/sci/ - Science & Math

I assign a sphere to the number line with its volume. Does that mean that sphere is a real number?

>>12673579
What position on the line does it have?

>>12673579
no. the sphere's volume is (it's 0).

>>12673538
>A real number is a number that can be assigned a location on the number line
Ugh, so what's the definition of the number line?

>>12673538
>A real number is a number that can be assigned a location on the number line.

Do you have a definition of "location on the number line" independent of that of a number?

>It doesn't matter how convoluted and impractical the number's exact calculation could be
Do you require that all real numbers are definable though?

>>12673822
The undefinable reals each correspond to specific positions on the number line, we just don't know what positions those are

>>12673538
Even accepting the infinitist framework of mathematics your definition is way too vague to be useful and to reflect the actual modern understanding of the real numbers.
This is not least because there is no a priori good reason why the number line should have the least upper bound property.
You could just as well take the number line to be the set [math]\bar{\mathbb{Q}}[/math] of algebraic numbers, or even the smaller set of constructible numbers and do Euclidean geometry on them perfectly fine.
But suppose you really want completeness, i.e. you really want your number line to have the least upper bound property, even though you can't justify it a priori.
Then the elephant that comes into the room is the Suslin problem: is the set R the only set fitting our "number line" intuitions? I.e. is it the only nonempty totally ordered set X such that
- X is unbounded in either direction
- X is dense as an order.
- X has the least upper bound property.
and a less intuitive technical requirement that
- Any collection of disjoint open intervals is countable.
...

>>12673910
To expand on the last property, it essentially says that your line is not too large, that it's tractable in some sense. The property is true for the reals because in every open set you can find a rational number. But you could define another line as "the long line", which is essentially uncountably many copies of the real line glued together. See
https://en.wikipedia.org/wiki/Long_line_(topology)
The last property eliminates such spaces from consideration.
You would think that these properties would uniquely characterize the set of real numbers up to order isomorphism, but in fact this assertion is independent of ZFC. What this means, in practical terms, is that you can take the axioms that there is another such a "number line" and never find a contradiction.
This definitely demonstrates the inherent vagueness of the concepts involved in this mythical "number line" because even with such an incredibly powerful theory as ZFC, all these seemingly rigorous properties are not enough to actually define the number line.
All of this definitely proves the absolutely futility of trying to define a real number as in the OP. The question then is, is the modern arithmetical definition of the reals using uncountably infinite equivalence classes of computable and uncomputable sequences of rational numbers any better (or take any equivalent definition, for example by Dedekind cuts or almost-homomorphisms)?
The answer is a resounding NO!

....

>>12673918
The first monster you face is the Continuum Hypothesis. We know view it as something less-than-mathematical, something only relevant to set theorists and not-so-real mathematicians, but this was not so. It used to be an absolutely central problem in mathematics, first on the list of Hilbert's 23 problems posed to set the direction of mathematics for the next century. Mathematicians genuinely believed that it had a definite answer, and that it was a genuine problem (albeit a bit esoteric in its statement, especially for those a bit sceptical of infinities), until Cohen came along and blew it all. Now the consensus is that it's a esoteric, set-theoretic problem that we don't really have to care about.
Ok, so let's forget such artificial statements as CH, as the Godel sentences and whatnot, and move on to do some analysis (a field for which set theory was invented), and focus only on natural questions!
In Hilbert spaces, a very natural and useful structure is the ideal of compact operators. When we quotient out by this ideal, we can ask: are all the automorphisms in the resulting space inner? A very natural question which in fact turns out to have physical relevance. The answer? Independent from ZFC!
Let's take a look at algebra. A very natural algebraic questions that arose is the Whitehead problem: is every compact path-connected abelian group a product of copies of the circle group, R/Z? You can probably already guess the answer. There is none! You can take any answer you like and you will never encounter an inconsistency. Of course you could argue that this problem isn't about the real numbers per se (even though the construction of R/Z relies on them), but you cannot argue against the fact that constructions involved in the questions are inextricably linked to those or real numbers and have the same kind of vagueness.
...

>>12673921
With this, and many more examples the position of people who say we have an adequate definition of real numbers and general infinite sets (even with the understanding that ZFC only tell us what IS a set, not what a set ISN'T, except in the exceptional circumstances that a contradiction is derived) is highly inadequate, despite decades of efforts to try to make it work.
In conclusion, your definition is inadequate and way too vague to be anything beyond a most general guidance towards what we should view as a number, without giving us any unifying system of numbers. But to your defense, the definitions that we currently have are not much better.

tl;dr Accept Wildberger as your lord and savior of burn in (finite) hell of undecidability and delusion.

>>12673922
You will never be a woman

>>12673910
>>12673922
So what do you suggest to do?

>>12673538
>A real number is a number that can be assigned a location on the number line.
That doesn't actually mean anything. You are trying to define an arithmetical number system in terms of a geometric object. Therefore, you've just passed the buck of what a real number is to what a "location on the number line" is. And while there are many subtle issues of what this exactly means (as >>12673910 mentions), the more obvious issue is that of avoiding circularity.
How are you going to define a line? If you stick with the usual delusion that a line is an infinite set of points, then what are points? Points today are defined in terms of a number from some base number system (rationals if you are practical and "reals" if you are fanciful). So if you don't change anything else other than your definition of reals, then you're essentially saying "real numbers are locations on the number line and a location on the number is a point specified by a real number." In other words, you haven't actually gotten anywhere.
And, if you leave a point as your undefined object, you're essentially taking the same approach to mathematics that Euclid had 2000 years ago. Introducing coordinates to geometry is really powerful since a lot of Euclid's postulates could now be theorems, since we can precisely define points, lines, etc. in terms of numbers and equations. If you abandon that, you basically lose everything from the 16-17th century onwards in terms of mathematics.
For example, how do you know whether two curves intersect? Today, you'd check to see if there is a point at which they meet, but you haven't actually defined a point so you're basically taking Euclid's approach of just drawing things somewhere and eyeballing.
Therefore, you basically have to "define" a real "number" arithmetically and not geometrically to avoid circularity and maintain a coordinate-based geometry.

In summary, accept your fate and bow down to the cult of the Berger.

>>12674124
Finite, computational mathematics (which is much more interesting anyway).

>>12674190
So do two lines not intersect if the common point is irrational or transcendental?

>>12673538
>if it has a position on the line it's a real number.
I agree. Also: nice!

Fractional Distance: The Topology of the Real Number Line with Applications to the Riemann Hypothesis
https://vixra.org/abs/1906.0237

>>12673579
You assigned the volume to the line and the volume is a real number. You have stupidly bestowed transitivity on OP's "assignment" when OP didn't and then you took the nonexistent transitive property of "association" to mean that nested association is equivalent to association of the first and last nested elements. This is stupid and you wrote it because you know it is stupid.

>>12673818
A line is a Hausdorff space of dimension one extending infinitely far in both directions. Because everyone who reads this feigns ignorance of or does not know what a Hausdorff space is, a Hausdorff space is a set such that all the elements of the set are on the interior of a ball which also belongs to the set. In 1D, a ball is in an interval. If a point is on the interior of an interval then some of the interval lies to the left of the point and some to the right. These are the two directions of infinite extent. To extend infinitely far in a direction means that the set does not contain an endpoint to the left or right of the given point. A number line is a line equipped with a chart and a metric.

Since 99% of people who read this definition of a line don't know what a Hausdorff space is, the other 1% usually pretends like they don't either and then they say this definition of a line is "vague and meaningless" because of the "crackpot nonsense" about directions and extending far. It's simple predation upon the ignorance of interested but inexpert parties. They wouldn't do it if it didn't work for their purposes and it obviously does work for those purposes perfectly well.

>>12674190
>That doesn't actually mean anything.
>Words don't have meanings
>predation upon the ignorance of interested but inexpert parties

>>12674505
>A line is a Hausdorff space of dimension one extending infinitely far in both directions.
So the set of rational numbers?

>>12674517
No, and that's a stupid suggestion. Extent is a property of connected sets. Any set that extends is necessarily connected. Are you ignorant or are you attempting to prey upon others' ignorance?

>>12674282
So no physics at all?

>>12674290
You're alluding to what I didn't mention in my previous post, which are the downsides of introducing coordinates to geometry. At the price of making things like curves, intersections, etc. more precise than drawings on a page, you are introducing something artificial to the problem that wasn't there originally, namely the coordinate axes and a special origin point.
Space looks pretty isotropic to us, so if we model intersection of curves by including some special directions that denote horizontal/vertical and a special origin point, it is not immediately clear that our spatial intuition is going to correspond with what is mathematically going on.
More precisely, by putting arithmetic before geometry, you are inherently introducing number-theoretical issues into the geometry that might not have been there originally. This is the reason why "real" numbers are so attractive/"intuitive"; in this scenario, they correspond to our intuition that if we draw two curves on the page and they *look* like they intersect, then they actually do intersect.
So do the curves x^2 + y^2 = 1 and y = x actually intersect? Most mathematicians today would say they do at a point "[1/sqrt(2), 1/sqrt(2)]". In reality, the answer is that these two curves actually do not intersect since their point of intersection requires completing infinite processes to complete (in one form or another) so talking about the completion of such a process is nonsensical; they do however intersect approximately, in that I can find an [x1, y1] that satisfies y = x, but can get arbitrarily close to satisfying x^2 + y^2 = 1. So when I write "[1/sqrt(2), 1/sqrt(2)]", I really mean this in the approximate sense that I just explained.
But if you can only procure approximations to "1/sqrt(2)", then why even talk about such a thing as if it were a precise quantity? The reason is because knowing that curves intersect if it *looks* like they do is very intuitive and we like that.

>>12674505
>>Words don't have meanings
Yes, if you define a real number in terms of a number line, and then define a number line in terms of points made of real numbers, that is inherently circular.
Try deluding yourself with infinite processes instead.

>>12674574
Yeah, this sounds like bullshit. We can draw arbitrarily many lines that all intersect but none will with this logic. A unit circle and many other forms don't even exist with this logic. Fuck that schizo.

>>12674583
>talks about the completion of infinite processes
>calls other people schizos
Okay buddy, you do you. You can pretend that your ideal realizations of curves (since such things are only approximate in physical reality in the first place) have properties corresponding to what your physical intuition thinks they should have, but feel free to entertain whatever delusions you want.

>>12674505
Hey uhh chief, you got a nice wall of text there.
But you do know a Hausdorff spaces is defined by its points being neighborhood seperable, right? By necessity a Hausdorff space is defined by the points it contains. So you want to define points on a number line that is already defined by the points on it? Seems pointlessly circular.

>>12674574
What did he mean by this?

>>12674576
Circular reasoning is not meaningless, imbecile. Do you even know the meaning of the word "meaningless?" Also, what OP did is not circular. You made it circular by adding, "then define a number line in terms of points made of real numbers." He didn't make it circular. That was you raising a straw man.

>>12674726
>So you want to define points on a number line that is already defined by the points on it? Seems pointlessly circular.
No, I don't want to. The numbers are the cuts in the line, not the points. Does 4chan know any fallacies besides the straw man? OP said, "Assign," and you chose to paraphrase that as, "Identify with points," which is the only possible meaning in a sea of ways to assign which would make OP's statement circular. It's like your reading it with intention to say that OP is retarded rather than reading it to try to get the gist of OP's idea. Your cognitive frame is inferior and it probably reflects the culture handed to you by your ancestors. Also, if you were in the mailing list for the rough draft of my paper which was never intended for publication then catastrophe is likely to befall you.

>>12674555
Alexandroff line as mentioned in >>12673918
is a Hausdorff 1d connected space, and yet is distinct from the real line.

>>12674569
Physics can use whatever they want, even horoscopes, for all I care, as long as they use it to build models that good predictions. Mathematical rigor is not an issue for the phyicist, but for a mathematician, whose job it is to make sure the abstract theory he talks about makes sense. That means no infinite woo

>>12674804
What is your point there? Are you deliberately ignoring that a circle only extends in both directions as far as its radius so that you can try to fool people who don't know any better into thinking that pic 2.1.1 is not unique and sufficient?

>>12674831
The alexandroff line extends infinitely to both directions.

>>12674789
I'd refrain from saying cuts, because those have an actual meaning, in the context of dedekind cuts. That is actually a valid way to construct the real numbers, going from the rational numbers. Of course, the rational numbers are constructed from the natural numbers, which are in turn constructed from ZF. So I doubt OP meant this.

And yes, the reasoning being circular is insufficient to descrfibe how bad it is. A hausdorff space requires the set theoretic notion of the real numbers to be correctly constructed. Therefore using a hausdorff space to then redefine the real numbers is actually improper.

Of course, you can be my guest and construct a hausdorff space without once invoking the real numbers, it would eb quite a sight.

>>12674830
Mathematical rigor is a fanclub that agrees to respect one person or another's opinion about that level of rigor which is said to be sufficiently rigorous. Opinions are subjective and not objective. An absolute standard of rigor would be objective necessarily. For this reason, there does not exist absolute rigor and everything is framed in terms of sufficient rigor. One can always move the bar of what rigor is to say, "No this other problem that was invented only to show an insufficiency of your thing is the real standard of rigor!" This is what Wildberger is always harping on without mentioning that it is only his opinion that makes the points he raises conflict with rigor. If there were some absolute standard of rigor, then Wildberger could show that it is not met but since rigor is 100% subjective, Wildberger can only say that the issues he identifies are his opinion of rigor even though most other people use a different standard of *sufficient* rigor. I criticize Wildberger because he acts like there is some objective standard of rigor to which he compares things, but there is not.

For instance, who knows of a source that says what the definition of being rigorous is? There is no source. It's always just someone's opinion and it's always just the opinion of other people to say, "OMG, So and So's balls are smart! If he says something is rigorous then it is and if he says it's not then it's not. Also, let's all make fun of Euler now and say he wasn't rigorous! Also, Riemann wasn't rigorous. No one except us is rigorous!" All these people who say, "Since you didn't define 'define' you're not rigorous," never themselves define the rigor which they cite.

Subjective rigor has some value because these people's opinions tend to be good opinions but when mental midgets frame "rigor" as if it was some objective and well-defined standard, they reveal themselves as fools. Sufficient rigor is an opinion and the consensus opinion is in constant flux.

>>12674835
No. The Alexandroff line is a circle. Circles have rightmost and leftmost points, or highest and lowest points. This is a property of circles.

>>12674738
You are just pretending that you can do an infinite number of things by fiat because of ZFC. If you want to engage in philosophical word games where you pretend something exists because you string phrases together, that's fine but that doesn't mean everyone else has to buy into it.

>>12674789
>Circular reasoning is not meaningless,
For making definitions it certainly is. OP did not define what he meant by a number line, so I assumed the logical thing and stated that if he just changed the definition of real number and kept everything else in mathematics the same, then the definition of a real number would be circular.
Nowhere did he mention Hausdorff spaces or anything of that form. I did not make any critique of your definition, since it involves """infinite sets""" and I don't buy into your ZFC word games anyways, so feel free to assume whatever you want.
And finally, if you define A in terms of B and B in terms of A, you haven't actually defined A or B, so I would state that your "definition" is meaningless in terms of actually defining something.

>>12674899
Ummm no it's not? It doesnt have a rightmost or a leftmost point. Why would you say it does?
Genuinely baffled as to what you're talking about.

>>12674844
You seem to ignore that Dedekind ended up with a specific type of cut named after him because he was already familiar with cuts from his primary education in Euclidean geometry. A Dedekind cut is just one specific type of cut like a honeycrisp apple is just one type of apple. A apple can still exist without being a honeycrisp. Does this surprise you? When you say the only "actual meaning" of a cut is in the Dedekind cut it's like you think Dedekind himself invented the notion of a cut in a line which is completely stupid. You can see Euclid making cuts in lines every dozen pages or so in his book. The reason Euclid didn't call them cuts is because that is an English word and Euclid wrote in Greek.

>>12674902
>You are just pretending that you can do an infinite number of things by fiat because of ZFC.
I don't even know what those axioms are and I have to look them up every time someone mentions them. I do not use ZFC at all.

>If you want to engage in philosophical word games
Isn't that what you're doing saying I'm citing ZFC when I did no such thing? Or are you just being fool to entertain yourself with your foolishness?

>For making definitions it certainly is.
Maybe it is you poor vocaublary that leads you to fail to distinguish between fallacious and meaningless?

>I assumed the logical thing
In my opinion you did not. IMO, the logical thing would be to give OP the benefit of the doubt that he he didn't use circular reasoning but instead you assumed the only possible interpretation which would allow you to cite a logical circle. You could have assumed OP didn't use a circle, which I think would have been logical in the absence of any explicitly stated circular reasoning, but you did the opposite of that.

>your ZFC word games
I literally do not even know what those axioms are and it seems like every post you have made in this thread involves you improperly paraphrasing someone and then criticizing your improper paraphrasal. That's called a straw man and it is the most kindergarten-tier of the classical fallacies.

>>12674914
Did you forget to write "sweaty" after you wrote "ummmm no?" Even if you want to use "the theta-hat" direction, the circle only extends from theta=0 to theta= +/- pi.

>>12674977
>>12674899

>>12674888
>>12674555

>>12674977
Dude, just read
https://en.m.wikipedia.org/wiki/Long_line_(topology)
It has absolutely nothing to do with circles and satisfies your definition.

>>12675053
>Uncountably many line segments
>Made of two long rays joined at the origin
How the fuck do long rays exist? They have an endpoint at the origin and a first element, the copy of 0,1 touching that origin. If they have a first element, they have a second element and so on. That is the opposite of uncountable and the whole idea is retarded.

>>12674917
The fact you are comparing geometric cuts in lines to a Dedekind cut shows that you have no idea what at least one of those is. Of course, we could go into the fact a line is fundamentally uncountable as a collection of points, where as a dedekind cut, a partition of the rational numbers, is a countable set. But I think at this point that would probably be a waste of time.

I'm just curious, do you actually have any idea what you are talking about, or do you just like insulting other people about their supposed lack of rigor and intelligence, while being blissfully unaware of the fact I'm laughing my ass off right now.

>>12674977
>I do not use ZFC at all.
Yes you do lol. You depend on it and your ignorance doesn't change that affect. Otherwise, you can't assert that there exists an infinite set: https://en.wikipedia.org/wiki/Axiom_of_infinity

>Isn't that what you're doing saying I'm citing ZFC when I did no such thing?
Once again, your entire "theory" depends on ZFC. You are implicitly citing it because you use it. If you do not rely on ZFC, then it is your job to outline a different, consistent theory of infinite sets.

>fallacious and meaningless
Yes, and the use of this fallacy causes the definition to become meaningless, in that definition-wise, there is no content that is conveyed by saying that A is B and B is A. You have not actually made any progress in defining either what A is or what B is. I would consider that meaningless as a definition.

>which I think would have been logical
In my opinion, it is logical to assume that when somebody introduces something new, you assume the current framework that exists, adjoined with their new definition. Therefore, if their proposed definition of a real number is a point on the number line, then I think it is completely reasonable to assume that the definition is circular since they mentioned nothing else and there was no indication that they had changed what the definition of a line was or that they were referring to Hausdorff spaces.

>I literally do not even know what those axioms
Once again, your ignorance does not excuse the fact that you inherently rely on and use these axioms. If you do not use ZFC, then the onus is on you to exactly clarify what you mean by the terms "set" and "infinite set".
>it seems like every post you have made in this thread involves you improperly paraphrasing someone
I am not improperly paraphrasing anyone. I am just correctly pointing out that every definition of real numbers currently relies on some infinitary concept (e.g. infinite sets), and the reason you think that is okay is by assumption.

>>12675053
Shitcunt, the long line has bounds in both directions. "Infinitely far" means "no bounds over there." You can define the lone ling on the interval [-pi/2, pi/2] with the chart x'=tan(x). The bounds make it not have infinite extent.

>>12675063
>How the fuck do long rays exist?
You can construct long rays with the x'=tan(x) construction given above. Each line segment [-pi/2,0] and [0,pi/2] is a long ray in the x' chart.

>>12675076
>The fact you are comparing geometric cuts in lines to a Dedekind cut shows that you have no idea what at least one of those is.
If I had done that, it wouldn't show that. However, I didn't make a comparison between them, Nice reading comprehension you've got there.

>>12675119
>Yes you do
This is wrong.

>You are implicitly citing it
I am not.

>it is your job to outline a different, consistent theory of infinite sets.
Please tell me more about your opinion. Also, please tell me about how something other than your opinion is the ultimate decider of what is or is not consistent.

>no content that is conveyed by saying that A is B and B is A.
This is so wrong that is ridiculous how wrong it is. What you wrote is basically the most textbook example of "meaning" that I could come up with. In your example, if A is B and B is A, then beyond the plain meaning of the two given facts, we also have additional meaning that "is" is reflexive between A and B, and countless other properties can probably be extrapolated as the meaning of your example phrase. Maybe you mean it's not useful? Your foot-stomping harumph with the word meaningless is stupid. I know what criticism you mean to levy with your harumph but your claim of meaninglessness is preposterous and absurd. How can you deny that "A is B and B and A" has the meaning that the "is" object is reflexive between A and B? This is the definition of reflexivity, you imbecile.

>you inherently rely on and use these axioms
I did not rely on them. You are wrong. I have my own volition separate from your desires for me to have used ZFC. You have not subjugated my autonomous initiative such that I have to do what you say. I didn't use or rely on ZFC, implicitly or otherwise, and your many insistent declarations do not amend the fact.

>I am not improperly paraphrasing anyone.
When you write, "You are just pretending that you can do an infinite number of things by fiat because of ZFC," that is a paraphrasal of what I have done. Once again, you are wrong and your wrongness probably stems from you ignorance of the meaning of the words you use.

>>12675076
>insulting other people about their supposed lack of rigor and intelligence
I never insult people over lack of rigor. I insult people over being wrong sometimes when they are pretending to some authoritative tone with their unqualified wrongness devoid of qualifiers like, "I think," or, "In mu opinion." Lack of rigor is mental midget's cop out when they can't find an error to cite or a contradiction to demonstrate but they still want to fling shit at someone's work or ideas. I cite errors and contradictions.

>>12675239
>Construct long ray by adjoining curves of [math]f^{-1}tan(x)[/math]
Hey dipshit, not only is each curve between each 2pi built of countable copies of 0,1 but the curves themselves are countable

>>12675318
>Construct long ray by adjoining curves of f−1tan(x)
Improperly paraphrase someone to erect a straw man with your own words and then call them stupid because what you wrote was written by an idiot.

>>12675261
>This is wrong.
No it is not.
>I am not.
Yes you are; if you are using sets, then you need a definition of sets, or at least some properties of it. Considering you have still not provided your own infinite set theory, you are implicitly using ZFC. Once again, if you use the word set in an un-traditional manner, it is up to you to define the word.

>Also, please tell me about how something other than your opinion is the ultimate decider of what is or is not consistent.
I still don't see a definition of the word set or any properties of a set from you, considering you don't rely on ZFC (even though you use the term Hausdorff spaces, which requires a definition of the word). It's not about consistency, it's first about definitions.
>reflexive between A and B
Yes, that would be a logical conclusion, *if* you have defined A and B. If you define a real number in terms of a number line, and a number line in terms of real numbers, then you haven't defined either, so reflexivity doesn't even come into question here.
>I didn't use or rely on ZFC,
Okay then, I'm not seeing your theory of infinite sets anywhere, so unless you've defined those words or outlined their properties somewhere, you are speaking out of your ass.
>Once again, you are wrong and your wrongness probably stems from you ignorance of the meaning of the words you use.
Your use of Hausdorff spaces relies on sets, and the most common usage of the word set mathematically comes from the ZFC axioms. If you are using some other axioms or you have some other properties that you endow on sets, the onus is on you to specify.

>>12675380
>nuh-uh
>yu-huh
You are not the decider of my actions.

Anyone in this thread who learned the ZFC axioms before they learned what a set was, please post onus.

>>12675336
Go on then, what's the method of taking the piecewise curves covering 2pi intervals of the integral of tan(x) and connecting them into a single line?

>>12675457
I don't know what you're talking about and even in the context of me not knowing, the interval I cited has length pi != 2pi. Are you replying to the right poster?

>>12675655
the graph of integral(tanx) has period 2pi

>Go on then

>>12675666
What is the point of telling me that?

>>12675687
You were arguing earlier in this thread that the long ray can be constructed by joining each 2pi interval of the graph of integral(tanx). I asked how and instead of answering you wrongly corrected me on the periodicity of integral(tanx).
Anyway, how the hell do you join each of the curves of integral(tanx) into an uncountable object? Each curve occupying a 2pi period on the graph of integral(tanx) can be divided into parts that can be put into a correspondence with N and there are countably many curves.

>>12675722
>You were arguing earlier in this thread that the long ray can be constructed by joining each 2pi interval of the graph of integral(tanx)
If that was true, the you could link to the post with that argument and you could copy the words I used into green text directly without replacing them with your own words. You cannot do that, however, because your claim is false.

What the fuck is the malfunction of these people's thinking?
>straw man
>straw man
>straw man
>straw man
>straw man
>straw man

>>12675739
You said it in >>12675239

>>12675751
>>12675739
>you could copy the words I used into green text directly
I did not.

>>12675765
Here's the quote of you
>You can define the lone ling on the interval [-pi/2, pi/2] with the chart x'=tan(x). The bounds make it not have infinite extent.
I was wrong, you weren't saying the curves could be joined up into something uncountable, you were saying that the segments in a single curve with a domain of 2pi length in integral(tanx) could be constructed to be uncountable on their own! When in fact, any selection of segments with a decreasing non-zero measure on a continuous asymptote is countable. That's an undergrad mistake, dude.
Take a break and when you come back, either formulate a construction of the long ray that isn't nonsense or disavow the silly concept.

>>12675790
>you were saying that the segments in a single curve with a domain of 2pi length in integral(tanx) could be constructed to be uncountable on their own!
No you wrote that. I didn't write that or mean that, and I don't understand what you mean when you write it. You're not making an undergrad mistake, you're making a preschool mistake.

>>12675801
That's what a chart is. You said
>the chart x'=tan(x)
You cannot construct anything uncountable from solely the chart of a continuous asymptote on any subset of [math]\mathbb{C}[/math]

>>12675833
I can tell your intention is to be retarded on purpose by your six times repeated insistence on misunderstanding me without asking me what I meant.

>>12675841
Okay. What did you mean by
>You can define the lone ling on the interval [-pi/2, pi/2] with the chart x'=tan(x). The bounds make it not have infinite extent.

>>12675854
If we take a generic line segment AB and equip it with a chart x such that the interval representation of the line segment is [-pi/2,pi/2], then we may equip it with another chart x'=tan(x) such that the interval representation of AB in the x' chart is [-inf,inf]. We have already constructed two intervals in the x chart which are [-pi/2,0] and [pi/2,0]. In the x' chart, these are the long rays [-inf,0] and [0,inf]. This answers the original question:>>12675063
>How the fuck do long rays exist?
The long rays are two halves of AB. That's how they exist: as two halves of a lines segment.

>>12675914
The resulting construction is just the real line, it's not longer in any way. All you've done is rename an existing continuous asymptote and it doesn't reach infinity any differently to how it did before you started.

>>12675933
The real line is (-inf,inf). I have constructed [-inf,inf] which is the long line. Adding a chart called x' does not execute an endpoint deletion operation but such an operation would be required to start with an interval of the form [a,b] and then end up with one of the form [c,d). Your words about "reaching infinity" are a non sequitur. The interval can't be closed in one chart and then half-open in the other chart. That would violate the bijectivity of the tangent function on the interval [-pi/2,pi/2]. Are you familiar with bijection?

>>12675971
I got all of this from Based Penrose, btw.

>>12675971
Based

>>12675971
>[-inf,inf] which is the long line
nope. https://en.wikipedia.org/wiki/Long_line_(topology)

>E = R
bold

>>12674403
No, I assigned a sphere to the line.

>>12673538

Good job triggering half /sci board. I always have a good laugh reading the R haters fucktards. Cheers on you, anon.

>>12675971
> I have constructed [-inf,inf] which is the long line
You have genuinely no idea what the long line is do you?
As I said, read the article. It's right there:
https://en.wikipedia.org/wiki/Long_line_(topology)
What you have constructed is topologically equivalent to the closed interval [0,1], which is very different from the long line.
For example, your space has a countable basis (by looking at intervals around rational points), while the long line doesn't.

>>12677406
>I always have a good laugh reading the R haters fucktards
Because you're a high schooler with no knowledge of foundations?

>>12677445
Cheers on you too. Never change please.

>>12674282
How do you do without formal languages then? Or do you use them in some way without accepting at most the countably infinite?
https://en.wikipedia.org/wiki/Kleene_star

>>12678203
How do you do mathematics without formal languages?
Open literally any mathematics paper or textbook that's not on foundations to see.

Yes, I have conflated the long line and the extended line in error. To construct the long line following my previous remarks, remove the interval [0,1] from [0,inf] and then operate with parity and translation. Let the parity operator be such that
P{[a,b]}=[b,a]

and let the translation operator be such that
T(x){[a,b]}=[a+x,b+x]

Then, using
[0, inf] = [0,1] union S

we obtain the long line as T(-1){P{S}}.

>>12679071
>P{[a,b]}=[b,a]
[a,b] only makes sense as an interval when a<=b. So your P only makes sense when a=b, when it doesn't do anything.
I think you meant something else.
You didn't specify what you meant by S, I assume you meant (1, inf]. In that case, what is P{S}? Why are you then translating by -1?
None of this seems even remotely similar to how the actual construction of the long line looks like.
Do you want to go through the construction together?
Let's take the first line:
>The closed long ray L is defined as the cartesian product of the first uncountable ordinal ω1 with the half-open interval [0, 1), equipped with the order topology that arises from the lexicographical order on ω1 × [0, 1). The open long ray is obtained from the closed long ray by removing the smallest element (0,0).
Are you familiar with ordinals, order topology, lexicographical orders?

Yes, I have conflated the long line and the extended line in error. To construct the long line following my previous remarks, remove the interval [0,1] from [0,inf] and then operate with translation. Let the translation operator be such that
T(x){[a,b]}=[a+x,b+x]

and let it be invertible so that T(-x)T(x) is the identity operation. Then, using
[0, inf] = [0,1] union S

we obtain the long line as T(-1){S}. I anticipate that the other poster will write, "This is not the long line because there are not uncountable unit elements concatenated." To refute that wrong idea, let
T(-1){S} = [x,y].

Since it is obvious that x=0, if this was not the long line then y would be a natural number. The naturals are closed under addition, for any other natural number n, the sum y+n is also a natural number. However, from the definition of the translation operator, we have
T(1){[x,y]}=[x+1,inf]

which implies that y+1 is not a natural number: a contradiction. T(-1){S} is the long line.

>>12679103
>a,b] only makes sense as an interval when a<=b
I noticed that already and corrected it here: >>12679116

>You didn't specify what you meant by S
I certainly did. I defined it as the complement of the unit interval in [0,inf].

>Do you want to go through the construction together?
No.

>Are you familiar with ordinals, order topology, lexicographical orders?
No. Are you familiar with fractional distance?

Yes, I have conflated the long line and the extended line in error. To construct the long line following my previous remarks, remove the interval [0,1] from [0,inf] and then operate with parity and translation. Let the parity operator be such that
P{[a,b]}=[b,a]

and let the translation operator be such that
T(x){[a,b]}=[a+x,b+x]

Then, using
[0, inf] = [0,1] union S

we obtain the long line as P{T(-1){P{S}}}. I anticipate that the other poster will write, "This is not the long line because there are not uncountable unit elements concatenated." To refute that wrong idea, let
P{T(-1){P{S}}} = [x,y].

Since it is obvious that x=0, if this was not the long line then y would be a natural number. The naturals are closed under addition, for any other natural number n, the sum y+n is also a natural number. However, from the definition of the translation operator, we have
T(1){P{T(-1){P{S}}}}=[x+1,inf]

which implies that y+1 is not a natural number: a contradiction. P{T(-1){P{S}}} is the long line.


>>12679103
>a,b] only makes sense as an interval when a<=b
I noticed that already and corrected it here with one extra P.

>You didn't specify what you meant by S
I certainly did. I defined it as the complement of the unit interval in [0,inf].

>Do you want to go through the construction together?
No.

>Are you familiar with ordinals, order topology, lexicographical orders?
No. Are you familiar with fractional distance?

>>12679128
>>You didn't specify what you meant by S
>I certainly did. I defined it as the complement of the unit interval in [0,inf].
You didn't tho lol. I looked over the previous post and your new post several times and couldn't find where S was defined. I can only infer it from the context.
>I defined it as the complement of the unit interval in [0,inf].
Which is (1, inf].
>No. Are you familiar with fractional distance?
No. But ordinals are a really fascinating topic that I think you will appreciate if you learn about them.
They're basically a generalization of the counting numbers. You declare w as the set of all natural numbers, then declare w+1 to be the union of w and {w}, in general for an ordinal a you define a+1 to be the union of a and {a}. The ordinal a is the set of all ordinals smaller than it.
You can keep doing this to get w, w+1, w+2, w+3, and then define w+w as the union of all these ordinals, and keep going.
The ordinals have the property that for any property which at least one ordinal satisfies, there is the smallest ordinal which satisfies it.
So talking about the least uncountable ordinal w_1 makes sense, since the power set of N is uncountable and by axiom of choice it can be well-ordered.

>>12679147
>I can only infer it from the context.
The equation that had S in it was the definition of S.

>>12679155
Not really, since it doesn't define it uniquely.
[0,1] union [1,2] = [0,1] union [1/2, 2] and yet
[0,1] != [1/2,2]

>>12679147
>So talking about the least uncountable ordinal w_1 makes sense
In fractional distance, the non-arithmatic numbers are like that.

>>12679161
That's true but you would have to be an asshole to pretend not to understand what I wrote.

>>12678357
You are right about that. But anyways you are missing infinite posibilities.

>>12679198
No I'm not lol. There is an infinite/unbounded supply of rational numbers, each one a different possibility.
The only thing I'm missing is illogical "completed infinite" onsense.

>>12679169
In what sense?

>>12679219
There's uncountably many of them and there's a least one.