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# /sci/ - Science & Math

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Uncountable sets are counter-intuitive and probably don't even exist. And axiom of countable choice makes more sense than full AC.
What is a better property, having all sets Lebesgue-measurable, or having some weird sets too big and too abstract to see, construct and comprehend?
Prove me wrong

 >> Anonymous Thu May 3 07:27:49 2018 No.9715502 >>9715496prove the countable AC "makes more sense". And sure there's uncountable sets. for example C. Uncountability is a fundamental concept in mathematics , and its for a reason.
 >> Anonymous Thu May 3 07:28:34 2018 No.9715503 File: 5 KB, 497x111, 1200px-Aleph0.svg.png [View same] [iqdb] [saucenao] [google] [report] Reminder that we don't know, and can't know, what the cardinality of so-called real numbers is, and saying that cardinality of R is continuum, which is defined as a cardinality of R, is just saying "the answer is whatever the answer is", so it isn't saying anything and is not proper maths.How can you tell R is a valid mathematical object if it doesn't even have well-defined cardinality?
 >> Anonymous Thu May 3 07:35:49 2018 No.9715510 >>9715502Firstly, AC leads to ridiculous results, like well-ordering theorem, Banach-Tarski paradox, existence of non-Lebesgue-measurable sets.Secondly, there's no way to even theoretically choose an element from uncountable family of sets. For sets not greater than $\omega$ it's at least theoretically possible to finish the process in finite time, with uncountable sets it will always take infinitely long if choosing an element takes non-zero amount of time.It's impossible to construct anything constructed by means of AC. We "know" there is well-order on reals, but it's impossible to construct it. We "know" we can cut a ball in five parts and reassemble them into two balls identical to the original one, but we can't possibly know how any of these parts looks like. AC brings nothing to mathematics, it just creates pathological counterexamples
 >> Anonymous Thu May 3 08:10:55 2018 No.9715548 >>9715496>Uncountable sets are counter-intuitiveOnly for brainlets.
 >> Anonymous Thu May 3 08:16:56 2018 No.9715556 >>9715503>How can you tell R is a valid mathematical object if it doesn't even have well-defined cardinality?Why is it not well-defined? What makes it less well-defined than the cardinality of the naturals?
 >> Anonymous Thu May 3 08:17:58 2018 No.9715557 >>9715510>For sets not greater than ω it's at least theoretically possible to finish the process in finite time, with uncountable sets it will always take infinitely long if choosing an element takes non-zero amount of time.Why are you invoking the meaningless notion of "time" into mathematics?
 >> Anonymous Thu May 3 08:18:31 2018 No.9715560 >>9715496How are you so brainlet that you think uncountable sets are counter intuitive and dont exist
 >> Anonymous Thu May 3 08:30:22 2018 No.9715574 >>9715556Are you serious? $\aleph_0$ is defined as sum of all finite ordinals, it's as well-defined a concept as it can get. $\mathfrak{c}$ is defined as $\mathfrak{c}$, which doesn't look like valid math to me.The fact we don't and can't know for what ordinal $\kappa$ $\aleph_{\kappa}=\mathfrak{c}$ means it's not well defined
 >> Anonymous Thu May 3 08:32:18 2018 No.9715577 >>9715574union of all finite cardinals*
 >> Anonymous Thu May 3 08:52:32 2018 No.9715599 >>9715574> ℵ0 is defined as sum of all finite ordinals, it's as well-defined a concept as it can get. That's not how it's defined, but if that's as well-defined as it can get then the definition of c as the cardinality of the reals is as well defined as it can get. Or we can define it as the cardinality of the powerset of the finite ordinals if that's not good enough for you.
 >> Anonymous Thu May 3 08:54:06 2018 No.9715603 >>9715574>The fact we don't and can't know for what ordinal k $\aleph_{\kappa}=\mathfrak{c}$ means it's not well definedWhy?
 >> Anonymous Thu May 3 09:15:55 2018 No.9715628 >>9715599>That's not how it's definedThen how? And if you say "as the cardinality of naturals", they're equivalent.We know what an ordinal number is, we know what a limit ordinal is, and about continuum? We know nothing, we only know that if it exists then it's strictly greater than $\aleph_0$
 >> Anonymous Thu May 3 09:21:07 2018 No.9715636 >>9715496>believes in countably infinite sets but not uncountably infinite onesSo you reject the power set axiom?
 >> Anonymous Thu May 3 09:23:27 2018 No.9715641 With AC I think OP might have a point (in the end it's preference), but I see no reason whatsoever that uncountability is a meme.
 >> Anonymous Thu May 3 09:47:23 2018 No.9715701 >>9715628>Then how? And if you say "as the cardinality of naturals", they're equivalent.They're only equivalent because the set of finite ordinals is countable, which is defined by the naturals.>We know what an ordinal number is, we know what a limit ordinal is, and about continuum? We know nothing, we only know that if it exists then it's strictly greater than ℵ0.This is like saying we know nothing about 0 because dividing by 0 is undefined. Wrong, we know plenty about c since it's the cardinality of the powerset of any countable set.
 >> Anonymous Thu May 3 15:11:14 2018 No.9716194 >>9715503What are you talking about? The cardinality of the reals is Beth-1
 >> Anonymous Thu May 3 15:34:57 2018 No.9716228 >>9715496infinity is a stupid jewish invention made by (((Georg Cantor))). Norman Wildberger is right,infinity doesnt exist at all, and breaks computational precison and needs to be purged from mathematics.
 >> Anonymous Thu May 3 15:44:13 2018 No.9716240 >>9716228>infinity is a stupid jewish inventionso is like half of mathematics>infinity doesnt exist at all, and needs to be purged from mathematicsfeel free to redefine calculus without it anytimealso, OP, please answer >>9715636
 >> Anonymous Thu May 3 15:55:11 2018 No.9716263 >>9716228*Wild(((berger)))
 >> Anonymous Thu May 3 20:09:30 2018 No.9716765 >>9715548So Gauss, Cauchy, and Kronecker were brainlets?
 >> Anonymous Thu May 3 20:14:36 2018 No.9716771 >>9716765>So Gauss, Cauchy, and Kronecker were brainlets?Yes.
 >> Anonymous Thu May 3 21:03:56 2018 No.9716857 File: 428 KB, 563x556, 1515711859443.png [View same] [iqdb] [saucenao] [google] [report] >>9715510>theoretically>theoretically possible>finish the process in finite time>it will always take infinitely long>non-zero amount of timeWhat do you mean?
 >> Anonymous Thu May 3 21:48:41 2018 No.9716946 >>9716240>feel free to redefine calculus without it anytimeAlready been done: https://en.wikipedia.org/wiki/Constructive_analysis
 >> Anonymous Thu May 3 22:04:21 2018 No.9716968 >>9716946That has nothing to do with infinity. Only absolute purists like norman wilberger reject infinity.
 >> Anonymous Fri May 4 07:52:01 2018 No.9717738 >>9715701>They're only equivalent because the set of finite ordinals is countable, which is defined by the naturals.not true>This is like saying we know nothing about 0 because dividing by 0 is undefinednot true either. Assuming AC all "reasonable" cardinalities are alephs, and because it's impossible to know which aleph is equal to continuum it's not a well-defined concept
 >> Anonymous Fri May 4 07:54:43 2018 No.9717742 >>9715636>>9716240Axiom of infinity and power set axiom are independent, so there exist models of set theory in which power set of countably infinite set is at most countably infinite (or empty)
 >> Anonymous Fri May 4 08:01:42 2018 No.9717751 >>9716857It takes some time to examine the set and choose an element from it. So, say we have countable family of sets $A_1,A_2,...$ and say it takes one unit of time to choose an element from $A_1$, half an unit of time to choose an element from $A_2$ and so on, so the process will end in two units of time. Now say we have a family $\left\{A_i\right\}_{i\in I}, |I|>\aleph_0$ and it takes $t_i>0$ units of time to choose an element from $A_i$, but for any choice of strictly positive $t_i$s $\sum_{i\in I}t_i=\infty$, therefore it's impossible to construct a choice function in finite time
 >> Anonymous Fri May 4 08:05:14 2018 No.9717756 >>9716968But wildberger and such rejecting infinity, or saying there's no infinity in their systems doesn't mean it doesn't really exist, right? Just like with ZFC, there are large cardinals which we "know" exist, but their existence can't be proven within ZFC, and same goes for wildberger, you can't prove within his system that infinite sets exist, but they do exist, and are just large cardinals, right?
 >> Anonymous Fri May 4 08:25:14 2018 No.9717777 >>9717751>It takes some time to examine the set>unit of timeSounds like a bunch of gibberish an engineer/CStard would say. Disregarded. >>9717756>there are large cardinals which we "know" exist>doesn't mean it doesn't really exist, right?What kind of existence" are you even referring to? Philosophy and retardation like platonism" belong elsewhere.>you can't prove within his system that infinite sets exist, but they do existIn what sense do they exist" if their existence is neither provable nor postulated (and even outright negated in some cases)? Does AC exist" in ZF since it's consistent with ZF?
 >> Anonymous Fri May 4 08:36:18 2018 No.9717793 >>9717738>not trueIn mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.>Assuming AC all "reasonable" cardinalities are alephs, and because it's impossible to know which aleph is equal to continuum it's not a well-defined conceptYou're just repeating the same fallacy. Any non-redundant axiom is "impossible to know" since it is independent from the other axioms, like the continuum hypothesis. So by your logic all axioms are not well defined.
 >> Anonymous Fri May 4 08:53:38 2018 No.9717823 >>9717738>Assuming AC all "reasonable" cardinalities are alephs, and because it's impossible to know which aleph is equal to continuum it's not a well-defined conceptEither the system is consistent or complete, you can only choose one as your definition of well defined. I would not call a self-contradictory axiomatic system well defined, so I think the choice is clear.
 >> Anonymous Fri May 4 08:55:58 2018 No.9717828 >>9717823>you can only choose one as your definition of well definedBeing well-defined has nothing to do with being consistent or complete.
 >> Anonymous Fri May 4 08:57:49 2018 No.9717833 >>9715510Give me two sets whose cartesian product is empty. I'll wait.
 >> Anonymous Fri May 4 08:59:30 2018 No.9717836 >>9717833Assume the existence of the empty set and take its cartesian product with itself.
 >> Anonymous Fri May 4 09:01:06 2018 No.9717839 >>9717836Now make them both non-empty.
 >> Anonymous Fri May 4 09:03:34 2018 No.9717844 >>9717839What axioms am I allowed to assume?
 >> Anonymous Fri May 4 09:10:06 2018 No.9717858 >>9717844Vanilla ZF.
 >> Anonymous Fri May 4 09:14:36 2018 No.9717869 >>9717833You don't need AC for the finite products
 >> Anonymous Fri May 4 09:21:06 2018 No.9717880 https://en.wikipedia.org/wiki/Kripke%E2%80%93Platek_set_theory>not being purist
 >> Anonymous Fri May 4 09:26:19 2018 No.9717887 >>9717793Construct a sequence of sets {}, {{}}, {{},{{}}},..., call them finite ordinals, take their union and here, you have an infinite ordinal without using the notion of natural numbers. And if you say construction of finite ordinals is just a construction of naturals, it's because naturals are defined to be finite ordinals.
 >> Anonymous Fri May 4 09:26:50 2018 No.9717889 >>9717828Then the continuum hypothesis being impossible to know has nothing to do with being well-defined. Good job.
 >> Anonymous Fri May 4 09:31:18 2018 No.9717902 >>9717880you mean (((Kripke)))–(((Platek))) set theory>puristthat's a weird way of spelling shabbos goy
 >> Anonymous Fri May 4 09:34:21 2018 No.9717909 >>9717858Let $(\mathcal{V}_{ZF},\models_{\diamond})$ be the non-standard forcing model given by the scheme valued sheaf $\mathcal{V}_{ZF}$ which is obtained by localizing the canonical "vanilla ZF" sheaf at the class of non-empty proper classes. The forcing relation is defined in the standard way, but keeping track of the fact that we have adjoined uncountably many non-standard sets $\mathfrak{X}_\kappa$ for every cardinal $\kappa$ with the property that $\mathfrak{X}_\kappa \in_{\text{meta}} \mathfrak{X}_\kappa^\bowtie \models_{ZF} \prod_{\kappa} \mathfrak{X}_\kappa^\bowtie = \varnothing$. Where $\bowtie$ is the imaginary modality operator induced by the canonical topology on $ZF + \mathrm{Con}(ZF) + \neg INF$. It is then a routine verficiation (using the results of Cohen and Chevalley) to see that $\mathfrak{X}_\kappa \in_{meta} \mathfrak{X}_\kappa^\bowtie$ holds in every model of ZF assuming the consistency of ZF.
 >> Anonymous Fri May 4 09:35:15 2018 No.9717912 >>9715496Without AC there exists a non-empty family of pairwise-disjoint inhabited sets $\left\{S_i|i\in I, |I|>\mathfrak{c}\right\}$ such that $\bigcup_{i\in I}S_i=\mathbb{R}$
 >> Anonymous Fri May 4 09:38:38 2018 No.9717924 >>9717889>impossible to knowThis isn't a well-defined notion to begin with. What do you mean by "impossible to know"? Impossible to know from within ZF? How does that make something not well-defined? >>9717912>Without AC there existsAny result of this form holds assuming AC too by basic set theory. See Cohen 1963.
 >> Anonymous Fri May 4 09:48:06 2018 No.9717939 >>9717924What do you mean by that? Assuming AC a set of cardinality $\kappa$ cannot be partitioned into more than $\kappa$ disjoint nonempty sets
 >> Anonymous Fri May 4 09:48:49 2018 No.9717940 >>9717912Proof?
 >> Anonymous Fri May 4 09:52:51 2018 No.9717945 >>9717939Not him, but that doesn't imply what's in your post. It's as retarded as claiming that without AC there provably exists a surjection which has no right inverse. That would immediately contradict the fact that AC is independent of ZF.
 >> Anonymous Fri May 4 10:05:57 2018 No.9717958 >>9717945>That would immediately contradict the fact that AC is independent of ZF.How so?
 >> Anonymous Fri May 4 10:08:22 2018 No.9717961 >>9717958Think.
 >> Anonymous Fri May 4 10:51:08 2018 No.9718028 >>9717924I'm not the one who claims AC isn't "well defined" >>9717738
 >> Anonymous Fri May 4 11:18:06 2018 No.9718083 >>9717909Could you explain the construction at a high level?
 >> Anonymous Fri May 4 19:53:07 2018 No.9719098
 >> Anonymous Fri May 4 19:56:44 2018 No.9719110 File: 50 KB, 488x398, Religion math.png [View same] [iqdb] [saucenao] [google] [report] >>9715496Atheists belong on leddit
 >> Anonymous Sat May 5 07:10:08 2018 No.9719871 >>9719110atheism dismisses countably infinite sets as well
 >> Anonymous Sat May 5 08:11:45 2018 No.9719934 File: 228 KB, 317x451, 1493773103762.png [View same] [iqdb] [saucenao] [google] [report] >>9715496>Uncountable sets are counter-intuitive>too abstract to see, construct and comprehend
 >> Anonymous Sun May 6 02:17:38 2018 No.9721657 >>9715496>Saying counterintuitive=wrong>Is intelligentPick only one.
 >> Anonymous Sun May 6 02:30:37 2018 No.9721676 >>9715496Set theory is a complete joke anyway. Name one concrete application it has in the real world.
 >> Anonymous Sun May 6 02:55:43 2018 No.9721710 File: 743 KB, 1080x1920, 1521525188713.png [View same] [iqdb] [saucenao] [google] [report] >>9715496>Grug dun unnerstun so dun exast
 >> Anonymous Sun May 6 03:04:14 2018 No.9721728 >>9716228>(((Georg Cantor)))>be devout lutheran>brainlets think you're jewish
 >> Anonymous Sun May 6 08:24:18 2018 No.9722126 >>9715496They're only counter-intuitive to you because you're a finite being and can't comprehend them. As someone with uncountably large IQ, I can easily grasp such concepts on an intuitive level.
 >> Anonymous Mon May 7 01:34:07 2018 No.9723898 bump
 >> Anonymous Mon May 7 02:28:52 2018 No.9723958 >>9721710/thread
 >> Anonymous Mon May 7 02:52:02 2018 No.9723974 >>9721710Holy shit that meme is so good
 >> Anonymous Mon May 7 20:38:40 2018 No.9725551 >>9715496bump
>>