The famous diagonal argument goes like this. Suppose you had a countable enumeration of some real numbers, then you could use the diagonal algorithm to produce a real number not in your enumeration. So far so good. The fallacy occurs when you then claim that this somehow shows that the set of reals is a bigger infinite than the set of naturals. Clearly, the diagonal argument made in the paragraph above contained no statement about comparing infinities, so it is logically impossible to deduce a statement comparing infinities from the diagonal argument in the above paragraph.You may complain "But not being able to find a countable enumeration is by definition the same as being an uncountable infinity". Well, you can make that definition if you want to but then you would be confusing epistemology for ontology. Just because a countable enumeration of all the reals can't feature in your proofs does not mean the reals are "uncountably large" (whatever that term is really supposed to mean) or that a countable enumeration does not "exist". I'm posting this because I too once believed in uncountable infinities like most simple-minded modern mathematicians, but the realization of this basic flaw in the argument destroyed my faith in set theory and now I think the entire field is built on a fallacy. This might help liberate others out there who are struggling with the unreasonable demands made by the set theory religion. Thanks.
>>16805540The diagonal argument is that you can't list a list of infinite lists into a list without missing part of the list. If you start from the natural observation that an infinite list can't exist in the first place, you've missed the point altogether.
>>16805543Did you read the post? I don't see what this is supposed to be addressing or where I said an infinite list can't exist.
>>16805544If you believe that a complete infinite list can exist, you can't complete an infinite list of your own completed infinite lists.
>>16805540But you do agree that there is no bijective correspondence between 1,2,3,... and all real numbers, right?
>>16805540Disregarding your idiotic "proof"Uncountable vs countable infinities are intuitiveYou are just a midwit for not immediately seeing this in the first placeThe proof just confirm what people with a brain intuitively understand
>>16805545I already said that in the first paragraph of the op though>>16805546It depends on what you mean by "there is no". I can agree that such a bijection can't be written down in ordinary proofs, but that's not the same as saying it doesn't exist.
>>16805548If you think they are intuitive, can you use your intuition to answer whether there are any infinities of size strictly between the reals and the naturals? This is a test of your claimed intuition. If you can't solve it, it means you don't have any intuition at all.
>>16805549>I already said that in the first paragraph of the op thoughSo how's it a fallacy if you said it yourself?
>>16805553Read the paragraphs after the first one.
>>16805554>Clearly, the diagonal argument made in the paragraph above contained no statement about comparing infinitiesIt makes the statement that you can't list a list of infinite lists into a list without missing part of the list. If you start from the natural observation that an infinite list can't exist in the first place, you've missed the point altogether.
>>16805559Okay, now read the third paragraph.
>>16805561>The first thing I wrote was retarded>The second thing I wrote was retarded>PLEASE READ THE THIRD THINGNo, you're a fucking idiot. It's as simple as believing that a complete infinite list can exist. If you believe that it can, you also can't complete an infinite list of your own completed infinite lists.
>>16805571If you didn't even read the post, you're not here to discuss things in good faith, so there's no point in discussing this with you.
>>16805573I give evil people two chances to prove they're not evil. You're quite obviously an evil person.
>>16805574Ok, hope you get better or whatever.
>>16805575>You may complain "But not being able to find a countable enumeration is by definition the same as being an uncountable infinity". Well, you can make that definition if you want to but then you would be confusing epistemology for ontology. Just because a countable enumeration of all the reals can't feature in your proofs does not mean the reals are "uncountably large" (whatever that term is really supposed to mean) or that a countable enumeration does not "exist".
Fake and gay
>>16805578>NIGG-MASTER5000
Fuck you go die in a hole bitch
>>16805582>Indian reply syntax
>>16805540>Clearly, the diagonal argument made in the paragraph above contained no statement about comparing infinities, so it is logically impossible to deduce a statement comparing infinities from the diagonal argument in the above paragraph.What do you think "comparing infinities" means? The only way to compare them is to establish a bijection between two infinite sets, or prove the impossibility of doing so. That's the meaning of comparing cardinalities, and that's what the diagonal argument does (it shows that you can't have a bijection between the naturals and reals).>You may complain "But not being able to find a countable enumeration is by definition the same as being an uncountable infinity". Well, you can make that definition if you want to but then you would be confusing epistemology for ontology. Just because a countable enumeration of all the reals can't feature in your proofs does not mean the reals are "uncountably large" (whatever that term is really supposed to mean) or that a countable enumeration does not "exist". That's not the definition. The definition is about finding bijections.>(whatever that term is really supposed to mean)What it means is very simple, I pointed it out above.
>>16805586>That's not the definition. The definition is about finding bijections.Finding a countable enumeration of R is the same as finding a bijection to N. Not sure why you're saying they're different.
>>16805552>if you think toroid wheels go faster than tetrahedron wheels, give me everty single possible wheel shape between the two
>>16805612>Can't answer or guess the most basic question arising from the theory of uncountable infinities>Claims it's intuitiveEmbarrassing
>no statement about comparing infinitiesit's implicit by virtue of taking things to their limit. there's no assumption here besides the idea that you can in fact form a bijection, yet doing so fails immediately by the intrinsic nature of having the diagonal entry not mapped. it's essentially a proxy for a supremum that will always violate the bijection when taken to the limit.
a finite list can be added onto, to a finite end, eventual, or deterministic in some eventual case.an infinite list cannot be added onto.the diagonal argument is incorrect because it assumes a new number can be created onto an infinite list, thereby increasing the size of the established infinite list; and it gets there by assuming the created number is not already part of the established infinite list, which is a bad assumption, because it already is part of the infinite list, it just wasn't part of the finite list representing a viewable perspective within the infinite list already. For example, Z=[1,2,3,4,5,...] (which is basically incorrect notation by assuming the end bracket ever comes), in the very least displays a finite viewable perspective which relates to the conception of the infinite list of integers Z, showing us "1,2,3,4,5".
>>16805540words words words words wordsyou know someone is a math hack when they make arguments without a single concrete mathematical statement
>>16805643>Argument from illiteracy
>>16805646more words
>>16805646>Argument from muh dikPreach nigger queen
>>16805540The fundamental problem with real numbers is that they include undefinable numbers, which means that MOST real numbers can not be represented by any finite expression. In other words, almost all “real” numbers are completely useless and imperceptible to us humans. It doesn’t even make sense to say they “exist” as there is no proof of them. It’s not even clear that there should be undefinable numbers at all. It could very well be the case that any string of digits after the decimal actually corresponds to some finite expression, but then that would mean all real numbers could be ordered by their lengths, and therefore countable, which would show that the diagonal argument has some other type of error that has gone unnoticed all this time.
>>16805662just because almost all real numbers are not computable doesn't mean you can't reason about them or make statements about their properties
>>16805664Again, it’s not proven that such numbers exist in the first place. And it makes no difference to reason about their properties since it’s ultimately irrelevant. Suppose I have two undefinable numbers a and b. Then does a + b equal b + a? Who cares? It doesn’t affect anything. And why should they be called “real” numbers anyway? Why not given them a different category? Can you prove that they in fact have the properties of the definable reals?
>>16805673in math, you have to formalize your argument so it can be analyzedyou seem incapable of formalizing your reasoning, meaning you don't have a well formed idea and instead are reasoning based on feelingsby trying to reason with your feely wheels, you will bend like a reed whenever presented with anything that demonstrates you are retardedoften times when one tries to formalize their ideas, they realize the limits of their understanding when they hit an impasseseriously, go ahead, try to form a mathematical argument demonstrating your idea, we're waiting
>>16805540>Clearly, the diagonal argument made in the paragraph above contained no statement about comparing infinities, so it is logically impossible to deduce a statement comparing infinities from the diagonal argument in the above paragraph.It's true that the paragraph doesn't use that term. But it does prove that no surjective function from natural numbers to real numbers can exist, where a a surjective function from real numbers to natural numbers does exist. That seems to me like a perfectly reasonable notion of size comparisons between infinite sets (it is a total order, for one thing, and agrees with notions of set size for finite sets). While that doesn't obviously make it *the* notion of size of sets, do you disagree that it is a plausible such candidate? If so, why?
>>16805678I just asked you to prove that undefinable numbers exist and that they have the properties of the definable reals and you refused to do it, because you can’t.
>>16805687it's your thread, sweethearti'm not going to lift a finger for you until you do some work yourself, starting with making the thread topic more concrete
>>16805636>bla-bla diagonal argument assumes bla-bla can be created onto an infinite list bla-lba increasing the size of the list bla-blano>assumes number is not already part of the established infinite listit not assumes, it literally constructs such a number. this number is different from all other numbers by construction.>>16805540we have established bijection. but we made such a real number that it doesn't have enumeration. thus, it's isn't a bijection. if it was, this number has to be in the list, but we are contstructing such a number so it will be different from all numbers in the list. OP, this is math 101 stuff, come on
>>16805540>1-to-1 correspondence, wut?Ngmi, Setless.
>>16805697It purports to construct the number in theory. If you disagree with me I urge you to name any number that satisfies the criterion. Pro tip: you cant.
>>16805540we have this thread every week. OP, you are just a midwit lacking any mathematical talent. go away.
>>16805740>name any number that satisfies the criteriongive me your suppose bijection to natural numbers theni think you don't understand two differences: 1. actual/existential infinity2. potential infinityin this example with diagonal argument it builds on the foundation of classical mathematics, in which we work with actual infinities (actual doesn't mean "physically real"). if you don't agree with that then you must use constructivist foundation. and i don't know well constructivistic mathematics to say what relationship it has with diagonal argument. anyway, i think your problem in diagonal argument lies in this. in the classical sense diagonal argument is OKso stop being some sophist and stop the demagoguery
>>16805746not OP, but I don't think that distinction makes any sense in math, or indeed at all. let's not dirty this thread with philosophizing. we talk about infinite sets, not special quantities or giod forbid numbers called 'infinities'.
>>16805540>Clearly, the diagonal argument made in the paragraph above contained no statement about comparing infinitiesWe compare the cardinality of infinities using bijections.The diagonal argument shows bijection is impossible.Therefore ones must be bigger than the other.
>>16805748no, it makes sense in math.if we speak in intuitionistic tradition (bishop, troelstra, markov's constructivism) then there is this distinction in potential and actual infinity. in constructivistic framework we only give an algorithm for constructing any natural number, but we don't have them all, unlike in classical mathematics where we literally have [math]\mathbb{N}[/math].your computer doesn't have all natural numbers, right?
>>16805540>You may complain "But not being able to find a countable enumeration is by definition the same as being an uncountable infinity". Well, you can make that definition if you want to but then you would be confusing epistemology for ontologylevels of retardation beyond human comprehension.
>>16805682start with n=1 [eqn]\\ a_n^i[b(f_n)] = b[f_{n1}, f_{n2}, f_{n3},\ddots,f_i]\\ a_n^{i}[b(\frac{1}{10^n})] = b[0.1,0.01,0.001,\ddots,\frac{1}{10^i}]\\ a_n^i\big[c(a_z^9[b_n(\frac{z}{10^n})]\big] = c\big[b_1[0.1,0.2,0.3,\ddots,0.9],b_2[0.01,0.02,0.03,\ddots,0.09],\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \enspace b_3[0.001,0.002,0.003,\ddots,0.009],\ddots, \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \enspace b_i[\frac{1}{10^i},\frac{2}{10^i},\frac{3}{10^i},\ddots,\frac{9}{10^i}]\big] =\\ c\big[_{n1}\overset{b1_1}{[0.1]}, \overset{b2_1}{[0.01]}, \overset{b3_1}{[0.001]}, \overset{b4_1}{[0.0001]},\overset{b5_1}{[0.00001]},\overset{b6_1}{[0.000001]}, \overset{bi_1}{\ddots}\\ \enspace _{n2}\overset{b1_2}{[0.2]},[0.02],[0.002],[0.0002],[0.00002],[0.000002], \ddots\\ \enspace _{n3}\overset{b1_3}{[0.3]},[0.03],[0.003],[0.0003],[0.00003],[0.000003], \ddots \\ \enspace _{n...},{\ddots}\\ \enspace _{n9}\overset{b1_9}{[0.9]},[0.09],[0.009],[0.0009],[0.00009],[0.000009], \overset{bi_9}{\ddots} \big] =\\ c\big[ [b1_1 \to b1_9] ,\to [b2_1 \to b2_9] ,\to [b3_1 \to b3_9],\to \ddots \big] \\ \sum_{n=1}^{\infty}c[n]= 4.999... \approx 5\\ \text{which is equivalent to}\\ \sum_{n=1}^\infty \sum_{x=1}^9 \frac{x}{10^n} \approx 5[/eqn]set c contains every possible decimal expansive part required to make any real number between 0 to 1. for example, the number [math] (\pi-3)[/math] is 0.14159..., also [eqn] \overset{+}{c}[b1_1, b2_4, b3_1, b4_5, b5_9, ...] [/eqn].pi is also summable from elements of c, for example:[eqn] \overset{+}{c}[b1_9, b1_8, b1_7, b1_6, b1_1,b2_4,b3_1,b4_5,b5_9,b6_2,...][/eqn]which is trivial because of the ability to construct any integer between 0 to 5 from elements of b1.since it is true that every number following the pattern [eqn]\frac{1\to 9}{10^1 \to 10^{\infty}} [/eqn] appears in this c set, essentially indexable by coordinates [x,y] where x is a number 19 and y is a number > 0
>>16805540rent free
>>16805552>can you use your intuition to answer whether there are any infinities of size strictly between the reals and the naturals?but tell us how you really feel about non-euclidean geometry
>>16805636>an infinite list cannot be added onto.do the naturals include 0?
>>16805697if you refer to the op image, it is using binary (0 and 1) to construct a binary number.s1 [0 0 0 0 0 0 0 0] = all 0's (equivalent to the minimum value)s2 [1 1 1 1 1 1 1 1] = all 1's (equivalent to the maximum value)s3 [0 1 0 1 0 1 0 1] = swapping from 0 to 1 to 0s4 [1 0 1 0 1 0 1 0] = swapping from 1 to 0 to 1s5 [1 1 0 1 0 1 1 0] = 11010 over and over (arbitrary)s6 [0 0 1 1 0 1 1 0] = 011 over and over, after an initial 0 (very arbitrary)s7 [1 0 0 0 1 0 0 0] = 1000 over and over (arbitrary)s8 [0 0 1 1 0 0 1 1] = 0011 over and over (arbitrary)s9 [1 1 0 0 1 1 0 0] = 1100 over and over (arbitrary)10 [1 1 0 1 1 1 0 0] = no determinable pattern given (exceedingly arbitrary)11 [1 1 0 1 0 1 0 0] = 11, 01,01,001,001 (arbitrary)the point is, the list is not meaningful. the additional lists are seemingly arbitrary, which means an arbitrary list could ever exist containing s = [1 0 1 1 1 0 1 0]s1 [0 0 0 0 0 0 0 0 0 0] = all zeross2 [1 1 1 1 1 1 1 1 1 1] = all oness3 [0 1 0 1 0 1 0 1 0 1] = 01 alternatings4 [1 0 1 0 1 0 1 0 1 0] = 10 alternatings5 [1 1 0 1 1 0 1 1 0 1] = 110 alternatings6 [0 0 1 0 0 1 0 0 1 0] = 001 alternatings7 [1 1 1 0 1 1 1 0 1 1] = 1110 alternatings8 [0 0 0 1 0 0 0 1 0 0] = 0001 alternatings9 [1 1 1 1 0 0 0 0 1 1] = 11110000 alternating10 [0 0 0 0 1 1 1 1 0 0] = 00001111 alternatings= [1 0 1 1 0 0 0 0 0 1] = could be another number added onto the list, or11 [1 1 1 1 1 0 0 0 0 0] = 1111100000 alternating, could also be added onto the list, or12 [0 0 0 0 0 1 1 1 1 1] = 0000011111 alternating, could also be added onto the listyou can add things onto the ends of lists, and then the list increases in size.a list of numbers, for example the integers, which affirms to try to be 'all' the integers, would be an ever-increasing list, and it's size infinite, as [for every n, n+1].there is no meaning to the op image for determining a new number that did not exist.
>>16805767oh wow, latex
>>16805777>you can add things onto the ends of listsyou do realize that the diagonal argument produces a number NOT ON THE LIST, right?, please do point at any mathematicians explanation of the diagonal argument right at the place in why they say "add things onto the ends of lists" or whatever the hell you niggerbabble could be understood as, thank you
>>16805780you have been diseasing this website for a long time. commit sudoku, retard-chan.
>>16805767what's the point of this post even? a lot of letters but no coherent conclusion
>>16805682>But it does prove that no surjective function from natural numbers to real numbers can existDoes it really? I think it only proves that there is no surjective function from N to R that you can write down. To give an analogy, Godel proves using a diagonal argument that there are statements of arithmetic which can be proved neither true nor false in some given proof system, but this is not the same as saying that truth values for those statements do not exist. I suspect something similar is happening here - what the diagonal argument is is a statement on the limitation on the sorts of functions that can appear in your proof system, rather than a statement about the size of infinities.Actually, if you go by this >>16805662 anon's suggestion, then the real numbers are, in some sense, countable since there will only be countably many finite expressions. So a bijection might exist, but you won't be able to write it down.>and agrees with notions of set size for finite setsThere are also places where it disagrees with finite sets. You can have a sets which have the same size as proper subsets according to this definition. Is the set of even numbers really the same size as the set of all naturals? It's obvious that the former is smaller than the latter.
>>16805794>It's obvious that the former is smaller than the latterprove to us, that the set of even numbers is smaller than the set of natural numbers. you know what will happen? you will not prove this. why? because there is a bijection between even numbers and natural ones: [math] f(n) = 2n, f^{-1}(m) = \frac{m}{2}[/math].>the limitation on the sorts of functions that can appear in your proof systemyou are telling us that some "functions" can't appear in our "proof system" because of this? do you even understand what do you mean by a "proof system", big brain? my fucking god it feels like anon learned about naive set theory and doesn't understand it's paradoxes and inconsistencies
>>16805811>prove to us, that the set of even numbers is smaller than the set of natural numbersIs the set {1, 2} smaller than the set {1, 2, 3}? Obviously yes because the former is a subset of the latter. Is the set {2, 4, 6, ...} smaller than {1, 2, 3, 4, 5, 6, ....}? Obviously yes because the former is a subset of the latter. >you are telling us that some "functions" can't appear in our "proof system" because of this?Yes, I am particularly talking about functions which exhibit a bijection from N to R. Those functions might exist, but they just won't appear in your proof system. This seems to be a much more reasonable conclusion to draw from the diagonal argument than the claim that, suddenly, this finite proof somehow shows the existence of uncountably many real numbers.
>>16805818>bla-bla thos functions might exist bla-blayou didn't answer the question about "proof system". i think you just have some kind of inner "inuition" about this "proof system", and this "intuition" is just wrong>subset means smallerwhen the sets are finite, then yes (proper subsets will be "smaller"). when the sets are infinite - no. just for you dumb-dumb, let's speak about even numbers. can i just use sign "1" to denote two? yes, it's just a fucking letter. can i denote four as "2"? the same logic. in the end the set of even numbers will look like [math]\{ 1, 2, 3, 4, 5, ... \}[/math]. and what it looks like? like fucking set of naturals.>b-b-but it's just letters!yes, it's fucking letters. the same logic can be used for naturals: we can denote them anyway we fucking want. for example, one as 2, two as 4, three as 6, and so on.
>>16805818>Those functions might exist, but they just won't appear in your proof systemThe mathematics we use is built on top of a set of axioms. We use those axioms to prove theorems.The real and natural numbers, as defined in our axiomatic system, cannot have a bijection, which is what the diagonal argument proves.This means that in our mathematical system, such a function cannot exist, which is the whole point of the theorem.You are free to create your own mathematical system and define the concept of natural numbers, real numbers, and bijections in a way that allows for them to exist.Point is: your system will be useless
>>16805830It looks like you're taking a purely formalist approach to mathematics. That's possible, but then you're not the target audience for my post. My post is intended for people who think there really are real numbers out there and they really are uncountably larger than the natural numbers. I think the real numbers exist too, but I just don't believe it's been shown that there are uncountably many of them.>>16805829>when the sets are finite, then yes (proper subsets will be "smaller"). when the sets are infinite - no.Under your definition of "smaller", sure. But you haven't convinced me that I should use your definition, since I already provided a very reasonable sense of "smaller" for which the even numbers are smaller than natural numbers.>. the same logic can be used for naturals: we can denote them anyway we fucking wantOkay, but I'm not talking about how you "denote" them. I'm talking about the actual sets themselves. It's like if I said 1 + 1 = 2 and you said "umm ackshually you can denote 1 by 2 and 2 by 3 and + by -, so you should ackshally say 2 - 2 = 3"
>>16805794>I think it only proves that there is no surjective function from N to R that you can write down.>the real numbers are, in some sense, countable since there will only be countably many finite expressions.You got that the wrong way around. The real numbers *that you can write down* are indeed countable. The set of all real numbers is not. The diagonal argument proves this, as it makes no assumption about being able to write down a real number; it only assumes the existence of a surjection.(Technically, the version you depict in OP shows the uncountability of the sequences of bits, not the real numbers. Applying that to the real numbers is an extra step. But not a difficult one, and the argument can indeed be adapted that way.)>There are also places where it disagrees with finite sets.Oh? Can you give an example?>You can have a sets which have the same size as proper subsets according to this definition.Not finite ones.>Is the set of even numbers really the same size as the set of all naturals?You will note that those sets are not in fact finite.
>>16805836dumb-dumb, let's use the evens that i denoted that way. what will be two plus two? four, right. 1 + 1 = 2, and just to remind you: '1' is two, '2' is four. coincidence? no, it works for ANY number. six plus eight = fourteen. 3 + 4 = 7, what is 7? fourteen. even numbers are isomorphic as a semigroup to naturals. so addition works literally like on naturals.>i'm not talking about how you "denote" thembijections are like denoting elements of a set differently (using a set bijective to it). this bijections doesn't mean isomorphism (conservation of operations)
>>16805839>it only assumes the existence of a surjection.Not quite. It assumes the existence of a surjection and it also assumes that you can treat that surjection like any other function and manipulate it as needed by the diagonal argument. It is only the conjunction of these premises that let you derive a contradiction and the diagonal argument doesn't itself tell you which of those premises you must discard.>Not finite ones.Well, you said it was reasonable since it agrees with corresponding statements for finite sets sometimes and I pointed out that it also disagrees with statements for finite sets sometimes. If P is the statement "If A is a proper subset of B, then there is no bijection from A to B", then it is true for finite sets but false for infinite sets.
>>16805836>My post is intended for people who think there really are real numbers out there and they really are uncountably larger than the natural numbersdude, nobody is the fucking target of your post then
>>16805847I don't think so. There are plenty of people who think the real numbers really are uncountably large.
>>16805852That's not what you just wrote
>>16805857? Explain.
>>16805858>My post is intended for people who think there really are real numbers out there AND they really are uncountably larger than the natural numbers
>>16805860The second statement already presumes the first anyway. How do you believe that the real numbers are really uncountably infinite and that they don't really exist?
>start with an undefinable infinite list containing undefinable infinite decimal strings and you will get an undefinable infinite decimal that’s different from all of them!woah..
>>16805864Correct. You're apparently one of the rare alpha retards who can formulate how the argument works into a valid sentence while still being so retarded that you have to add a sarcastic element proclaiming your own retardation.
>>16805861Because math is made up? Are you dense?
>>16805868If you think the real numbers are made up, this thread is not for you like I already said.
>>16805871Nobody thinks real numbers exist "in the wild" as physical things. Your thread is about nothing
>>16805874Nobody mentioned anything about "physical things" other than you. If you believe the real numbers really exist, then you believe that they are really uncountably large or you don't. If you don't believe in real numbers, you are free to ignore the thread.
>>16805876>Nobody mentioned anything about "physical things">>My post is intended for people who think there really are real numbers OUT THEREout there where? in the real physical world. you don't even know what you wrote 5 minutes ago
>>16805878>in the real physical worldNice addition you just made which wasn't in my post at all. Now can you stop derailing the thread, please?
>>16805880what does out there mean then, come on make up some excuse about how it means "out there in the magical realm of mathematics"
>>16805883Like I'm saying for the third and final time, if you don't believe in the real numbers, you are free to ignore the thread.
>>16805884You have proven you are incapable of clearly expressing yourself even in simple english. You got exposed and now you're deflecting.If real numbers don't exist in the real world, then the diagonal argument proves that the reals are bigger than the naturals within the axioms we all use for our mathematics.If real numbers exist in the real world then you first have to tell me to which physical objects they correspond before any discussion could be had
>>16805886>the diagonal argument proves that the reals are bigger than the naturals within the axioms we all use for our mathematics.No, read the OP.
>>16805888>no rebuttalLMAO>read the OPnothing in the OP rebukes what I wrote>Clearly, the diagonal argument made in the paragraph above contained no statement about comparing infinitiesthis statement is already wrong
>>16805888>contained no statement about comparing infinitiesNTA, the diagonal argument literally says that all the indexed s are complete and infinite and that the index is complete and infinite. You're apparently one of the beta retards.
>>16805889Apart from addressing it in the OP, I've already addressed it in several posts. You think being able or unable to find bijections is the same as comparing sizes and I am saying that facts about sizes of sets are or should be independent of being able to find bijections in your axiomatic systems.
>>16805890Reread the OP or see >>16805892.
>>16805892>facts about sizes of sets are or should be independent of being able to find bijections in your axiomatic systems1) Why?2) What alternative that works better or makes more sense do you propose?
>>16805893>contained no statement about comparing infinitiesThis is from your own OP. You can either try to rewrite it or be wrong forever. As it stands, you're simply wrong.
>>16805894Because a fact is true or false regardless of the axioms you choose.I'm inclined to think all sets are countable but I have not yet fully developed a system around this guess and I wouldn't post it here if I did.
>>16805896>contained no statement about comparing infinitiesAnd this "fact" is false.
>>16805895It's factually correct. The first paragraph just doesn't contain the words like "comparing infinities" or anything like that.
>>16805896>Because a fact is true or false regardless of the axioms you choose."Parallel lines never intersect" is a true fact in the axioms of Euclidean geometry and a false fact in the axioms of spherical geometry, so what you said is demonstrably wrong
>>16805906If they never intersect then they were never parallel to begin with. Chew on that one, nerd.
>>16805899>The first paragraphOf what? Are you like a gamma retard or something? This is the diagonal argument paper.https://www.digizeitschriften.de/id/37721857X_0001%7Clog29?tify=%7B%22pages%22%3A%5B83%5D%2C%22view%22%3A%22info%22%7D#navi
>>16805906The axioms are about different things here (Euclidean vs spherical), so your example is not relevant.
>>16805909That's not what parallel means
>>16805910Seriously, why bother posting if you didn't even read the OP? I mean the first paragraph of the OP.
>>16805912>the axioms are about different thingsif you change the axioms you get a different thing, exactly. which is why a fact can be true under certain axioms and false under others, thus your previous statement us false
>>16805913You might say that they seem parallel but at no section on the sphere are they truly parallel. They only approximate being parallel starting at a certain point, but points can’t be parallel. As soon as you add another point onto the lines, they cease to be parallel.
>>16805915You can't point out a fallacy in someone else's paper by pointing out a fallact your own retarded cliff notes about it. Fucking lol.
>>16805920No, then those would just be facts about different things. 1 + 1 = 2 is true regardless of what axioms you use.
>>16805923I tried to engage with you in good faith but you seem to be genuinely mentally challenged, sorry.
>>16805922What is the definition of parallel, then?
>>16805926You said your own "first paragraph just doesn't contain the words like "comparing infinities" or anything like that." So? Cantor is dead. You're not Cantor, you retard.
>>16805925I can easily create an axiomatic system where 1 + 1 = 1, so again you are wrong
>>16805931No, you fucking cannot, not unless you think all mathematics is nonsense.
>>16805927If I select any point on line A and find the shortest distance to a point on line B, the distance of this line should be the same regardless of which point I start with. Which is to say that the distance between the lines never changes, and that they never intersect.
>>16805933Why would an axiomatic system where 1 + 1 = 1 be nonsense?
>>16805937Because 1 + 1 = 2 and 1 is not equal 2.
such a retard thread at this point
>>16805933in trivial group this is true, braindead
>>16805939Why is 1 + 1 = 2 better than 1 + 1 = 1?
>>16805942>>16805943If you cannot agree to such basic things like 1 + 1 = 2, this thread is not for you. You should take up your issues with other mathematicians. Sorry, I won't discuss this any further. You are just derailing my thread.
>>16805940Of course it's a retard thread. The only legit criticism of the diagonal argument is to agree with it sarcastically, like this alpha retard did >>16805864
>>16805949>your axioms are nonsense because they don't agree with my axiomsok? something more sustantial? I could say your axioms are nonsense just as well
>>16805949Are you implying that 1 + 1 = 2 is true because some mathematicians say so?
TLDR:>OP believes mathematics should follow how the real world works and human intuition, but he cannot admit it ITT because then everyone would leave laughing
>>16805969Infinity doesn’t exist within human intuition, it’s more of a divine concept.It’s more defined by what it’s not, and types of infinities are also defined similarly.What op is missing is that we’ve already done something unintuitive which is begin in a space we dont fully comprehend or understand, and we applied a test that failed meaning our assumption the test works are wrong which is why it implies the existence of uncountable setsIf we started with a more intuitive space like a countable set we wouldn’t really progress our knowledgeThat’s a big deal, and frankly I think it demonstrates the fragility of intuitionism. Of you lack understanding your set of truth is smaller than the real set of truthsPeople’s personal gnosis will always be limited by genetics and educationAnd they will always push against you, but formalism also kills them because it demonstrates truth without understandingSo even a machine could be convinced about a truth. But machines have no souls so there is no personal development in formalism it’s mostly just for sanitySo in general intuition developed from a divine level like the pedagogical figure of platonism works better since it begins sight from above the ground rather than beneath it, but you’ll need both to transverse towards knowledge andunderstanding.it’s like this: I seriously doubt a god would understand what it’s like to be a worm, a man strives to know of the worm and god equally, the god and the worm seek to know nothing equallyThus man is not a slave like gods or worms, he is born freeThis is why I’m highly against excessive abstraction that claims to unify all issues, as well as excessive practice which claims to increment knowledgeBoth are slaves to the knowledge of nothingwhen I already know nothing, so that I might seek to know somethingNot slave nor god slave, I am the king of my gardenTL;DR: You’re burning in hell for being retard, meanwhile I am in heaven for being based
>>16806121T. 30IQ btw, so even a retard like me can be free and not enslaved, what’s your excuse OP?Here’s how I think about it. Sophia is Athena, and she has many virgin daughters in touchable by men, and they have many daughters of lower rank, some touchable. The daughters of math are my concubines, meanwhile you are a concubine-less faggot.
>>16806121>divine conceptstop reading after that
>>16805931Would your proposed axiomatic system be able to prove "if x + y = x + z then y = z"?
>>16806132in trivial group this works, dumbass. starting reading your fucking books, anon
>>16806128My iq is sub 30 and I read the Bible. Problem with that?Sorry you can’t into figure and will remain a sup 100 iq science bitch made faggot.https://youtu.be/X_jmEOGvriwHere’s the proof using the Bible, imagine god had an infinite name. Take the set of all names of god and enumerate it, there is a name of god distinct that exists via construction not in the enumeration.SOIENCE BITCH BTFO’D BY A SUB30 IQ NEOPLATONIST CHRISTIAN. Common science bitch L
>>16806124>>16806131>>16806140No one cares about your shitty IQ foid
>>16806143So uhhh are you just going to keep being off topic or actually try to refute me?Maybe >>>/lit/ is more your speed
>>16806159>Maybe >>>/lit/ is more your speedsays the anon that does all his "math" in paragraphs
>>16806132>Would your proposed axiomatic system be able to prove "if x + y = x + z then y = z"?It depends, but how is it relevant?
Who the fuck are these retards and why are they derailing my thread?
>>16806163Because if you're trying to axiomatically treat addition then it should, but then you can easily get 0 = 1 if 1 + 1 = 1.Of course if your point is that you can make *some* axiomatic system where 1 + 1 = 1 as long as + (and 1 and =) are allowed to mean whatever you want, then yeah but that's not an interesting observation
>>16806171because your thread is stupid
>>16806175Fuck this place. I'm never posting here again.
>>16805540its impossible to actually finish this proof fyi
>>16806180small dick energy
>>16806171>Indian reply syntax
>The famous diagonal argument goes like this.>Suppose you had a countable enumeration of some real numberswrong and facetious to make the argument seem weak>The fallacy occurs when you then claim that this somehow shows that the set of reals is a bigger infinite than the set of naturals. also wrong. mathematical induction is not a claim it's a logical extension of the proof on the base case.>it is logically impossible to deduce a statement comparing infinities from the diagonal argument in the above paragraph.see above. you are struggling with this because you don't understand the laws of truth.>You may complainfabrication to discredit>Well, you can make that definitionfabrication to discredit>Just because a countable enumeration of all the reals can't feature in your proofs fabrication to discredit>I'm posting this because I too once believedfabrication to appeal>I think the entire field is built on a fallacy.irrelevant>This might help liberate others out therefabrication to discreditlow quality reasoning levels coupled with malicious layering of false pretenses to frame yourself as dealing with a weaker version of the argument. a juvenile toolkit that's neither intriguing nor valuable.
>>16805794By completeness theorem any statement that is true in every model has to have a proof. If your "proof system" leaves behind all this biyective functions out there, aka there is no proof of any of their properties, then that means there has to be one model in which their existence is true and also one in which its false since your "proof system" doesn't speak about them and you could add them to your axioms. Build a model in which cardinality of natural numbers is the cardinality of R (there exists a biyective function f:N->R) and then prove that given T consistent then T + [N]=[R] is also consistent
>>16805794>I think it only proves that there is no surjective function from N to R that you can write downWhere in the diagonal argument is it assumed that the function is "written down"?
>>16805552>can you use your intuition to answer whether there are any infinities of size strictly between the reals and the naturalsYes, there are.
>>16806121>the fragility of intuitionismi don't like the attitude of a lot of intuitionists, but i don't think that what they are about applies to what you are writing, it is a really piss poor name that they chose, so that's where the confusion might have arisen, other than that yeah you are right
>>16806275my dude dropped gold in a pile of shit. good job
>>16805540The diagonal argument is much more informal than Cantor's Theorem. Any questions of its rigor are irrelevant.
>>16806593Search for lawvere theorem. Even though you may label diagonal arguments as "informal" it allows you to see many theorems as tentacles of the same lovecraftian being that was out there all this time just in the same way that everything was hodge star operator all this time. The core cause of godel is the same as turing, tarski etc: a problem that arises when a system tries to talk about itself, manifested as a fixed point theorem and caused by a difference in size between the set and all the possible truth values
>>16805751talk to me when lim {n->actual infinity} a_n starts differing from lim {n->potential infinity} a_n. until then it's philosophical masturbation.>your computer doesn't have all natural numbers, right?Inductive nat : Set :=O : nat| S: nat -> nat.
don't they kind of get less important though?
why do all these threads turn out to be troll threads?
>>16806870oh yes, it's literally storing every natural number, great job, dumbassyou do understand that limits in intuitionistic/constructivistic analysis are different? i, not as fucking arrogant like you, don't know how they define limits with their notion of infinity
>>16805540>so it is logically impossible to deduce a statement comparing infinities from the diagonal argument in the above paragraphNot really, the diagonals + the linears is always going to be a larger set than the linear values themselves when you have established that there are in fact unique values that appear in the diagonals that cannot appear in the originals.
op just wanted to say there are not multiple infinities or varying sizes of multiple infinities.there is just the one.
>>16805689>'m not going to lift a fingerClearly a liar since you had to lift your fingers dozens of times to make that retarded post.
>>16807569We all know what retard OP said. The small problem is that he just claimed it's true without proof, meanwhile a proof of the contrary exists in the maths we all use
>>16805740>I urge you to name any number that satisfies the criterion.It can not only be named through enumeration, but there is an algorithm to generate infinite numbers that satisfy the criterion.
>>16806121>infinity ooga boogashut the fuck up, inf has a definitionInfinity is an unbounded quantity greater than every real number.
>>16806953can't people be retarded without being trolls?
>>16807649well then, he ain't going to curl a dumbell, is that better?
>>16807671mind explaining your pic?
>>16807805OC is a numberonce A & B touch you are at infinity
