infinity

[sushi]

do any of you know how to visualize infinity other than numbers or examples of infinity

like infinite stairs or infinite ladder

or scrolling down the screen without ever reaching the bottom.

what appears to be infinite other than space or time? or any visualization, like the number of stars etc

i have started to look into number theory and how numbers progress into infinity

 

[dr froyd]

maybe the mandelbrot fractal zoom or something. That's more of a illustration of the infinitesimal rather than the infinite though.

 

[sushi]

i think population is a form of infinity


when we think of population we only refer to humans

but it can applied to number of stars, planets , galaxies, etc

direction can also be a form of infinity, a direction with no end

 

[sushi]

in terms of mathematical operations , its almost always +1 (sum) or x1 (product)

there is probably more mathematical operations than this though

 

[scorpiomover]

do any of you know how to visualize infinity other than numbers or examples of infinity

Infinity is the number that is bigger than any finite number, and any finite number is any number you can measure, and thus visualise. So if you can visualise something, then it's probably the case that you are visualising a finite number, and so infinity is the number just beyond that.

 

[onesteptwostep]

A mobius strip? There's also the division of segments ad. infinitum. Tautological statements could be conceptual infinities. I think the first statement in the OP is literally a tautology.

It says, how to visualize infinities other than other examples of infinities? How is there an infinity beyond infinities themselves.

Man, this makes me want to go read Bertrand Russell again and make fun of him.

 

[ZenRaiden]

Anything beyond number three or four is hard for me to understand really.
I can visualize higher numbers though.
When it comes to infinity its like what scorpiomover and onesteotwostep say ergo in relation to other numbers.
Ergo its always something that is beyond the beyond.

 

[birdsnestfern]

May be nightmarish to try.
In a pediatricians exam room - the doctor had mirrors facing each other on opposite walls, so that a baby could see itself and also see itself getting smaller and smaller in the reflections. Eventually, the light reflected image gets too small to see. But, there is likely an infinity of things getting smaller and smaller also.

I imagine the same thing happens in expanding views, the mind can't fathom it or see it after a point.

Physics Tutorial: Other Multiple Mirror Systems

Multiple mirror systems are merely the extension of what we have already learned about plane mirrors. The locating of images is an extension of the principle that the image distance to the mirror is the same as the object distance to the mirror. Drawing ray diagrams for multiple mirror systems...

www.physicsclassroom.com

Alfred Hitchcocks Psycho movie is a mobius of these two smiles:

Psycho's scariest moment isn't a scream, but a smile

The shower scene in Alfred Hitchcock's masterpiece is famous, but a pair of scenes were two characters delight at doing evil are even more chilling.

www.syfy.com

 

[sushi]

do any of you know how to visualize infinity other than numbers or examples of infinity

Infinity is the number that is bigger than any finite number, and any finite number is any number you can measure, and thus visualise. So if you can visualise something, then it's probably the case that you are visualising a finite number, and so infinity is the number just beyond that.

a ladder or staircase/escalator could be theortically be infinite

so is the infinite hotel.

a road towards the horizon could theortically be infinite

 

[ZenRaiden]

Infinity is like a concept that can shape our thinking of numbers in some sense, but knowing infinity is like knowing number 100.
We kind of understand that 100 is a number, and in relation to other numbers we kind of can understand that number has certain numerical relations.

The problem though is when I think 100 I just cannot envision it as something separate as a given entity that has some quality to it.
Its merely the numerical side of 100 I understand.

When it comes to number one for instance this number is easy to understand.
Because oneness has many relatable qualities.

Twoness also has many qualities.

Threeness can be seen in triangles, but also understood in some other fashion than mere number. Like geometry.

When it comes to infinity I can think of a very infinitely long line going for ever on.
Like if I were God and had infinite amount of speed and time and wanted to see the whole line from start to finish I would never find its starting point or end point.
IT would just go on forever.

 

[scorpiomover]

do any of you know how to visualize infinity other than numbers or examples of infinity

a ladder or staircase/escalator could be theortically be infinite

so is the infinite hotel.

a road towards the horizon could theortically be infinite

Looks like you answered your own question.

 

[sushi]

just post anything related to the concept of infinity


examples is also good

 

[sushi]

Infinite | Internet Encyclopedia of Philosophy

iep.utm.edu

 

[ZenRaiden]

I think that if one were to divide infinity in half, one would get half of infinity.
We know however that half of infinity is still infinitely long compared to other numbers.
Which means that half of infinity cannot be a finite number.
It must be a number approaching the mid section of infinity. If infinity were represented as infinite line half of that line would still be too far.
Which means at any given point infinity is twice as big as half of infinity.
However if I were able to stand on half of infinity I would be able to cover the other half of infinity just as well.
This being though only possible if I were to cover the half distance to infinity.
Which meas I would have to approach half of infinity half as much as infinity it self.
The problem is if one were to split infinity into three parts, those parts would be 3 way still infinite. Meaning the third of infinity if we were approaching it would have to be smaller than half of infinity, but still infinitely far away.

So this means infinity can be viewed in two ways. Either a infinitely long value that goes on forever, or some value something can approach.

So for instance I could view the line as something that keeps going.
Other way to think of it if I were going and walking the line Id be closer to its value the more I walk from any given point, but never half or third of its distance.
In fact not even close.
The only one could ascertain a finite value in infinity would be to thus walk to a finite point in infinite value, which would approach a finite numerical value. Thus the only way to escape infinity is to define a specific value that one approaches during the walk over infinity.

Ergo approaching the value of half of infinity by the number 10 000 which is not even close to half of infinity, but closer than number 1.
Which means the number 10 000 is truer to infinity than 1, by infinitely small fraction.
But how would we know the fraction so infinitely small in relation to infinity, well we would have to for example compare its ratio of numbers approaching 2 other numbers. Ergo if the numbers were 20 000 and 30 000 we would know that the ratio of infinity to 10 000 between numbers of 20 000 and 30 000 is infinitely small.
It is thus conceivable that any two numbers other than 20 000 and 30 000 would also be infinitely small compared to infinity, but if their value is greater or smaller we would know that given their value we could know whether something is going to be approaching infinity or not approaching infinity.

But what does not approach infinity means it has to come close to a more finite value. Ergo we know that 2000 and 3000 approaches more finite value than 20 000 or 30 000 compared to half of infinity.

But here we might ask would this still be same as whole infinity and the answer is yes, 2 twice as much.

 

[ZenRaiden]

Just kidding.

 

[sushi]

i just realize there is some way to conceptualize it

a piece of paper=1

there are two ways a stack of paper can reach infinite, either by sum or by mulitiplication

paper+1 more+ 1 more so on to infinite

paper x 2 x 2 x2 x2 to infinite

its either + a constant, or multiply by a constant

as for exponential i will explain further later in terms of paper

 

[sushi]

other than paper analogy

money is also a useful idea

give sum amoney M , how much operations does it take to turn the number and operations into infinite amount

M+1+1 +1+ 1=?

Mx 2x 2x2 x2 x2=?

MxMxM xM xM=?

the whole infinity is related to the idea of continuum , where each succession number is greater than the last

 

[sushi]

 

[sushi]

law of 0
law of one (anything to do with 1,+1 or multiply by one)
law of 2 (+2 , multiply by 2)
law of 3
law of 4
law of 5
law of 10
law of insert random number

 

[dr froyd]

View attachment 7276

that's actually not true

if we're talking about integers for example, it can be show than 0 to infinity and 20 to infinity are sets of same size:

match 0 in the first one to 20 in the second
then 1 -> 21
2 -> 22
3 -> 23
and so on infinitely, i.e. you can make a one-to-one mapping between them, by which you can argue they have equally many elements (both are so-called countably infinite and there's no meaningful way of distinguishing between their sizes)

however there are cases where one cannot make a mapping like that, so some infinite sets are of a bigger infinity than others (like the set of real numbers vs integers). This was famously shown by Cantor via the "diagonal argument"

 

[sushi]

View attachment 7276

that's actually not true

if we're talking about integers for example, it can be show than 0 to infinity and 20 to infinity are sets of same size:

match 0 in the first one to 20 in the second
then 1 -> 21
2 -> 22
3 -> 23
and so on infinitely, i.e. you can make a one-to-one mapping between them, by which you can argue they have equally many elements (both are so-called countably infinite and there's no meaningful way of distinguishing between their sizes)

however there are cases where one cannot make a mapping like that, so some infinite sets are of a bigger infinity than others (like the set of real numbers vs integers). This was famously shown by Cantor via the "diagonal argument"

i kind of understand what you are saying, because i have studied it , but can you draw the chart? its hard to visualize it.

its just to demonstrate that larger sets are more inclusive than smaller sets

 

[ZenRaiden]

Visual is a well that is infinitely deep.
Whether you throw the stone into the well now or 20 feet later the stone keeps falling for ever.

 

[dr froyd]

i kind of understand what you are saying, because i have studied it , but can you draw the chart? its hard to visualize it.

its just to demonstrate that larger sets are more inclusive than smaller sets

same concept but using a comparison of integers and even numbers:

Cantor's diagonal argument - Wikipedia

en.wikipedia.org

as it says in that article, "An infinite set may have the same cardinality as a proper subset of itself, as the depicted bijection f(x)=2x from the natural to the even numbers demonstrates."

so, you can remove objects from a set, but the size of the set doesn't change. Dealing with infinity can lead to confusing results.

what Cantor showed is that there's different sizes of infinity, countable and uncountable. The diagonal proof shows that even if you have an infinitely long list, you cannot list all real numbers, i.e. the set of real numbers is uncountably infinite (in contrast to natural numbers)

 

[Tomten]

I think dealing with infinite sets is confusing because the axiom of infinity in set theory is itself incoherent and therefore wrong. The other axioms in set theory are insufficient to prove the existence of infinite sets, so to talk as though they do exist, one needs to assume that at least one infinite set exists - namely, the set of all natural numbers - and this assumption is the axiom of infinity. The "set of all natural numbers", to be clear, isn't a strictly self contradictory concept. But I maintain that it is an incoherent concept, like the concept of a married bachelor. Once you understand the meaning of the terms "married" and "bachelor", it's obvious that married bachelors can't exist. Yet it's not a strictly self contradictory concept, as is the concept of an unmarried married man.
Why do I say "the set of all natural numbers" is also an incoherent concept? When you count the natural numbers, beginning from 0, you'll never finish counting them. When you've counted up to and including n, there's the subsequent number n+1, that you have not yet counted. Yet for the set of *all* natural numbers to exist it would have to be possible to finish counting the natural numbers, beginning from 0. One could counter this by saying that the set of all natural numbers exists even though it is impossible to construct. But that would be a highly dubious assumption, and a theory built on such an assumption would be a joke.

 

[dr froyd]

When you've counted up to and including n, there's the subsequent number n+1, that you have not yet counted. Yet for the set of *all* natural numbers to exist it would have to be possible to finish counting the natural numbers, beginning from 0. One could counter this by saying that the set of all natural numbers exists even though it is impossible to construct. But that would be a highly dubious assumption, and a theory built on such an assumption would be a joke.

the fact that you can always go n + 1 is almost literally the definition of a countably infinite set

so your argument is, in a way, self-contradictory. You dont like an infinite set because there's always an n + 1, yet any set where you cannot go n + 1 is a finite set. I.e. you're almost arguing it should be illegal to add 1 to a number once it gets too high.

not trying to be argumentative, just thought it was an amusing thought

a lot of definitions of infinity in mathematics can be though of as just that - actions that can be repeated without end. For example I can divide a number by 2 as many times as I like; there's no final number where I am forced to stop. Thus the sequence of these divided numbers is an infinite set. The entirety of calculus is based on this idea.

 

[sushi]

what is the logic of infinity

is keep adding one (n+1)is the only way to infinity
or keep adding 2 (n+2) the way

or mulitiply by a constant like 10or 2 (nx2) (nx10)
or multiply by itself, like expontential n power 2, n power 3

also sets are inclusive

a larger set means it include more numbers than a smallter set

(0, 2, 4 , 6, 8, 10)
(0, 2 4 6)

 

[sushi]

is addition and multiplication the only road to infinity

or is there any other way to increase a number other than mulitplication and addition
which we have not discovered. (i think power rule is a form of mulitplication, like self)

 

[Tomten]

When you've counted up to and including n, there's the subsequent number n+1, that you have not yet counted. Yet for the set of *all* natural numbers to exist it would have to be possible to finish counting the natural numbers, beginning from 0. One could counter this by saying that the set of all natural numbers exists even though it is impossible to construct. But that would be a highly dubious assumption, and a theory built on such an assumption would be a joke.

the fact that you can always go n + 1 is almost literally the definition of a countably infinite set

so your argument is, in a way, self-contradictory. You dont like an infinite set because there's always an n + 1, yet any set where you cannot go n + 1 is a finite set. I.e. you're almost arguing it should be illegal to add 1 to a number once it gets too high.

not trying to be argumentative, just thought it was an amusing thought

a lot of definitions of infinity in mathematics can be though of as just that - actions that can be repeated without end. For example I can divide a number by 2 as many times as I like; there's no final number where I am forced to stop. Thus the sequence of these divided numbers is an infinite set. The entirety of calculus is based on this idea.

There's a 2000 year history in the philosophy of mathematics of making a distinction between actual and potential infinities ("potential infinity" is kind of a misleading name though, because such an "infinity" isn't really infinite), and for most of that history mathematicians accepted potential infinities, but rejected actual ones. The latter largely changed in the aftermath of Cantor's work on set theory. Basically all mathematicians now accept the existence of both.
You seem to have failed to appreciate the distinction between the two concepts. Or it may be that we don't share some presuppositions about something like what it means for a mathematical object to exist.
I reject the existence of actual infinities, but accept the existence of potential ones (some mathematicians even reject the existence of potential infinities, and so would indeed claim that there is a biggest number or at least withhold judgement as to whether there is one. And that's not as crazy as one might think because it's a sort of down to earth attitude they have. "It may not be physically possible to construct just any number."




"The fact that you can always go n + 1 is almost literally the definition of a countably infinite set

so your argument is, in a way, self-contradictory. You dont like an infinite set because there's always an n + 1, yet any set where you cannot go n + 1 is a finite set. I.e. you're almost arguing it should be illegal to add 1 to a number once it gets too high."




If I were to interpret what you said charitably, I would assume what you mean by "can always go n+1" is that for any n in the set, n+1 is also in the set, and not that you can always add another element n+1 to the set (because you can't add an element to any set, whether it's a finite set or a supposedly infinite set). But the problem for you is that given the first interpretation, it's actually harder to see where the supposed self-contradiction lies.


The fact that I accept that the process of trying to count the natural numbers is unlimited, does not mean that I have to accept that there exists a countably infinite set like the set of all natural numbers. Let's get into some details. The set of all natural numbers can be extracted from the axiom of infinity - which says: "A set exists that contains the empty set and the successor set to every element in the set. The successor set to a set A is the union of A and the set containing just A." The axiom implicitly gives us rules for generating lists of sets, and each set can be defined to be a unique natural number: "If you haven't jotted down the empty the set, then do so, otherwise keep jotting down the successor set to the set you previously jotted down":
{}=0
{0}=1
{0,1}=2
{0,1,2}=3.
...

The process above is an example of a potential infinity because it can generate an unlimited number of lists and lists unlimited in size, but the process is never complete, each list is created in a finite number of steps. And so a list can never be made infinitely long this way nor will one ever create anything but a finite number of them.
I accept that the process is unlimited, and so I accept the existence of potential infinities. Someone who rejects the existence of potential infinities would say the process has a limit (or perhaps something else) or that they merely are not convinced that it is unlimited.

What I reject about the axiom of infinity is that despite the impossibility of generating an infinitely long list using the two rules mentioned above, it basically implies that there already is such an infinitely long list and claims that a set exists which contains all the sets in that infinitely long list.




a lot of definitions of infinity in mathematics can be though of as just that - actions that can be repeated without end. For example I can divide a number by 2 as many times as I like; there's no final number where I am forced to stop. Thus the sequence of these divided numbers is an infinite set. The entirety of calculus is based on this idea."



Here you're presupposing that there already is an infinite sequence like a, a/2, a/4, a/8... which forms a completed totality and exists as a given object - an actual infinity.
The notion that such an object exists does not follow from the fact you can begin a sequence with a, a/2, a/4 and keep going without any limit - which is a potential infinity.
An unlimited process can not be completed, so why would it follow that such an object exists?

Edit: I'll try to make a steel man argument here, which is seemingly plausible (maybe?) but makes a conceptually flawed implicit assumption:
If you can keep jotting down rational numbers without any limit, then there can't be a finite number of rational numbers, because if there were, you'd eventually have to stop because you would run out of numbers to pick from.
So there is an infinite set of them.

Here's the response: there not being a finite number of rationals does not mean there's an infinite set of them.
There isn't a finite number of rational numbers, but that's because there is no fixed number of them. Consistent with that I would simply say that the rational numbers are potentially infinite: they are generated through an unlimited process.

 

[dr froyd]

An unlimited process can not be completed, so why would it follow that such an object exists?

this is the root of your misunderstanding. If you can prove that a sequence cannot be completed, then you have proved it forms an infinite set. It's a matter of definition.

if you feel the need to know that an infinite set 'exists' in the sense that it's a physical box you pull numbers from, this has nothing to do with mathematics.

 

[scorpiomover]

There's a 2000 year history in the philosophy of mathematics of making a distinction between actual and potential infinities ("potential infinity" is kind of a misleading name though, because such an "infinity" isn't really infinite), and for most of that history mathematicians accepted potential infinities, but rejected actual ones. The latter largely changed in the aftermath of Cantor's work on set theory. Basically all mathematicians now accept the existence of both.

Cantor's Theorem shows that there are multiple potential infinites.

You seem to have failed to appreciate the distinction between the two concepts. Or it may be that we don't share some presuppositions about something like what it means for a mathematical object to exist.
I reject the existence of actual infinities, but accept the existence of potential ones (some mathematicians even reject the existence of potential infinities, and so would indeed claim that there is a biggest number or at least withhold judgement as to whether there is one. And that's not as crazy as one might think because it's a sort of down to earth attitude they have. "It may not be physically possible to construct just any number."

I did Real Analysis in my 1st year of uni. The definitions are very interesting, because all sequences and series whose parameters diverge to infinity, such as n^2 -> infinity as n -> infinity, are defined in such a way as to avoid acknowledging even the concept of potential infinities.

So in reality, the way maths works, is to use infinity as a theoretical concept. But when it comes to doing real-world calculations that we have to rely on, the maths that proves those real-world calculations avoids the question of if actual infinities exist, and even avoids the question of if potential infinities exist.

Even Cantor's own proof never really requires that potential infinities exist. His proof is only about sets in a way that doesn't distinguish between finite sets and trans-finite sets.

So if you want to do maths, while not believing in actual infinities, or even potential infinities, you can.

The fact that I accept that the process of trying to count the natural numbers is unlimited, does not mean that I have to accept that there exists a countably infinite set like the set of all natural numbers.

Yes. But if you want to prove some general rules about arithmetic that apply to all natural numbers, you're either going to need to assume such a set exists, or prove it in a different manner. It can be done. But you'll still probably need to state something like "for all n s.t. n is a natural number", or you can say "for all n e N where N is the set of all natural numbers", which is basically the same thing for the most part.

So I think that for the most part, it's a matter of semantics.

I could be wrong, though.

Is there an instance where it would make a difference?

 

[scorpiomover]

the fact that you can always go n + 1 is almost literally the definition of a countably infinite set

so your argument is, in a way, self-contradictory. You dont like an infinite set because there's always an n + 1, yet any set where you cannot go n + 1 is a finite set. I.e. you're almost arguing it should be illegal to add 1 to a number once it gets too high.

not trying to be argumentative, just thought it was an amusing thought

It's a tricky concept, because infinity doesn't exist in any way we can directly physically measure, and yet it produces results that can be measured, like the results of calculus, and so it does exist in ways that we can indirectly measure.

But some of the rules of regular arithmetic don't hold, when you get into trans-finite arithmetic.

That in turn leads to the question of "Is there an infinity of infinities?" or "Is there a master number/set which is the one to rule them all?", i.e. Is there a "set of all sets"? A lot of mathematicians don't like the set of all sets, because it breaks even more rules and even breaks one of the most fundamental rules of logic, that something cannot be both true and false at the same time. The set of all sets must contain every other set by definition. Yet Cantor's Theorem proves there must be a bigger set.

Everyone is bound to have some problem with these concepts.

a lot of definitions of infinity in mathematics can be though of as just that - actions that can be repeated without end. For example I can divide a number by 2 as many times as I like; there's no final number where I am forced to stop. Thus the sequence of these divided numbers is an infinite set. The entirety of calculus is based on this idea.

Calculating areas via integration would be much more difficult to prove without the idea of infinitely-long sets. If you want to integrate an area, you have to break it down into strips, and then make the strips smaller and smaller, which means more and more strips, until there's an infinite number of those strips.

In addition, there's another problem: what happens if you dissect the area in 2 ways and get 2 different results? So in order for calculus to hold, you have to be able to prove that the results of your integral work out to be the exact same value, no matter how you choose to dissect your area into smaller and smaller areas.

So to calculate an area via integration normally assumes that you can make almost any set of an infinite number of smaller areas. So it kind of assumes that you can make almost any set of an infinite number of elements.

It might be possible to integrate areas without that. But the argument would be much more tricky, as you would have to leave everything about your dissections as a statement of limits, and thus would likely also require that everyone who learns integration, also learns the concept of limits, which lots of people find really difficult, because limits aren't formuale or numbers. Limits are logical statements. So then every high school student who learns calculus, probably needs to learn to do complex logic as well.

 

[Old Things]

No, but I would ask Neo.

 

[sushi]

its not only n+ 1, could be n+2, n+ 3,. n+ C

or nx2 , nx3, nx4, nx5

or n^2， n^4 , n^5

or some unknown increase function and process we havent discovered yet that lead to increase by a factor of

thats why i said all the laws involving the number 1 , such as +1, x 1, divide by 1 as F(1)
or all the laws involving the number 2 F(2)

if a set is larger than another set, it means it already includes all the numbers of that set.

 

[sushi]

although one can argue that n+1 set already includes the numbers of all other types of increasing sets

n+1 has all the odd numbers and even numbers

 

[dr froyd]

Calculating areas via integration would be much more difficult to prove without the idea of infinitely-long sets. If you want to integrate an area, you have to break it down into strips, and then make the strips smaller and smaller, which means more and more strips, until there's an infinite number of those strips.

yes exactly

the entirety of calculus is based on infinite sequences and finding what those sequences theoretically converge to. For example, one cannot derive any clear definition of a derivative without talking about what happens at the limit, i.e. when taking an infinitesimally small step. The only exceptions are the very trivial cases like straight lines where the answer is obvious without taking limits.

 

[dr froyd]

although one can argue that n+1 set already includes the numbers of all other types of increasing sets

n+1 has all the odd numbers and even numbers

yeah, but the n + 1 set is exactly as big as the set of odd numbers

 

[sushi]

although one can argue that n+1 set already includes the numbers of all other types of increasing sets

n+1 has all the odd numbers and even numbers

yeah, but the n + 1 set is exactly as big as the set of odd numbers

i am curious is there any other process other than mulitiplication or addition as increasing function that we havent discovered yet.

 

[scorpiomover]

i am curious is there any other process other than mulitiplication or addition as increasing function that we havent discovered yet.

Power function (2 ^ 5 = 2 * 2 * 2 * 2 * 2).

Other functions like matrix addition and matrix multiplication. Also determinants. Also norms. But in mathematics, when they are used, they get labelled as either a type of addition or a type of multiplication.

 

[dr froyd]

i am curious is there any other process other than mulitiplication or addition as increasing function that we havent discovered yet.

keep in mind you don't need an increasing function to have an infinitely big set

the set of real numbers between 0 and 1 is infinite, for example

 

[sushi]

i am curious is there any other process other than mulitiplication or addition as increasing function that we havent discovered yet.

keep in mind you don't need an increasing function to have an infinitely big set

the set of real numbers between 0 and 1 is infinite, for example

0 is still increasing to 1 though, its not decreasing to negative one

the increase is just incrementally small.

 

[sushi]

i am going to name very large numbers, supernumbers