff prior is relative to a choice of Universal Turing machine.>Second, the Solomonoff prior is not computable.>However, there are responses to both problems.>Different Solomonoff priors converge with more and more data.>Further, there are computable approximations to the Solomonoff prior.>I argue that there is a tension between these two responses.>This is because computable approximations to Solomonoff prediction do not always converge.
>>16898070>and Anna's Archivetry .li instead of .org
>>16898142>try .li instead of .orgIt worked thank you, the internet lives a little longer. Half the reason I'm interested in type theory is to make it easier to encode meshnets.
>>16898050How is the Axiom of Union sieved out, may I ask?
>>16892148If not then the symbols are just scribbles on paper, a meaningless hallucination of something thought meaningful
>>16898070>>16898236go on their wikipedia page for the newest links
>>16899757>go on their wikipedia page for the newest linksWoah you're right it says .li and .pm and .in; I'm suprised wikipedia is even allowed to do that. Thank goodness for the 3rd world keeping information free.
