So what's wrong with treating the number line as a big circle with infinity opposite zero where passing either switches parity?
That's
https://en.wikipedia.org/wiki/Real_projective_line
and has it's uses.
>>7680696
If you mean using it in replacement of the ordinary number line: you cannot possibly set a scale. All numbers will be infinitely close to 0 and the number circle becomes useless.
>>7680696
No room for hyperreals
>>7682256
Why can't you put them at 0 and infinity?
>>7680696
Bit of a math pleb here but recently learning about this stuff, would this have any effect on set theory/contiumm hypothesis stuff if we made it a circle instead of a line? I'm thinking no but just asking.
>>7682253
Kek, of course you can set a scale.
X|-> X/(1+abs(X)) for x on the real line and with + infinity represented by 1, etc.
>>7682268
I doubt it. This circle has the same cardinality as the real line.
>>7682300
Right ok I figured. What would be the value of representing it like this then? Just easier to visualize?
>>7682304
It has some application in metric spaces, I guess. It's probably just interesting.
It's isomorphic to any closed, real, interval, so maybe you could carry some theorems over.
>>7682312
Kinda off topic now but what is the general opinion on the contiumm hypothesis now? I know its still not proven true or false but do they think its "probably" true or "probably" false?
>>7682319
It's an undecidable proposition. It's not like it's simply unproven. It can't be proven or disproven.
At least if I remember correctly. It's not really in my branch of triple integrals.
>>7682327
Well I know its proven to be undecidable within ZFC but that doesn't necessarily mean it is totally undecidable.
Even Gödel thought it could be solved, not within ZFC, but he thought it could be solved.
>>7682340
I wouldn't worry about it until something besides the ZFC is accepted.
![1448511381942[1].jpg](https://i.imgur.com/iZpX2Kl.jpg "1448511381942[1].jpg")
I got you a moral dillema /sci/
Say you have to be somewhere that you want to be and on your way there you see 2 car crash when clearly it was one guy's fault only(probably girl's, but whatever, it's irrelevant). Noone else is there to see it so you have to testify. You receive no consequences if you don't go to testify, but the driver at fault may get away with it. You don't and most probably won't know any of the two people involved. If you go testify you'll spend 5 to 10 hours in a bureaucratic setting.
Two questions;
-What's the logical thing to do?
-What would you do?
>>7682367
oh I didn't create a new thread. My bad.
>>7682300
not a good test at all. there are many nonisomorphic models of other theories which have the same size. for example, there are uncountably many nonisomorphic countable linear orders.
also, i feel like this fucking happens on sci all the time. your question was answered, accurately, here >>7680731 immediately, and desu you should just go read about properties of projective geometry before doing anything else, or you're just shitposting, because there is a clear followup to this idea, and there is already a wealth of knowledge about it.
>>7682319
continuum hypothesis is independent of ZFC, meaning you can choose either to have it or not, and it will be consistent with ZFC. Even "worse", asserting the truth of the continuum hypothesis does not mean that all "continuum hypotheses" hold (i.e. it does not imply that all X<Y<2^X X infinite imply Y=X or Y=2^X). I believe you actually need infinitely many independent "continuum" hypotheses to make all cardinals strictly those formed by exponent taking.
KEK, is this what people are being taught in STEM? Random meaningless shit about circles that has no application in real life? Buhahahahahahahahahahaha, what a waste of fucking time lmao.
>>7682363
Ah ok. So we haven't really made any significant progress on it then.
It may or may not have a solution but its definitely not in ZFC and we don't have anything better yet. That's still where we're at.
>>7682385
>unless it can be used to make me new toy its useless
Nice meme
>>7682391
Yup, sounds about right
>>7682385
projective geometry has applications in physics, optics, control theory, and number theory (so encryption), and in many ways complex projective space is the most natural algebraic geometry setting, more fundamental than the real number line. obv not interesting if you dgaf about history of science or philosophy (or any of the applications), but if you like to think of yourself as educated in geometry at all, you probably want to know that this exists (bezout's theorem is probably the most fundamental result of algebra and geometry).
>>7682253
>If you mean using it in replacement of the ordinary number line: you cannot possibly set a scale. All numbers will be infinitely close to 0 and the number circle becomes useless.
It appears as a normal number line at any given points, since an infinitely large circle has next to no curvature.
>>7682373
OP here. I was reading some on projective geometry, but I'm having a hard time wrapping my head around how you would graph functions on this.
>>7682418
Form an isomorphism from the extended real line to the projected version, and apply that isomorphism to the original graph of the function.
>>7682418
just to start, just think of it like normal, but add "points at infinity" where necessary (how much can you "really" picture the infinitely far points of a regular graph anyway?).
To convince yourself something geometric is happening, use the regular projection picture (hinge a line onto the circle, then associate each other point on the circle with the unique point it passes through on the plane, which always exists unless the line is tangent - i.e. the point at infinity). Convince yourself that this is just the "normal" real line with a single point infinitely far away from any point on this line. Convince yourself of the same for a sphere sitting on a plane, with the line hinged on the sphere on the point opposite from where the plane is tangent. When the line is not parallel to the plane, you will again have a point on the sphere corresponding to a point of the plane, otherwise the point at infinity.
If you use homogeneous coordinates taking values in a field, you can always scale a fixed coordinate to 1 and use the homogeneity.
Practice just drawing the distortion onto the circle ((unions of) line segments go to arcs on the circle), and then sphere. Basically, you can see behavior at infinity as part of the regular geometry, distorted to act the same way generically "far from the origin".
>>7682445
Came up with an idea where basically, you've got the two axes which basically comprise the surface of a sphere, and the x-coord is distance from the origin and y-coord is distance 'up' from the x-coord. Has the interesting property of (infinity,infinity) being at (0,0).
>>7682500
Sorry, but someone beat you to it. Look up the Riemann Sphere
>>7682503
Well that's disappointing, but not surprising.
Seemed like the only satisfying way to set it up, so it figures someone else thought of it before me.
Since we're on the topic of projective geometry, can somebody recommend a good beginner textbook?
I've taken up to Abstract Algebra if that helps.
>>7682376
Isn't this what the generalized continuum hypothesis is for?
this is fucking cool
>>7682997
What is anon?
>>7682304
The projective line is the "compactification" of the real line. This just means that any convergent sequence has its limit point in the projective line (because we've added a "point at infinity").
In algebraic geometry, everything is easier to state projectively, precisely because of these limit points. You know how generally you expect any two lines in the plane to intersect, but parallel lines ruin this would-be-theorem? You don't get this pathology in projective space -- any two lines at all in a projective plane intersect! I can't speak for anything outside of algebraic geometry, because this is my field.
>>7682531
Shafarevich: Basic Algebraic Geometry I
>>7682686
it's why you need it independently if you want it
