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Owing to the work of Dedekind, people today assume that mathbb{R} is a set which makes mathcal{R} a complete ordered field. However, the axiom that mathbb{R} has this property is not only (a) not motivated, in fact it is (b) not motivated at all and only assumed because it is convenient.

Indeed, it is circular to to claim that mathbb{R} is the set which makes mathcal{R} a field because mathcal{R} uses mathbb{R} in its construction. Therefore, we need to come up with some other definition for what mathbb{R} is.
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>>10799269
In this sense isn't [eqn]mathbb{R}\subset mathcal{R}[/eqn]? Because it's saying the reals are contained within the set which I'm assuming is the real field?
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>>10799269
>>10799299
Oh wait that's what you said nvm. But idk mathbb R is just the set? Why would it need further definitions?
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Pic related, might it be completely retarded not to make a distinction between "the real numbers" and "a model of the real number system" including the real and two operations?

>>10799299
Depending on the assumptions you make about how things are defied, you can definitely get things organized so that mathbb{R} is a subset of mathcal{R}. However, is there any reason to prefer that organizational scheme over another one?
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>Few mathematical structures have undergone as many revisions or have been presented in as many guises as the real numbers. Every generation reexamines the reals in the light of its values and mathematical objectives.

F. Faltin, N. Metropolis, B. Ross and G.C. Rota. The real numbers as a wreath product. Advances in Mathematics, 16(1975), 278–304.
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The less abstract things get, the more civilized people become. It's a fucking shame because I'd really like to have a discussion about dimensions and spacetime and shit without some self important fuckwit calling me a shit eating retard 12 times when he doesn't even understand the topic. Yet, I have no patience for math. How do I into math better?

It's my first day on /sci/ but I've read books and college and stuff so I'm not just running around making IQ threads and asking about Elon Musk. What's the culture here? Answer in the form of an interpretive math equation if you don't feel like disturbing the flow of conversation.
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>>10799387
Well if your wanting to talk maths more formally that heavy dose of abstraction is needed because it's useful for generalizing an idea rather then going through an exhaustive list of non abstract ideas. Abstraction allows us to look at the bigger picture. So idk if your not comfortable with the abstractness then mathematics strictly shouldn't be your convo topics. Perhaps physics ideas are more your taste.
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>>10799314
Sorry the first post just ignore that. So to define the reals on its own would consequently mean to define the other sets contained in R right? I mean that's the only reason why the reals exist anyway. So perhaps it's looking at the sets contained in R and defining them. Which for the most part are very straightforward definitions wise.
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>>10799400
Abstract is what I'm good at. My brain moves around too much to focus on the tiny steps which is why seeing bigger pictures feels so much easier for me. The problem seems to come - even outside of this discussion board - when I encounter others willing to converse but unable to see the same picture I do. They seem to get angry and mean about it.

I also wish I was more patient for math. Is that an acquired skill? I just passed my trig class by the skin of my teeth with a 90 so I'm not completely retarded but if I had three wishes two of them would be to be an expert at calculus. I don't think it's an intelligence issue for me I just think it's an attention span issue. Or I could be making excuses for being a tard, who knows?
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>>10799400
I was really just commenting on how nobody is calling eachother a retard or shitting on eachother in this thread and how it's nice, and I'd like to find conversations like that somewhere.
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>>10799269
>However, the axiom that mathbb{R} has this property is not only (a) not motivated, in fact it is (b) not motivated at all and only assumed because it is convenient.
Isn't that the whole point of axioms?
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>>10799439
>Abstract is what I'm good at.
>The problem seems to come when I encounter others willing to converse but unable to see the same picture I do.
If everyone else sees a different big picture than you, did you consider the hypothesis that perhaps you are not in fact good at abstract analysis?
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>>10799314
>Pic related, might it be completely retarded not to make a distinction between "the real numbers" and "a model of the real number system" including the real and two operations?
No, it might not. "The unique model of <axiomatic system>" is exactly what mathematical structures mean, in the end.
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>>10799447
I happen to be the select few on sci to not question people's intelligence seeing as I'm insecure of my own. However this board does have its moments of when there's actual good conversation. If it's percentage based then I would say about 10% of the time the threads are civil. I probably being to generous with that number honestly haha
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>>10799466
I would tend to agree, as long as the manner with which I'm disagreed is cogent and clearly expressed with definition. That is to say, I'd very much like to comment on the intelligence of people I've had abstract conversations with, but let's instead just say that the above requirement for a rebuttal I'd accept usually isn't met. Or attempted.
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>>10799503
It's really hard to call a significant number of people idiots when trying to defend an undefined statement without seeming like I'm trying to make a claim to intelligence I think I have but actually don't.
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>>10799503
>just say that the above requirement for a rebuttal I'd accept usually isn't met. Or attempted.
I'm sure that's true, for 90% of everything is crap; I'm sure 90% of the people trying to rebut you make a complete mess of their supposed counterarguments. But if the minority that is sensible and cogent ALSO sees things differently from you, that should be quite a hint.
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>>10799387
>What's the culture here?
4chan is the most vile website on the world wide web. The people who heckle me here are child molesters, human slavers, and torturers.
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>>10799439
90 is literally good enough. Where people struggle is below 70 that's when you can tell they need help. If your getting 80 or higher in math classes then you mostly understand what's going on. As for the abstract thinking thing.. when I talk to people about abstract topics (usually math) they say "ugh my head hurts stop!" or "no more talking about this" basically any statement that decapitates the conversations. Immediately shut down. Now if you are talking to people with an interest and they have a different version of what your trying to explain then idk involve more people and come to some sort of conclusion? That would need more of an example seeing as that could vary depending on the conversation topic.
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>>10799513
I've had some good conversations that changed my mind or gave me pause, I'm not saying it's 100%, but it definitely feels like 85-90%. If those statistics were to be factored in I'm inclined to believe I'm not too shabby at abstract analysis.
>>10799514
I meant just /sci/. I've been around other boards for an indeterminate number of years.
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>>10799448
>Isn't that the whole point of axioms?
I suppose so, to an extent. However, it seems like an overstepping of convenience to say, "The entire set of real numbers is such that the real field is
[math] \mathcal{R}=\{ \mathbb{R},+,\times \}[/math]"

when a much more general statement, "The entire set of R contains some some subset R_0 such that the real field is
[math] \mathcal{R}=\{ \mathbb{R}_0,+,\times \}[/math],"

generates every advantageous analytical condition give by the former statement and still preserves the geometric notion that a number is a cut in a line.

So... that point you make is, indeed, the purpose of axioms but axioms which are needlessly restrictive are not great axioms. What is gained by requiring that all of R satisfies the field the axioms when defining R such that it contains a subset satisfying the field axioms gives all the same tools and also preserves the geometric notion of what a number is?
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>>10799527
I'm talking more about open discussion areas like here where a general statement is made and the main idea is disregarded because they don't understand one of its pillars.

For example, one guy recently tried to argue with me that a singular point on a graph has spatial dimensions. Like, the point on the graph itself should be measurable, by itself, with no accompanying lines. Like it's a circle. He said if couldn't be then it wouldn't exist because it's "nothing".

That's an example of what I run into the most, and what has me feeling down and dumpy enough to interrupt a comfy math thread so civil logicians can give me a shoulder to cry on.
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>>10799466
>If everyone else sees a different big picture than you
What if everyone in your sample of "everyone else" is a member of some vile community known for child porn and snuff porn, and you are on a glorious SJW quest against such things, and it is their intention to shit on whatever you do wherever you go because, deep down, they know you are trying to kill them?

>>10799476
>No, it might not.
You don't think it might be retarded to fail make a distinction between "numbers" and "number systems?" Why give them two different names if not because they are two different things which need to be distinguished?
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>>10799549
Ah I see, well yeah it's rare to come across threads like these haha so enjoy it while it lasts
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>>10799527
No. Everyone who gets a B in any university math class really deserves a C. Everyone who gets below a B really deserves to fail. Sorry but it's true. The homework grade and the curve are already saving you, as well as any other extra shit your instructor does to help, such as open note tests, projects, and dropping your lowest scores. If you get a B you likely understood a few basic, easier concepts near the beginning of the term. If you get a C, you likely didn't understand jack shit. But I guess you wouldn't know this, since you're still taking high school level math.
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>>10799816
>Sorry but it's true.
it is true
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>>10799816
This place sure is edgy. I think I've seen your posts in every thread though, there's always one guy in every thread who is invariably exactly this toxic with this sort of tone and weak, angry prose. I'll assume it must all just be you.

And just so you know, I can tell you don't quite fit in here. You're not bad at faking topic knowledge but to anyone who actually is having a discussion and is knowledgeable of it, you're getting get spotted from a mile away. Not sure if you're capable of shame but if you are the best way to avoid it might be to troll on a different board.
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Has anyone ever bothered to do the definition purely from ZFC?
Define N as some sequence of nested sets.
Define Q as equivalence classes of NxN
Define R as equivalence classes of Q^N (Cauchy equivalence).
Define all the necessary functions (+, *) and prove they are well defined.
prove they satisfy the field axioms
Define the order on R, prove it is a total order.
Do any of these steps not really work? Am I done here?
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>>10800434
Defining R as equivalence classes of cauchy sequences is quite common and yes imo it's much more natural than dedekind cuts (and in most countries that aren't the US this is the more common definition taught).
In short, yes you're done here.
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>>10800462
>>10800434
>>10799269
Also, this is your answer OP, this is the natural way to define R.
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>>10800429
You think everybody who thinks you're an idiot is just one guy? You're delusional.
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>>10800434

Better than ZFC, CIC: http://corn.cs.ru.nl/.
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I think I can use pic related to build the Cauchy sequence definition of R for reals in the nbhd of infinity
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>>10799269
Jon it doesn't even matter how you define R, your argument has been disproved in multiple ways. See https://warosu.org/sci/thread/S10782732#p10784713

Your only response was to say that the quotient is defined for x/x, which doesn't show that x can be a line segment, nor does it explain how to determine that AB = 0.5XY without referring to length. You then ignored the main arguments.
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>>10799269
BOOORING
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>>10801062
>nor does it explain how to determine that AB = 0.5XY
did you miss the statement:"f we define AX = 0.4AB then he quotient (AX) / AB =0.4 ?"

>doesn't show that x can be a line segment
yes it does

I see what you are doing here but your posi-neg pseudo-trolls are going to earn you about the same thing the others get, and it would be better for you to write nothing at all than to write a posi-neg. If you have more respect for the gay troll etiquette rules of the literal worst website on the internet than for the laws and decrees I set before your ancestors then you are not going to like your rewards.
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>>10801394
>if we define AX = 0.4AB
If I say that cats and bears are special iff cats = 0.5*bears then that doesn't tell me how to determine that cats and bears are special, because the value of cats and 0.5*bears are undefined in general.

>yes it does
Wrong.

>I see what you are doing here but your posi-neg pseudo-trolls are going to earn you about the same thing the others get
Keep your delusions to yourself schizo.

Ahhhh another post where you fail to respond to my arguments. Feels good.
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>>10801709
>doesn't tell me how to determine that cats and bears are special
Please state your reasoning for why you think it should tell you this when the definition has nothing to do with that? I have also not told you how to determine the price of tea in China, and I don't see how either of these two non-specifications is a problem. For some reason, however, you feel entitled to be able to determine the "specialness" of cats and bears from this definition:
X is Y iff Z=0.5.

Why do you care if cats and bears are special but you don't care if China has a bulk discount on Earl Grey? Neither of things is claimed to be specified in the definition.

> the value of cats and 0.5*bears are undefined in general.
Like way "x" and "0.5y" are undefined in general until you give them "a definition" such as:
x = 0.5y ?

>my arguments
You didn't make any arguments. You stated that you can't determine the specialness of cats and bears. This is not an argument. Neither did you argue for why you should be able to determine their specialness from a definition that has nothing to so with specialness.

In your earlier post, you stated "nor does it explain how to determine that AB = 0.5XY without referring to length" but you did not make argument for why this definition should do that? Do you think my definition claims to convey this information? If so, please cite the string of words which leads you to this belief.

My argument is that those things you say are not problematic because the definition does not claim to define those things. The definition says "X is Y iff Z = 0.5." I think your grasping for something to whine about if you don't think that's a perfectly valid definition. Lets say x and y are real numbers and I have a definition which says
"x is special iff x/y = 0.5"

This definition doesn't tell you how to compute x/y. Do you feel like this is a malformed definition? You must, because it is the same as the other definition which you claim is malformed.
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>>10801709
>Ahhhh Feels good.
This is not a very good quip. I give it 0.1 / 10 meme zinger rating. Here's my meme zinger:
YOU DIDN'T MAKE AN ARGUMENT!!! What meme zinger rating do you give me?
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>>10802203
>Please state your reasoning for why you think it should tell you this when the definition has nothing to do with that?
Did you forget what this is all about Tooky? Go read the thread.

>Like way "x" and "0.5y" are undefined in general until you give them "a definition" such as:
>x = 0.5y ?
Yes, variables and operations on those variables need to be defined.

>You didn't make any arguments. You stated that you can't determine the specialness of cats and bears.
Did you read the rest of the post I linked to or did you think only responding to the first paragraph was sufficient?

>Lets say x and y are real numbers and I have a definition which says
>"x is special iff x/y = 0.5"
>This definition doesn't tell you how to compute x/y.
Are you fucking retarded? Division and equality are well defined for real numbers. That's exactly my point.

You are failing to understand the basic argument of my first point and completely ignoring the rest, because you have no answer and you know it.
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>>10803487
>Division and equality are well defined for real numbers. That's exactly my point.
Ok then, genius. Let x and y be two real numbers. How do you determine the quotient
x / y = z?

What does z equal?
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And really, just to we can cut to the chase, the argument that this individual or bot
>>10803487
is that pic related isn't a perfectly well-posed math problem.
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>>10803558
Construct the reals by Dedekind cuts. Division is defined for positive reals X and Y as

X / Y = { x / y: x ∈ X ∧ y ∈ ( Q \ Y) }

Where Q is the set of rational numbers and Q \ Y is the relative complement of Y in Q.
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>>10803567
It's not, retard. in most cases we would assume that these variables are numbers. In your case they are line segments, not numbers.

And you are still ignoring the rest of the post. I'll post it again so that you can't hide anymore:

>2) Hilbert's discarded axiom guarantees two circles with two points of intercestion. This is property of line segments that exists independently of the metric.
That is insufficient though, what you need are two circles with equal radius. As I already explained, this is only independent of the metric on actual line segments (finite with two endpoints). If you are going to use a "line segment" that is infinite with two endpoints then the metric matters because a circle with infinite radius centered at a point infinitely far away from 0 is indeterminate.

>You argument is that Hilbert was wrong.
Why are you lying Jon? I did not say Hilbert was wrong, I said you misinterpreted him. If you need to resort to lying then you have already lost the argument.

>define c to be that number.
There is no such number since c/inf = 0 =/= 0.5.

>I choose both.
Then your argument contains a contradiction and fails. Thanks, that was easy.

>The points do exist. That's why I used the two charts. If they exist in one, then they exist in the other.
Incorrect, as I've already proved.
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>>10800434
Lastly you need to show that R satisfies the LUB property.
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>>10803607
For someone who goes on about me ignoring you, you didn't answer my question: "What does z equal?"

>That is insufficient though, what you need are two circles with equal radius.
Pic related, use Hilbert to arrange ABCD, then define AC = BD, then use AC and BD as the two equal radii. The metric has nothing to do with it, and the only reason you mentioned the metric is because you needed a red herring to try to hide your straw men with.

>I did not say Hilbert was wrong
It seemed to me like you were saying it is only possible to arrange ABCD per Hilbert when you attach a certain metric. To the contrary, Hilbert's axiom says that it is always possible. Do you agree that it is always possible or do you think Hilbert is wrong?
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>>10803613
>There is no such number since c/inf = 0 =/= 0.5.
Nice claim you have there, try proving it.

>>10803661
>Lastly you need to show that R satisfies the LUB property.
I've proven that it does not have this property. However, if you throw away the definition of R that was used for thousands of years about the geometric notion of what a number is and then you use the Dedekind construction of R, then everything I'm saying is wrong and the point is irrelevant.

However, the point I'm advocating is that it was not a good idea to get rid of the geometric notion of what a number is. Also, another irrelevant point since I will destroy your dev team, you are using the "see earlier" prose construction too much to be believable.
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>>10799830
Did you just make a reaction gif of yourself?
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>>10803804
>For someone who goes on about me ignoring you, you didn't answer my question: "What does z equal?"
I did. What a moron.

>Pic related, use Hilbert to arrange ABCD
ABCD don't exist as you've described them. You need two circles with equal radius. As I already explained, this is only independent of the metric on actual line segments (finite with two endpoints). If you define AC = BD then AD cannot be infinite. If you are going to use a "line segment" that is infinite with two endpoints then the metric matters because a circle with infinite radius centered at a point infinitely far away from 0 is indeterminate.

>It seemed to me like you were saying it is only possible to arrange ABCD per Hilbert when you attach a certain metric.
No, nothing I said implies that. The problem is in you trying to have ABCD centered on a midpoint that doesn't exist on an infinite line. On an infinite line, AB and CD cannot be equal. Hilbert's discarded axiom is completely irrelevant towards proving your claim.

>>10803811
>Nice claim you have there, try proving it.
https://en.wikipedia.org/wiki/Extended_real_number_line#Arithmetic_operations
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>>10803811
>However, if you throw away the definition of R that was used for thousands of years about the geometric notion of what a number is
>>10799547
>the geometric notion that a number is a cut in a line.

Rewriting history, eh? We've been over Euclid's Elements before, with the magnitudes and ratios, and the integer multiples.
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>>10803869
No, I made it last year

>>10803876
>You need two circles with equal radius.
Axiom has nothing to do with circles. Axiom is about points four points in a line.

> If you define AC = BD then AD cannot be infinite.
Nice opinion you have there, try proving it.

>a midpoint that doesn't exist on an infinite line
Nice opinion you have there, try proving it.

>https://en.wikipedia.org/wiki/Extended_real_number_line#Arithmetic_operations
not a proof.

>Rewriting history
It is you all who rewrite history when you say the thing from 1872 is part of the problem that was defined in 1859.



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