/sci/ - Science & Math

Dedekind cuts are a undergrad-tier thing, Jon. Reals are defined as limits of series; open up your baby Rudin.

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.

>>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?

>>10123578
there is an even better way, pic related

>>10123652
>a real number is a real number

>>10123578
There's literally nothing wrong with the decimal construction of the reals.

>>10123668

>>10123554
1. 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" [math] \mathbb{Z} \to \mathbb{Z} [/math], and B is the set of functions [math] \mathbb{Z} \to \mathbb{Z} [/math] whose range is bounded. Where a function [math] f [/math] from [math] \mathbb{Z} [/math] to [math] \mathbb{Z} [/math] is said to be an almost homormophism iff the function [math] d_f [/math] from [/math] \mathbb{Z} \times \mathbb{Z}[/math] to [math] \mathbb{Z} [/math] defined by [math] d_f(p,q) = f(p+q) - f(p) - fq [/math] has bounded range, i.e., for some integer [math] C, |d_f(p,q)| < C [/math] for all [math] p, q \in \mathbb{Z} [/math]
Here: Arthan, R. D. (2004). The Eudoxus real numbers. arXiv preprint math/0405454.

3. Construction of Real numbers as infinite decimals. That is: [math] \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\} [/math]
where [math] d = \pm a_k a_{k-1} a_{k-2}\dots [/math]
[math]\mathbb{T}_9 [/math] stands for decimals ending with infinitely many 9's and [math] \mathbb{T}_0 [/math] for decimals ending with infinitely many 0's
And a jump goes from [math] \mathbb{T}_9 [/math] to [math] \mathbb{T}_0 [/math] or from [math] \mathbb{T}_9 [/math] to [math] \mathbb{T}_0 [/math] for decimals that are sufficiently close.
Klazar, M. (2009). Real numbers as infinite decimals and irrationality of [math] \sqrt {2} [/math]. arXiv preprint arXiv:0910.5870.

>>10123891
4. 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 [math] b [/math], containing finitely many elements, such that [math] a \leq b [/math]. 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 axiomatization
Wikipedia 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=867304826

Leibniz 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 [math] \mathbb{Z} [/math], Knopfmacher-Knopfmacher's various constructions, Pintilie's construction by infinite series, Arthan's irrational construction, Conway's Surreal numbers
Weiss, I. (2015). Survey Article: The real numbers–A survey of constructions. Rocky Mountain Journal of Mathematics, 45(3), 737-762.

>>10123956
And the nonstandards approaches
Cutland, N. J., Di Nasso, M., & Ross, D. A. (2006). Nonstandard methods and applications in mathematics (Vol. 25, pp. 1-262). AK Peters.

>>10123578
Assume 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.

I demand an imaginary number thread

>>10124212
You're already in one, bub.

>>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,... tn
gonna need a proof for that, bucko

>>10123574
Dedekind cuts are in baby Rudin you poseureous shutbag

>>10124212
they're just the rotation of real numbers

>>10124284
are 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.

>>10123668
>nothing wrong with the decimal construction of the reals.

>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

>>10123668
>>10123817
https://www.youtube.com/watch?v=tXhtYsljEvY

>>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)

>>10124025
This is one of those answers where I don't know if you're trolling or braindamaged.

>>10125143
You resort to insults because you're facing your contradictions. Last gasp of a dying theory.

>>10125143
I 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.

>>10125158
non-sequitur

>>10125189
did 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.

>>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.

>>10125306
wow

>>10125306
Wew

>>10125036
Nice 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.

>>10125628
Ignoring 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.

>>10125689
What 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?

>>10123891
>>10123956
>>10123959
wow. just wow.

>>10125689
It'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.

>>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.

>>10124394
> poseureous

>>10125794
>choose an origin and another point
sorry 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

>>10125689
the fact that the real number line is even mentioned in that particular definition of a real number.

>>10125861
I'm happy to use Riemann's definition. Which one did he use?

>>10125861
also, how did Riemann choose to continue the domain of the Dirichlet function?

>>10123554
they 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.

>>10124025
We 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.

>>10126833
>by taking the cauchy sequence as a function of 1/n

which 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

>>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 wait
Simple, 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.

>>10123554
Dedekind?

>>10123652
Wtf. Cuts are defined on the rational numbers, not the real numbers. That would be circular.

>>10126869
That 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.

>>10126898
then you can't check the numbers past n either

>>10126898
If 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.

>>10126899
If you can choose an arbitrary n then I can choose an arbitrary m>n.

>>10126916
My 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.

>>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.

>>10126893
You are confusing, I believe, Dedekind cuts with cuts. Dedekind cuts are a subset of all cuts: a subset defined on the rationals.

>>10126934
my 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

>>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.

>>10123652
>a real number is a real number
tripfag and namefag. Can't make this shit up

>>10126967
>>10127005
So 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.

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.