/sci/ - Science & Math » Thread #14162173

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Extended real numbers

Anonymous Sat 29 Jan 10:07:39 2022 No.14162173 View Reply Original Report

Quoted By: >>14168184 >>14169441

In this thread we discuss the concept of extended real numbers and their properties
> topological compactification of R
>sup/inf always exist
> no need to carry around edge cases of convergence in definitions around, we can now converge to infinity
> limits of converging sequences just add together (unless one is infinity and the other negative infinity)
> "point/line at infinity"
> makes finitist seethe

are they based?

Anonymous

Anonymous Sat 29 Jan 2022 16:58:59 No.14163365 Report

Quoted By: >>14163372

>>14162513
They can be treated as such and are incoherent at P=0 and N=-1 because 0=-0 is a logical contradiction.

Anonymous

Anonymous Sat 29 Jan 2022 17:02:07 No.14163372 Report

Quoted By: >>14163376 >>14163383

>>14163365
>0=-0 is a logical contradiction
it's not, you're coming to that conclusion by meandering between math statements and verbal statements to weave a virtual discontinuity across contradictions which only manifest through you incorrectly mapping mathematics to words and vice-versa
you have no idea what you're talking about, you're not onto anything, and spamming this asinine conclusion in every thread with an equals sign is only going to make people stop listening faster

Anonymous

Anonymous Sat 29 Jan 2022 17:03:38 No.14163376 Report

Quoted By:

>>14163372
He is actually right and he's proven you wrong. He's a trooly intelligent poster and you're a sheep. Imagine thinking that mathematics is different from words.

Anonymous

Anonymous Sat 29 Jan 2022 17:06:27 No.14163383 Report

Quoted By: >>14163737

>>14163372
>meandering between math statements and verbal statements
I am going between arithmetic's logic and its parent, natural logic which disproves arithmetic as a completely logical system without self contradiction because it is built on a contradictory element that is its own opposite.

In mathematics, an opposite number is defined though additive inversion, if something is its own opposite, it is self contradictory, 0 is its own opposite, the resulting number system is a numerical explosion from that initial paradoxical value.

El Arcón

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El Arcón Sat 29 Jan 2022 17:06:59 No.14163385 Report

Quoted By: >>14163397

Topological Closure of Extended Real Relations

For all of the bar raising that gets levied against my work and my definitions, I'd have a nice look at Equation (1) in this perfectly respectable, scholarly paper.

El Arcón

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El Arcón Sat 29 Jan 2022 17:09:59 No.14163397 Report

Quoted By:

>>14163385
I formatted my arithmetic tables based off of the nice style in this other paper, by the way.

Anonymous

Anonymous Sat 29 Jan 2022 17:13:16 No.14163410 Report

Quoted By:

the cris of /sci/

Anonymous

Anonymous Sat 29 Jan 2022 17:14:52 No.14163415 Report

Quoted By: >>14163541 >>14163564

>>14163176
It is strange that nobody has yet been able to exhibit a single point in a
definition of real numbers that even mentions sets, let alone infinite sets.

Anonymous

Anonymous Sat 29 Jan 2022 17:56:33 No.14163541 Report

Quoted By: >>14163564 >>14163646

>>14163415
I have the same experience. All the criticisms I've seen of how R is defined seem to not be specific criticisms of how R is defined but rather foundational criticisms of ZF in general which try to attach themselves to R as a proxy punching bag but fail to land any punches.

Anonymous

Anonymous Sat 29 Jan 2022 18:06:20 No.14163564 Report

Quoted By: >>14163586 >>14167773

>>14163415
The definition of a sequence as a function as a set of pairs.
>>14163541
ZFC was invented to give a foundation for real numbers. Without real analysis there wouldn't be ZFC. The completed infinite nature of real numbers forced mathematicians to come up with a bullshit justification for why that's legit math, and they called it ZFC.

Anonymous

Anonymous Sat 29 Jan 2022 18:18:02 No.14163586 Report

Quoted By: >>14163679 >>14163991 >>14164162 >>14167773

>>14163564
Why does an infinite sequence need to be completed in order to prove that it converges? I understand that most sequences can't be proven to converge but isn't R only concerned with those that can?

Anonymous

Anonymous Sat 29 Jan 2022 18:36:06 No.14163646 Report

Quoted By: >>14163696 >>14166972

>>14163541
yes, the fundamental problem with R is that rests upon ZFC. if you have a better definition of R, feel free to share.

Anonymous

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Anonymous Sat 29 Jan 2022 18:42:10 No.14163679 Report

Quoted By: >>14163717 >>14168328

>>14163586
A sequence is said to converge, iff their limsup and lininf coincide (the lower and upper envelope, so to speak, as the index n goes to infinity). This makes sense for bounded sequences (an oscillating sequence of -1,+1 will not converge by that definition). When a sequence shoots to infinity, say the sequence of Natural numbers, they converge by that definition too if we admit infinity as a value for limsup and liminf (which will coincide in that case).

Anonymous

Anonymous Sat 29 Jan 2022 18:45:39 No.14163696 Report

Quoted By: >>14163819 >>14166972

>>14163646
What part of the definition of Cauchy equivalence classes is spoiled by ZFC? I had a very thoughtful discussion recently (with a poster who seemed quite well-versed in set theory) which from my perspective went in a circle from proving by ZF that we can define a sequence with two mutually exclusive equivalence classes to being unable to prove by ZF that the duality mattered or could even be logically describe in terms on an unary predicate on sets.

Anonymous

Anonymous Sat 29 Jan 2022 18:50:59 No.14163717 Report

Quoted By: >>14163858

>>14163679
I understand your criticism of extended R (and it is particularly fair in the context of this thread) but my curiosity is specifically to open R.

Anonymous

Anonymous Sat 29 Jan 2022 18:56:32 No.14163737 Report

Quoted By: >>14163749

>>14162248
>>14163383
true or false: 0 + -0 = 0

Anonymous

Anonymous Sat 29 Jan 2022 19:00:09 No.14163749 Report

Quoted By:

>>14163737
The axioms of arithmetic confirm that logical contradiction, but the laws of logic say logical contradiction such as that always leads to a deductive explosion of trivial truths within that illogical system of axioms, 1=1, 2=2, ... etc such that arithmetic value identity is completely incoherent at the poles 0 = 0 + 0 + ... inf = inf + inf + ...

Anonymous

Anonymous Sat 29 Jan 2022 19:21:02 No.14163819 Report

Quoted By: >>14163881

>>14163696
No, the problem is not that the sequence defines two equivalence classes at the same time. It defines precisely one equivalence class, but you can't compare it to other classes. And not because of an insufficient mathematical advancement, but because it's impossible in principle. When you ask a mathematician how to compare two real numbers, he'll tell you something similar to "pick two representants of the respective equivalence classes, take the difference, check whether it converges to zero or whether it stays above or below". That sequence was just a simple example to show how naive it is to expect such a thing to be possible in general. Whenever you talk about a Cauchy sequence, you picture something "nicely" defined such as $\sum_{n=0}^{\infty} \frac{1}{n!}$ and you consider the example in question to be pathological. But why exactly is it pathological, because you can't compare it to 0? "Most" Cauchy sequences cannot be even defined by a predicate like that. If you pick a Cauchy sequence at random, there's literally 0% chance you'd be able to check even the most basic properties (of course besides the ones which require only a finite amount of work). From this point of view, the nice Cauchy sequences you're used to work with are actually much more pathological. Here's a challenge. Work through the definition of R, but everytime a Cauchy sequence is mentioned, don't picture a nice sequence defined by a closed formula. Picture something like that CH example while keeping in mind that "most" Cauchy sequences are even worse. Do you consider it to be a solid basis for a number system?

Anonymous

Anonymous Sat 29 Jan 2022 19:32:10 No.14163858 Report

Quoted By:

>>14163717
not having +-infinity will not pose a problem in general as far as I know. The definition will just change, we then say that there is divergence in cases where the series wanders to infinity (or we say convergence to infinity but this is a seperate definition from the usual convergence and not to be mistaken by that). It is certainly possible to exclude such cases from your definitions but then you will have to carry around all of those edge cases manualy. A limit contained in R will not be guaranteed to have it's limit in R again (noncompact) unless it is bounded, but then were does the series go to? Compactification by the extended reals just saves you all that hassle of adding artificial defintions and rules to make sure everything stays in R. Really the important point for extended reals is to ensure the existence of sup/inf, the rest of your definitions from analysis are based on those and can be stated in form of monotone sequences.

Anonymous

Anonymous Sat 29 Jan 2022 19:39:18 No.14163881 Report

Quoted By: >>14163965

>>14163819
>It defines precisely one equivalence class
Yes, I understand this aspect and if you're the same poster thank you again for being so thoughtful and descriptive.
>but you can't compare it to other classes
Why not? Can you describe an unary predicate on sets that would disallow that comparison?

I've read your whole post but it will color my understanding of it and prevent me from saying ignorant things about it if we can clarify it together point by point.

Anonymous

Anonymous Sat 29 Jan 2022 19:59:26 No.14163965 Report

Quoted By: >>14163986

>>14163881
I mean you can compare it in the sense that exactly one of <0, =0, or >0 is true. It's undecidable which one.

Anonymous

Anonymous Sat 29 Jan 2022 20:08:25 No.14163986 Report

Quoted By:

>>14163965
How would you contradict this argument in terms of a Cauchy sequence not being satisfied by the property described by the argument?
>P(x) is an unary predicate about sets, in this case P(x) ? x is a Cauchier sequence. {a_n} is equal either to (1,1,1,..), or to (0,0,0,...), thus ¬P({a_n}) implies ¬P((1,1,1,..)) or ¬P((0,0,0,...)) and this is inevitable. There doesn't exist a property of sets which would be satisfied by both (1,1,1,..) and (0,0,0,...), but not by {a_n}.

El Arcón

El Arcón Sat 29 Jan 2022 20:10:58 No.14163991 Report

Quoted By: >>14164000 >>14166477 >>14166972

>>14163586
>isn't R only concerned with those that can?
{0,1,0,1,0,1,0,1,...}

Anonymous

Anonymous Sat 29 Jan 2022 20:13:22 No.14164000 Report

Quoted By: >>14164106 >>14164162 >>14164962

>>14163991
What Cauchy equivalence class does this sequence belong to?

Anonymous

Anonymous Sat 29 Jan 2022 20:44:55 No.14164106 Report

Quoted By: >>14164122

>>14164000
Not a cauchy sequence. R can be defined as the set of all equivalence classes made from equivalence relations on cauchy sequences, or it can be defined axiomatically.

Anonymous

Anonymous Sat 29 Jan 2022 20:48:31 No.14164122 Report

Quoted By: >>14164135 >>14164162 >>14164962

>>14164106
>equivalence classes made from equivalence relations on cauchy sequences
What Cauchy equivalence class contains {0,1,0,1,0,1,0,1,...}

Anonymous

Anonymous Sat 29 Jan 2022 20:52:56 No.14164135 Report

Quoted By: >>14164143

>>14164122
None, because it is not a cauchy sequence. Quote from wikipedia

'Two Cauchy sequences are calledequivalentif and only if the difference between them tends to zero. This defines anequivalence relationthat is compatible with the operations defined above, and the setRof allequivalence classescan be shown to satisfyall axioms of the real numbers."

The equivalence is set up between cauchy sequences.

Anonymous

Anonymous Sat 29 Jan 2022 20:55:18 No.14164143 Report

Quoted By: >>14164167 >>14164962

>>14164135
Then if {0,1,0,1,0,1,0,1,...} doesn't belong to a Cauchy equivalence class, what does it have to with R?

El Arcón

El Arcón Sat 29 Jan 2022 21:03:07 No.14164162 Report

Quoted By: >>14164188

>>14164000
>>14164122
One and zero are both real numbers and R is concerned with them. If you don't know that the answer to your question is that there is no such equivalence class, you might want to brush up on your ignorance.
>>14163586
>I understand that most sequences can't be proven to converge but isn't R only concerned with those that can?

El Arcón

El Arcón Sat 29 Jan 2022 21:03:47 No.14164167 Report

Quoted By: >>14164188

>>14164143
It is a sequence of real numbers, you retard.

Anonymous

Anonymous Sat 29 Jan 2022 21:10:12 No.14164188 Report

Quoted By: >>14164962

>>14164162
>>14164167
>It is a sequence of real numbers
Cauchy means the [rational] numbers of the sequence get arbitrarily close to each other. In the case of {0,1,0,1,0,1,0,1,...} the epsilon is capped at 1 and so that algorithm is not Cauchy by definition.

El Arcón

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El Arcón Sun 30 Jan 2022 00:50:32 No.14164962 Report

Quoted By: >>14165197

>>14164188
You have clarified 1-0=1, but you have not clarified whether or not you understand that some sequences of x?R are not Cauchy or in an equivalence class.
>>14164000
>What Cauchy equivalence class does this sequence belong to?
>>14164122
>What Cauchy equivalence class contains {0,1,0,1,0,1,0,1,...}
>>14164143
>Then if {0,1,0,1,0,1,0,1,...} doesn't belong to a Cauchy equivalence class, what does it have to with R?
I assume you do know that not every sequence is Cauchy, however, and I am not asking you for clarifications on the material from the first week of undergraduate analysis. Rather, my intention was to point out to the person I replied to (pic) that non-converging sequences are often interest in R, which was what was asked.

Anonymous

Anonymous Sun 30 Jan 2022 01:45:10 No.14165197 Report

Quoted By: >>14166210 >>14166367 >>14166932

>>14164962
>non-converging sequences are often interest in R
I'm the Anon in your pic (and also I'm every Anon you've quoted in your post). I don't understand how R is supposed to be affected by sequences that don't converge?

Anonymous

Anonymous Sun 30 Jan 2022 10:19:57 No.14166210 Report

Quoted By: >>14166477 >>14166932 >>14166972

>>14165197
You can CONSTRUCT the system of real numbers by forming equivalence classes of cauchy sequences (made from rationals). Just read the wikipedia article. I don't understand what you mean with "affected", but the sequence you provided should play no part in the construction of R as it is not a convergent sequence. Take the sequence of N so {1,2,3,...} You would agree that its elements are contained in R, so quoting you "it has something to do with R". However it is not in any equivalence class during the construction, because it is not a cauchy sequence of rational numbers.

El Arcón

El Arcón Sun 30 Jan 2022 11:42:05 No.14166367 Report

Quoted By: >>14166477

>>14165197
It's not affected by them at all. That's like asking how R is affected by melty ice cream.

Anonymous

Anonymous Sun 30 Jan 2022 12:56:17 No.14166477 Report

Quoted By: >>14166731

>>14166367
Then why did you reply by this >>14163991

>>14166210
>the sequence you provided
That sequence was provided to me, not by me.

Anonymous

Anonymous Sun 30 Jan 2022 15:22:44 No.14166731 Report

Quoted By: >>14166758

>>14166477
That wasn't me, I am OP. I wanted to have a fruitful discussion about the extended numbers without 'finitists' derailing this thread with their refusal to learn basic analysis concepts, but you went along and asked multiple times how the non-cauchy sequence {0,1,0,1...} is related to R, even after explaining to you in detail that only cauchy sequences of rational numbers are used for the construction of R. At that point I suspect that your question is posed vaguely on purpose to instigate some sort of meaningless conversation, just like
>>14162248 who clearly doesn't know what he is talking about.

Anonymous

Anonymous Sun 30 Jan 2022 15:34:43 No.14166758 Report

Quoted By:

>>14166731
I didn't ask you anything. You jumped into a side conversation that another poster initiated by replying to me that R is concerned with the noncovergent sequence {0,1,0,1...}. Normally I'd understand how you might be confused—given that everyone is Anonymous—but in this case the other poster is not Anonymous so your confusion about this is really quite baffling.

Anonymous

Anonymous Sun 30 Jan 2022 16:46:39 No.14166932 Report

Quoted By: >>14166972

>>14165197
> I don't understand how R is supposed to be affected by sequences that don't converge
>>14166210
Me answering

Prior to this in
>>14165197
you claimed that each of the post was made by you. Each of those posts was asking the relationship between R and this sequence. Now you claim you have never asked a question.

Anonymous

Anonymous Sun 30 Jan 2022 17:00:03 No.14166972 Report

Quoted By: >>14167533

>>14166932
None of those were questions to you. They were all directly asking the other poster to explain by what logic he thought the sequence {0,1,0,1...} was related to R. That itself was a tangent from my original question (starting here >>14163696 ) asking >>14163646 to explain by what logic he thought ZFC spoiled the definition of R. This poster >>14163991 then jumped in, derailing the original conversation, just as you later jumped in, derailing the derailed conversation with your own triviality here >>14166210. Now do you actually have anything of substance to add, either to my own original question about ZFC or to your own original thread about extended R, or are you intend to simply keep bumping your own thread with confused banalities?

Anonymous

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Anonymous Sun 30 Jan 2022 19:41:19 No.14167533 Report

Quoted By: >>14167773

>>14166972
I am refering to the question
> Isn't R only concerned with sequences that can converge
which was vaguely stated by your side and therefore admits several ways of interpretation, all of which were explained to you. If anything, you derailed your own initial conversation by not only asking that question in such unclear fashion but also asking the same question again and again after being provided with an answer. Then somehow I was supposed to distinguish between all Anonymous posters, which I was unable to understand because all of the several last posts were made solemnly by you. I do admit however that there might have been some confusion going on from my part as well and you are right that the best course of action would be to discuss the topic at hand. I can only hope that the question I am refering to has been answered, if not I again recommend the wikipedia page of the construction of R, which shows a direct way of construction by rational cauchy sequences which serves an alternative to the axiomatic definition.

Anonymous

Anonymous Sun 30 Jan 2022 20:40:47 No.14167773 Report

Quoted By: >>14168149

>>14167533
So you're talking about this post? >>14163586
>Why does an infinite sequence need to be completed in order to prove that it converges? I understand that most sequences can't be proven to converge but isn't R only concerned with those that can?
And you don't understand that the post is replying to the poster directly above it >>14163564 who asserts that ZFC spoils the definition of R on account of a "completed infinite nature of a real number"? Wew. Okay, , is it easier for you to read a different language maybe? French or Spanish?

Anonymous

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Anonymous Sun 30 Jan 2022 22:25:14 No.14168149 Report

Quoted By:

>>14167773
I'd prefer the english language and I also don't concern myself with what that someone else has posted before, as I don't understand the argument he is trying to make with the incompatibility of ZFC and R. I replied to "your" post and addressed "your" questions specifically, by giving you both the definition of convergence and later on the nature of sequences used in the construction of R, which I hope have clarified both of your questions. You suggested we end this aimless exchange and I agreed upon that, so it would be best if we focused on your specific questions (in case you are not satisfied by my answers).

Anonymous

Anonymous Sun 30 Jan 2022 22:37:25 No.14168184 Report

Quoted By: >>14168240

>>14162173
> no need to carry around edge cases of convergence
> limits of converging sequences just add together (unless one is infinity and the other negative infinity)
> unless one is infinity and the other negative infinity
> unless

Anonymous

Anonymous Sun 30 Jan 2022 22:53:14 No.14168240 Report

Quoted By: >>14168253

>>14168184
Yes but this is a problem of arithmetic or the lack of operations that are permitted for infinity. Infinity - Infinity is not defined. This is not a problem of extended reals, since you are running into the same type of problems when you look at division of two limits, that will converge to the division of both limits unless both converge to 0. Same with a diverging sequence multiplied by an infinitesimal quantity, which can converge.

Anonymous

Anonymous Sun 30 Jan 2022 22:58:02 No.14168253 Report

Quoted By: >>14168328

>>14168240
Yeah, but you're still carrying around the same edge cases of convergence minus one.

Anonymous

Anonymous Sun 30 Jan 2022 23:12:36 No.14168328 Report

Quoted By:

>>14168253
That is true, you will not magically get rid of all edgecases, however in regards to
>>14163679 it strengthens and simplifies the definition of convergence. The existence of sup/inf is really strong and beautiful in my opinion, as both of these operations build the fundamental cornerstone of real analysis. Regarding the problem, we cannot have equivalence of lim x + lim y and lim(x+y), as the first sum has to exist to carry out the implication to the left.

Anonymous

Anonymous Mon 31 Jan 2022 03:53:55 No.14169235 Report

Quoted By:

Pseud thread

Anonymous

Anonymous Mon 31 Jan 2022 05:35:15 No.14169441 Report

Quoted By: >>14169596

>>14162173
They aren't based so much as they are sometimes contextually a good space to work in. It really just depends on what you're doing. If you want to do analysis, then sometimes you want certain sequences or functions to be in your space, but your space is defined by convergence, so you add points that they converge to, shit like that. It doesn't really encode any especially new or interesting information since diverging in the reals is the same as converging to $\pm\infty$ in $\bar{\mathbb{R}}$ , it's just useful for some closure properties it provides.

Anonymous

Anonymous Mon 31 Jan 2022 07:14:11 No.14169596 Report

Quoted By:

>>14169441
>diverging in the reals is the same as converging to ±? in R¯
No it isn't.

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