Infinity

[granite maple]

Can you explain the problem again? I'm afraid I don't understand

[copper vale]

like what infnity actualy is ?

[trim prawn]

infinity is a concept not a number

it just means "as u get rlly big" or something like that

like if i take the limit to infinity, it just means as x gets rlly big

[copper vale]

imagine there is a hotel with infinte rooms and each room is booked like

[granite maple]

Oh this question

[copper vale]

in room1 =1st men

in romm2 = 2nd man

soo on to infinte

now a new man come a get a room how?

[granite maple]

You tell everyone in the hotel to move down one

At least thats the intended solution

It doesn't quite make sense if you try to think about it realistically but the whole point of the hotel question is so that it's a simplified version of explaining bijections( comparing the sizes of two sets by matching up elements)

[copper vale]

exactly we make infnity smaller by of room by adding more man in the same infnity

[granite maple]

Uh, not quite. I don't think it will make infinity smaller per say

[trim prawn]

infinity+1 is still infinity because infinity is not a number but a concept

[brave vortex]

infinity can be seen as an accumulation point of values

in the riemann sphere sense 0 is farthest from infinity

[trim prawn]

i believe the hotel problem also introduces countable vs uncountable infinity

very trippy to think about for the first time

[copper vale]

think like a angle of circle now infinte angle is at zero

[brittle egret]

infinity is an entire class of numbers which people like to all give the same symbol to.

[granite maple]

The only reason the angle of a sector goes to 0 as you split it into infinitely many segements is because you are taking a final value ( 360 degrees) and divide it among infintally many portions so they all get a very small amount. Here, on the other hand, we have a number and we a simply adding one( so just normal addition).

[copper vale]

A runner wants to reach a wall 10 meters away.

But first he must go half the distance.

Step 1: run 5 m

Step 2: run 2.5 m

Step 3: run 1.25 m

Step 4: run 0.625 m

This keeps going forever:

he have to take infnite step to reach the zero

[granite maple]

you mean the other side?

[granite maple]

He may take infinite number of steps but he will travel a very finite distance

in this instance

[copper vale]

when will it reach the oter side ?

[granite maple]

Well, it depends on how fast he is. For example if he traveled 1 meter per second it will take him 10 seconds to reach the other wall.

[copper vale]

ok

infnity-1 = infnity right?

[granite maple]

Yea

[brittle egret]

in an infinite amount of steps,
the step length is 1/2^n and the time is also 1/2^n. because these are identical infinities they cancel and leave us with a normal number

[copper vale]

so infnity - any number = infnity right ?

[granite maple]

Yea

[brittle egret]

infinity - 1 = a different infinity
there's a symbol for this but i forget what its called, (omega mabey??)

[copper vale]

infnity - (the no. one less then that infinity) = infnity or 1

[granite maple]

Aleph null? Also I'm not sure if it's a different infinity. It probably has the same cardinality

[brittle egret]

why can concepts not be used in equations?
they should function just like unknown variables.
as far as i understand math the point is that its logically consistent and doesnt need arbitrary walls of what can and cant be done.

[rough ridge]

why directly use infty in context of field of real numbers

proposing this is hard to justify without limits

[copper vale]

so math also have errors it can't define infnity

[granite maple]

yea ok you right. I was saying the wrong thing.

[rough ridge]

the most consistent way you could define it here is the limiting value of monotonic, unbounded functions

[rough ridge]

the extended and hyperreals define it

although the extended just unions R with {+-infty}

ion think it's that important

[copper vale]

why normal rule aren't applicable on infnity ?

[rough ridge]

so surely it isn't a number closest to 0

it's a sequence term

although that's still close to 0

[rough ridge]

because you don't work with it here w/o limits

i. e. lim n = lim n + c = infty

[copper vale]

so infnity is just finte veryyy large number

[sand skiff]

no, it's not, the point is that infinity is not finite

[brittle egret]

it could be like moving your reference point from 0 to the point chosen by a function that goes to a place real numbers cant get to, then counting normally starting from there?

[brittle egret]

shouldnt math be able to do any operation, even if the results arent very useful?

didnt understand everything in the later messages, were you saying 'use the (infinity + (the symbol that says add 1 to infinity))'?
dont understand what the limits are supposed to mean here

[umbral epoch]

n-1 is the closest to n, if we take n as infinity, each infinity is different, and 0 cant be the closest , that would mean that infinity is like a loop of sorts, if you mean that 1 is your infinity than that could be true but not otherwise

[idle thunder]

Bro infinity is a concept to show a number so large that it might be beyond imagination... For 2 ppl infinity may be different... For instance for me infinity may begin with 11111....... And for u it may be 999...... But ur infinity may be finite at a point making my imagination number bigger than urs

[brittle egret]

infinity is a real concept, and must be grounded in mathematics

[cyan steppe]

yes

[cyan steppe]

not exactly, infinity-1 is the same infinity talking about cardinals ($\aleph_0, \aleph_1, \dots$), but a different one when talking about ordinals ($\omega,\omega+1,2\omega\dots$), althougth in ordinals $\omega-1$ is not defined.

[leaden vault]

Tangent

[cyan steppe]

also, usually $\inf$ is $\aleph_0$ or $\aleph_1$ depending on context

[leaden vault]

Tangent

[brittle egret]

i can believe this, set theory squishing all the addition and subtraction done to infinity would avoid breaking math, and would allow for the shifting pairs to make more space in infinity proof to make sense

[cyan steppe]

all is set theory

[night pivot]

Infinity is nothing, you can never reach it

[rough ridge]

no and it isn't even an element of the real numbers

[arctic current]

infinity is not a definite number
its just a behavior that means getting bigger without bound

[rough ridge]

the limits enable you to actually do the arithmetic

how would you take infty from say N or R if {infty} is not an element of either

you could obtain it from a limit of say an unbounded, monotone function

[rough ridge]

also the proposition you're talking about doesn't really apply here because how would you justify using infty outside of nsa and ra context

you can do it, it just takes some extra work

[idle thunder]

Just search Infinity by Ramanujan

[night pivot]

Infinity is not defined but still its something we can never reach. And 0 as an example of that is wrong probably

[brittle egret]

infinity and 0 share many properties. if you multiply either of them by a number, you still get 0 or infinity.

i think of 0(and infinity) as a classification of numbers that happen to occupy the same spot on the number line rather than a single number.

[rough ridge]

why the same spot

[brittle egret]

all the 0s share a spot, all the infinities share a spot (metaphorically opposite) to the 0s

[rough ridge]

i don't really think "all the infinities" would really be well-defined in this context

because usually we mean different infinities whenever we talk about something called "cardinalities"

which is just how many elements are in a set

i. e. it would make sense in this way:

there's something called aleph numbers

for example

the cardinality of N is aleph 0

the cardinality of R is 2^(aleph 0) iirc