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# /sci/ - Science & Math

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If the Dedekind cut is the only rigorous definition of a real number as a cut in the real number line, then how were Riemann and them able to carry out rigorous analyses with non-rigorous numbers? How is the Riemann tensor, the Cauchy-Riemann equations, or the Riemann sphere when it is based on shit definitions of real numbers that even a kindergartner can see the guy is crank viXra viXra viXra?
>brontosaurus

 >> Anonymous Wed Nov 7 00:14:21 2018 No.10123574 Dedekind cuts are a undergrad-tier thing, Jon. Reals are defined as limits of series; open up your baby Rudin.
 >> Anonymous Wed Nov 7 00:16:52 2018 No.10123578 That's not the only construction of a real number that can be used. You can also construct the real numbers as equivalence classes of cauchy sequences.
 >> The Landlord Wed Nov 7 00:56:07 2018 No.10123652 File: 83 KB, 857x376, TIMESAND___762++145rhwrss3ggege4y5788rtetyhinr48641.png [View same] [iqdb] [saucenao] [google] [report] >>10123574>open up your baby Rudin.What year was Rudin born? When did he first define reals as limit of a series? Was it while Riemann was still alive?>>10123578there is an even better way, pic related
 >> Anonymous Wed Nov 7 01:08:15 2018 No.10123667 File: 459 KB, 956x960, 1537776069641.png [View same] [iqdb] [saucenao] [google] [report] >>10123652>a real number is a real number
 >> Anonymous Wed Nov 7 01:08:28 2018 No.10123668 >>10123578There's literally nothing wrong with the decimal construction of the reals.
 >> Anonymous Wed Nov 7 02:52:08 2018 No.10123817 File: 4 KB, 100x100, njw.jpg [View same] [iqdb] [saucenao] [google] [report]
 >> Anonymous Wed Nov 7 04:11:09 2018 No.10123891 >>101235541. Another way to define the real numbers is as the final coalgebra of the functor X · ω, ordinal product with ω.Pavlović, D., & Pratt, V. (1999). On coalgebra of real numbers. Electronic Notes in Theoretical Computer Science, 19, 103-117.and Freyd, P. (2008). Algebraic real analysis. Theory and Applications of Categories, 20(10), 215-306.2. You could also represent them by the eudoxus reals which is the quotient group S/B where S is the set of all "almost homomorphisms" $\mathbb{Z} \to \mathbb{Z}$, and B is the set of functions $\mathbb{Z} \to \mathbb{Z}$ whose range is bounded. Where a function $f$ from $\mathbb{Z}$ to $\mathbb{Z}$ is said to be an almost homormophism iff the function $d_f$ from [/math] \mathbb{Z} \times \mathbb{Z}[/math] to $\mathbb{Z}$ defined by $d_f(p,q) = f(p+q) - f(p) - fq$ has bounded range, i.e., for some integer $C, |d_f(p,q)| < C$ for all $p, q \in \mathbb{Z}$Here: Arthan, R. D. (2004). The Eudoxus real numbers. arXiv preprint math/0405454.3. Construction of Real numbers as infinite decimals. That is: $\mathbb{R} = \mathbb{D}/\sim \; = \{\{0\}\} \cup \{\{d\} \; | \; d \in \mathbb{D}\setminus (\mathbb{T}_0 \cup \mathbb{T}_9)\} \; \cup \; \{\{c, d\} \; | \; c < d \; is \; a \; jump\}$where $d = \pm a_k a_{k-1} a_{k-2}\dots$ $\mathbb{T}_9$ stands for decimals ending with infinitely many 9's and $\mathbb{T}_0$ for decimals ending with infinitely many 0'sAnd a jump goes from $\mathbb{T}_9$ to $\mathbb{T}_0$ or from $\mathbb{T}_9$ to $\mathbb{T}_0$ for decimals that are sufficiently close.Klazar, M. (2009). Real numbers as infinite decimals and irrationality of $\sqrt {2}$. arXiv preprint arXiv:0910.5870.
 >> Anonymous Wed Nov 7 05:16:44 2018 No.10123956 >>101238914. Or also a positive real number is an equivalence class of aggregates which have finite value. Where an aggregate a has finite value if there exists an aggregate $b$, containing finitely many elements, such that $a \leq b$. Where aggregates are collections of integer reciprocals (that are not evaluated).Tweddle, J. C. (2011). Weierstrass’s construction of the irrational numbers. Mathematische Semesterberichte, 58(1), 47-58.5. The abstract algebraic synthetic approach as a Dedekind-complete compatibly linearly ordered field (not exactly the same as dedekind cuts)Hall, J. F. (2011). Completeness of ordered fields. arXiv preprint arXiv:1101.5652.Tarski's axiomatizationWikipedia contributors. (2018, November 4). Construction of the real numbers. In Wikipedia, The Free Encyclopedia. Retrieved 10:08, November 7, 2018, from https://en.wikipedia.org/w/index.php?title=Construction_of_the_real_numbers&oldid=867304826Leibniz completeness, Hilbert completeness, Bolzano completeness, Heine-Borel completeness, Cantor completnesses, Hall, J. F., & Todorov, T. D. (2011). Completeness of the Leibniz field and rigorousness of infinitesimal calculus. arXiv preprint arXiv:1109.2098.Bachmann's construction by refinement of rational nests, Bourbaki's approach, Maier-Maier's variation of Dedekind cuts, Shiu's construction by infinite series, Flatin-Metropolis-Ross-Rota's wreath construction, De Brujin's construction by additive expansions, Rieger's construction by continued fractions, Schaunuel et al's construction using approximate endomorphisms of $\mathbb{Z}$, Knopfmacher-Knopfmacher's various constructions, Pintilie's construction by infinite series, Arthan's irrational construction, Conway's Surreal numbersWeiss, I. (2015). Survey Article: The real numbers–A survey of constructions. Rocky Mountain Journal of Mathematics, 45(3), 737-762.
 >> Anonymous Wed Nov 7 05:17:50 2018 No.10123959 >>10123956And the nonstandards approachesCutland, N. J., Di Nasso, M., & Ross, D. A. (2006). Nonstandard methods and applications in mathematics (Vol. 25, pp. 1-262). AK Peters.
 >> Anonymous Wed Nov 7 06:11:00 2018 No.10124025 >>10123578Assume there are more than 2 real numbers, and let x and y be two of them. let s be any cauchy sequence in the class x. For any integer n, there exists a cauchy sequence t in the class y such as s0, s1,... sn is equal to t0, t1,... tn.Therefore x = y, which contradicts the assumption that there are at least 2 different real numbers.
 >> Anonymous Wed Nov 7 09:27:32 2018 No.10124212 I demand an imaginary number thread
 >> Anonymous Wed Nov 7 09:43:27 2018 No.10124241 >>10124212You're already in one, bub.
 >> Anonymous Wed Nov 7 10:25:35 2018 No.10124284 >>10124025>let s be any cauchy sequence in the class x>For any integer n, there exists a cauchy sequence t in the class y such as s0, s1,... sn is equal to t0, t1,... tngonna need a proof for that, bucko
 >> Anonymous Wed Nov 7 11:22:16 2018 No.10124394 >>10123574Dedekind cuts are in baby Rudin you poseureous shutbag
 >> Anonymous Wed Nov 7 11:29:40 2018 No.10124421 >>10124212they're just the rotation of real numbers
 >> Anonymous Wed Nov 7 11:42:13 2018 No.10124473 >>10124284are you dense?take any t in y, switch all the first n terms to match those of s, then that new sequence is still a cauchy sequence in the class y by definition.
 >> The Landlord Wed Nov 7 15:27:15 2018 No.10124976   File: 42 KB, 220x249, TIMESAND___762++145rhwrssdwfewrtetyhinrtubiet6uq8641.png [View same] [iqdb] [saucenao] [google] [report] >>10123668>nothing wrong with the decimal construction of the reals.
 >> The Landlord Wed Nov 7 15:37:34 2018 No.10124994   File: 20 KB, 220x165, TIMESAND___76266725q75q788q7878q782456jt35u34548uy4p861.gif [View same] [iqdb] [saucenao] [google] [report] >I hope, even if the above discussion falls short of a complete proof, that I have given enough detail to convince you that there are natural definitions of addition and multiplication and ordering of infinite decimals, and that with these definitions they form a complete ordered field. https://www.dpmms.cam.ac.uk/~wtg10/decimals.html
 >> Anonymous Wed Nov 7 15:56:08 2018 No.10125036
 >> Anonymous Wed Nov 7 16:07:26 2018 No.10125054 >>10124212>let a,b,c,d be real numbers>define (a,b), (c,d) be complex numbers with addition and multiplication defined as follows>(a,b) + (c,d) = (a + c, b + d)>(a,b) x (c,d) = (ac - bd, ad + bc)
 >> Anonymous Wed Nov 7 16:55:35 2018 No.10125143 >>10124025This is one of those answers where I don't know if you're trolling or braindamaged.
 >> Anonymous Wed Nov 7 17:01:05 2018 No.10125158 >>10125143You resort to insults because you're facing your contradictions. Last gasp of a dying theory.
 >> The Landlord Wed Nov 7 17:08:39 2018 No.10125189   >>10125143I too could not parse what he put in that post, and I think he knew most people wouldn't be able to when he wrote it.>>10125158non-sequitur
 >> Anonymous Wed Nov 7 17:26:11 2018 No.10125251 >>10125189did you fail freshman year or do you just act dumb tripfag? take two "cauchy sequence equivalence classes", one class corresponding to a real number x, and one to a real number y different from x.Get it so far?Now pick any cauchy sequence s in the class of cauchy sequences that converge towards x (you should be able to do so, if you believe in real numbers).Now pick any sequence t among the cauchy sequences that converge towards y. Pick any integer n, as big as you want. Replace t's nth first terms with the nth first terms of the sequence s. This doesn't change the fact that t converges towards y. So t is still a cauchy sequence in the equivalence class that we took it from, yes? If two sequences have the same terms, they're equal, right? Then I claim x=y. Don't believe me? Take a sequence s from x, and an integer n. I can come up with a sequence t in y that matches the sequence s up to its nth term at least. Therefore s is also a sequence in y, therefore x=y.
 >> The Landlord Wed Nov 7 17:48:53 2018 No.10125306   >>10125251>take two "cauchy sequence equivalence classes",>Get it so far?No. I don't know what the definition of an equivalence class is. Maybe if you say precisely what you mean instead of using the jargon labels I will be able to understand you better.
 >> Anonymous Wed Nov 7 18:19:56 2018 No.10125365 >>10125306wow
 >> Anonymous Wed Nov 7 18:22:32 2018 No.10125372 >>10125306Wew
 >> The Landlord Wed Nov 7 20:40:41 2018 No.10125628 File: 17 KB, 520x520, TIMESAND___762sdiwft45df4sg4sf3g4er3h4rrky34r3h4rrky34r3h4rrky343ul4iu3p4uip3.jpg [View same] [iqdb] [saucenao] [google] [report] >>10125036Nice video. I like how he did the exposition of the pros and the cons. Very informative. What are the shortcomings of the following definition?A real number is a cut in the real number line.A cut in a line separates one line into two pieces.
 >> Anonymous Wed Nov 7 21:16:56 2018 No.10125689 >>10125628Ignoring the circular definition, it's not always so easy to actually construct the dedekind cuts other than for textbook examples like sqrt(2). For example, try to find a dedekind cut for sqrt(2) + pi + e, or even just pi.
 >> The Landlord Wed Nov 7 22:13:17 2018 No.10125794 >>10125689What is circular about the definition? A number is a cut in a line. The number is real if it's a cut in the real line. A line is the real line if I say it is.To construct those numbers, choose an origin and another point, then define the second point as that number. What's the problem?
 >> Anonymous Wed Nov 7 22:18:57 2018 No.10125805 >>10123891>>10123956>>10123959wow. just wow.
 >> Anonymous Wed Nov 7 22:45:08 2018 No.10125852 >>10125689It's not that hard to make dedekind cuts for e or pi. There are plenty of series which converge monotonically to those numbers, and then you just make your cut the set of all upper (or lower) bounds of whatever series you used.
 >> Anonymous Wed Nov 7 22:49:27 2018 No.10125856 >>10125251>Take a sequence s from x, and an integer n. I can come up with a sequence t in y that matches the sequence s up to its nth term at least. As you said, that doesn't change the fact that t converges to y and s converges to x. Two equal sequences can't converge to different values. Counterproof: For any arbitrary n there is an m>n such that the mth term of s does not equal the mth term of t, otherwise they would converge to the same value. Thus the two sequences are never equal.
 >> Anonymous Wed Nov 7 22:51:36 2018 No.10125858 >>10124394> poseureous
 >> Anonymous Wed Nov 7 22:54:01 2018 No.10125861 >>10125794>choose an origin and another pointsorry but the Axiom of Choice wasn't formulated until 1904 so if we're only allowed to use Riemann-era math you'll need to fix your definition
 >> Anonymous Wed Nov 7 22:56:17 2018 No.10125866 >>10125689the fact that the real number line is even mentioned in that particular definition of a real number.
 >> The Landlord Wed Nov 7 23:03:38 2018 No.10125874 File: 78 KB, 1280x705, TIMESAND___762sdiwdftw7dghhtt5s3dfhjitsfsfgs8qwr123gs32wdzzdw5s55s59s59s.jpg [View same] [iqdb] [saucenao] [google] [report] >>10125861I'm happy to use Riemann's definition. Which one did he use?
 >> The Landlord Wed Nov 7 23:16:08 2018 No.10125887 File: 930 KB, 628x720, TIMESAND___7626g468wg467323468e4w68w4t6w4t68we4twet4w8y4u864o8uy4p861.png [View same] [iqdb] [saucenao] [google] [report] >>10125861also, how did Riemann choose to continue the domain of the Dirichlet function?
 >> Anonymous Thu Nov 8 00:19:57 2018 No.10125999 >>10123554they got the right intuition, you don't need know the full structure of real numbers to do the math they did. You only need an object that would satisfy the basic intuitive assumptions on geometry and topology of the euclidean space to reproduce their results. There is nothing inherently wrong with their theory, they don't touch the problematic things that appear in set theory. Analysis=/= set theory.
 >> Anonymous Thu Nov 8 12:11:46 2018 No.10126833 File: 92 KB, 1440x2392, inverted cauchy sequence.png [View same] [iqdb] [saucenao] [google] [report] >>10124025We can easily see this is false by taking the cauchy sequence as a function of 1/n. This function converges to x as 1/n approaches 0 from the right. Your argument is akin to saying that changing the function at some point from 1/n to to 1 changes the convergence. It doesn't since the only thing that matters is to the left of 1/n, not to the right.
 >> Anonymous Thu Nov 8 12:20:18 2018 No.10126844 >>10126833>by taking the cauchy sequence as a function of 1/nwhich they are not.Show me a procedure to test the equality of 2 sequences and I'll show you that every cauchy sequence is equal. I'll wait
 >> Anonymous Thu Nov 8 12:32:29 2018 No.10126869 >>10126844>which they are not.It doesn't effect the convergence since the order of terms is perserved (simply reversed). So it's an equivalent construction.>Show me a procedure to test the equality of 2 sequences and I'll show you that every cauchy sequence is equal. I'll waitSimple, all terms of the sequence must be equal. Saying that they are equal from 1 to n does not mean they are equal past n.
 >> Anonymous Thu Nov 8 12:33:18 2018 No.10126871 >>10123554Dedekind?
 >> Anonymous Thu Nov 8 12:46:37 2018 No.10126893 >>10123652Wtf. Cuts are defined on the rational numbers, not the real numbers. That would be circular.
 >> Anonymous Thu Nov 8 12:49:07 2018 No.10126898 >>10126869That is not a proper definition of equality, since you assume you can check that all terms are equal. But you can't in finite time. So this is not a valid procedure.
 >> Anonymous Thu Nov 8 12:50:15 2018 No.10126899 >>10126898then you can't check the numbers past n either
 >> Anonymous Thu Nov 8 13:00:24 2018 No.10126916 >>10126898If they are unequal then we will find an mth term that is not equal in finite time. But this is irrelevant anyway since the burden of proof is on you to show that they're equal, not me.
 >> Anonymous Thu Nov 8 13:01:27 2018 No.10126920 >>10126899If you can choose an arbitrary n then I can choose an arbitrary m>n.
 >> Anonymous Thu Nov 8 13:03:57 2018 No.10126924 >>10126916My point is that they don't exist, not that they're equal. You can't prove equality or inequality because you can never stop checking infinite sequences. If you stop before the end there might be a difference.So as far as I'm concerned if real numbers are defined as cauchy sequences, they are all equal.
 >> Anonymous Thu Nov 8 13:09:16 2018 No.10126934 >>10126924>My point is that they don't exist, not that they're equal.Then your argument that x=y fails. And I don't agree with your second argument that time is somehow relevant to a mathematical definition.>You can't prove equality or inequality because you can never stop checking infinite sequences.Inequality is already proved by the fact that they don't converge to the same value, so this is irrelevant.>So as far as I'm concerned if real numbers are defined as cauchy sequences, they are all equal.Doesn't follow. You're handwaving at this point.
 >> Anonymous Thu Nov 8 13:21:48 2018 No.10126956 >>10126893You are confusing, I believe, Dedekind cuts with cuts. Dedekind cuts are a subset of all cuts: a subset defined on the rationals.
 >> Anonymous Thu Nov 8 13:30:42 2018 No.10126967 >>10126934my argument is reductio ad absurdum you cocksucker. You can define a thing that is equal to two different things if you want, but that won't make me use it.Anyway you're obviously in way over your head here so... Keep assuming infinite sets exist
 >> Anonymous Thu Nov 8 13:53:20 2018 No.10127005 File: 48 KB, 600x467, 001.jpg [View same] [iqdb] [saucenao] [google] [report] >>10126967>my argument is reductio ad absurdum you cocksucker. State the premises and conclusion.>You can define a thing that is equal to two different things if you want, but that won't make me use it.They aren't equal by your first premise and no one cares what math you use.
 >> Anonymous Thu Nov 8 15:55:08 2018 No.10127233 >>10123652>a real number is a real numbertripfag and namefag. Can't make this shit up
 >> Anonymous Thu Nov 8 18:20:38 2018 No.10127539 >>10126967>>10127005So still no answer. I guess you gave up the argument since both your claim that x=y and that we need "infinite time" to check that they are not equal don't follow.
 >> Not Hodor Fri Nov 9 11:59:44 2018 No.10128767 File: 21 KB, 751x440, TRINITY___dimensionlessconstants.png [View same] [iqdb] [saucenao] [google] [report] Look how the three most important real numbers of physics, the classical electrical coupling constant, the quantum electrical coupling constant, and the gravitational coupling constant all come from the same simple model which is taught to 3000-level undergrads in physics: the 2D box.Note that the 8pi term is the stress energy tensor and it contains the same f^3 terms as Planck's law for the energy density. Obviously this is just numerology because anyone can take any foundational problem in physics and it use it to derive these three numbers in a way that includes the dependence of planck's law int he energy term, but I haven't seen anyone else do it. Since it's meaningless numerology I know there's 900 other ways to crank out the same numbers but I can't remember where I saw those other 900 wrong demonstrations which show that this result isn't unique and worthy of further close study.
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