Is infinity a quantity?

• 25
I'm of the opinion an actual infinity cannot exist.

I believe that beyond time and space, that infinity is only a mathematical construct.
We can not deal with infinite time or space, and nothing else in our world becomes infinite.

We measure the totality of energy by the rate of electrical flow times the amount of time, giving us Kilowatt hours.
If we multiply an infinite universe times the infinitely small, moving point in time in which it exists, perhaps the infinities cancel. This seems like a comfortable, perhaps pathological workaround to the notion that nothing can be infinite.
• 1.4k
1. A quantity is a specified amount of something. It has a limit. The infinite is that which has no limits and so cannot be quantified. Therefore, not a quantity as not quantifiable.
I would say that it simply isn't countable or computable. Yet it does in my view quite clearly define a quantity.

The problem is that math starts from counting. And hence we have all the problems with infinity.
• 25
If different infinities have different values then is it really a definite quantity?

Ammount of real numbers = infinite
Ammount of even real numbers = infinite
Does this mean that infinity divided by 2 equals itself?

Does this not violate the definition of a real number? Seems like there was something on this in "analytical geometry"...
• 25
Ammount of real numbers = infinite
Ammount of even real numbers = infinite

I meant whole, or natural numbers, not real numbers.
• 763
If different infinities have different values then is it really a definite quantity?

Different levels infinities have different sizes. They're size is definite. "Infinity" is not one value.

Amount of real numbers = infinite
Amount of even real numbers = infinite
Does this mean that infinity divided by 2 equals itself?

The real numbers have a cardinality (size) larger than that of the even numbers. The even numbers have the same cardinality as the natural numbers, aleph-null, and so are "countably" infinite. The real numbers cannot be put into a one-to-one correspondence with the natural numbers, being that the reals are a larger infinity. Hence the reals are "uncountably" infinite.
• 378
Infinity is a Mathematical fiction and should be applied carefully to the World of Physical Things. For example we can say that there are an Infinite number of Natural Numbers. Natural Numbers are Mathematical concepts. But there can not be an infinitely large Pencil in the Universe. A good old fashioned Pencil is made out of a core of Lead or graphite (lets just say Lead). surrounded by a tube of Wood and then a coat of Paint. Take a point exactly in the center of the Lead and then let the Pencil grow in size to Infinity. You will have a Universe that is completely filled with Lead. You can never get to the Wood no matter how far you travel away from the center point (assuming we can travel through Lead). There will be no Wood or Paint in this Universe. The Pencil will become something less than it was when it becomes Infinite. You can not really have an Infinite Pencil.
• 1.4k
Infinity is a Mathematical fiction and should be applied carefully to the World of Physical Things. For example we can say that there are an Infinite number of Natural Numbers. Natural Numbers are Mathematical concepts.
Well, infinity is a very useful mathematical concept then. After all, the number "3" doesn't physically exist either.
• 378
Infinity is a Mathematical fiction and should be applied carefully to the World of Physical Things. For example we can say that there are an Infinite number of Natural Numbers. Natural Numbers are Mathematical concepts. — SteveKlinkoWell, infinity is a very useful mathematical concept then. After all, the number "3" doesn't physically exist either.ssu

3 things can exist but an infinite amount of things can not.
• 763
I don't see how that's a given. Space is infinitely divisible. Whether or not space counts as a "thing" or not I don't think matters, but it's infinite.
• 378
↪SteveKlinko I don't see how that's a given. Space is infinitely divisible. Whether or not space counts as a "thing" or not I don't think matters, but it's infinite.
Space is definitely a Thing. There can be the 3D Space that we are familiar with, but there can also be 4D Space for example. 4D Space is a whole different Thing than 3D Space. If Space can be different Things then there can be no Space. That would be Absolute Nothingness.

It is not known if Space is Infinite or not. It depends on what the value of a particular constant in Physics is found to be. The current thinking is that our 3D Space is Finite but Unbounded in the sense that it curves back around on itself in some way.
• 763
If space is a thing, it's not the same as the natural understanding of a thing. That's what I was talking about.

You're other point wasn't what I was talking about. I'm not saying space is infinite in breadth, but it can be infinitely divided without hitting some kind of base unit or boundary point.
• 378
If space is a thing, it's not the same as the natural understanding of a thing. That's what I was talking about.

You're other point wasn't what I was talking about. I'm not saying space is infinite in breadth, but it can be infinitely divided without hitting some kind of base unit or boundary point.

I wouldn't be so sure that Space can be Infinitely divided. I'll give that a Maybe. It might be subject to the Planck Constant. I don't think you should assume that a Physical Space is the same as a Mathematical Space.
• 1.9k
Actual Infinity is not a quantity:

- There is no number X such that X > all other numbers
- Because X+1 > X

Space is discrete that’s why we get paradoxes when we assume it’s continuous (Zeno’s paradoxes).
• 378
Actual Infinity is not a quantity:

- There is no number X such that X > all other numbers
- Because X+1 > X

Space is discrete that’s why we get paradoxes when we assume it’s continuous (Zeno’s paradoxes).

The Zeno paradox, where to walk any finite distance you have first walk half the distance, then half of the remaining distance etc. is a paradox because Zeno is throwing in a false assumption that you are going across each Half in the same amount of time. This is artificially slowing you down. Of course this would make it impossible to cover the total distance when you consider an Infinite amount of Halves. Zeno forgets that at the same time the Half distances are going to Infinity the number of Halves you are traversing (at constant velocity) is going to an Infinite amount of Halves per second. The Infinite Halves per second and the Infinite number of Halves are compensating Infinities that cancel the paradox.
• 1.9k
The way you are solving the paradox uses the undefined quantity ‘infinity’ but I acknowledge there are other ways out of Zeno’s paradoxes other than discrete space.

Still I’d argue for discrete spacetime on the grounds:

- there is no such distance as 1/oo mathematically.
- Imagine a particle moving over a finite period of time. Continuous spacetime would require the particle to have occupied a actually infinite number of states which is nonsensical.

Still even if space is continuous, that would only be a potential infinity rather than actual infinity.
• 659
There is a mathematical relation between the various transfinite numbers, and these relations are analagous to size and quantity, but these are not "quantities" in the exact same sense as the quantities of individual real (or natural) numbers.
• 378
The way you are solving the paradox uses the undefined quantity ‘infinity’ but I acknowledge there are other ways out of Zeno’s paradoxes other than discrete space.

Still I’d argue for discrete spacetime on the grounds:

- there is no such distance as 1/oo mathematically.
- Imagine a particle moving over a finite period of time. Continuous spacetime would require the particle to have occupied a actually infinite number of states which is nonsensical.

Still even if space is continuous, that would only be a potential infinity rather than actual infinity.

I say that Infinity is a Mathematical Fiction that only exists in the world of Mathematics. But since the construction of the Zeno Paradox uses Infinity as the basis of the argument we must accept the premise and argue from that. I think the compensating Infinity argument is the best way out of the Paradox.

If a particle only occupied discrete states then according to your theory it would have to jump from position to position while moving. It would necessarily have to stop at each position for the time it would take to continuously travel between two of the positions. This is as nonsensical as a continuous movement with Infinite intermediate positions. These are both nonsensical and serve to illustrate the problems you can get into when you think a little more Deeply about things. Good Thoughts however.
• 1.9k
If a particle only occupied discrete states then according to your theory it would have to jump from position to position while moving. It would necessarily have to stop at each position for the time it would take to continuously travel between two of the positions.

A good point. It depends on your view of time as to whether you think the particle exists in an actually infinite number of states:

- Presentist. The past does not exist. So the particle does not exist in an Actually Infinite number of states, just one state, the present.
- Eternalist. The past exists so continuous time implies the particle must exist in an actually infinite number of states.

Presentism leads to paradoxes, so that suggests Eternalism. But time must be discrete for Eternalism to be free of Actual Infinity (which I class a paradox).
• 378
If a particle only occupied discrete states then according to your theory it would have to jump from position to position while moving. It would necessarily have to stop at each position for the time it would take to continuously travel between two of the positions. — SteveKlinko
A good point. It depends on your view of time as to whether you think the particle exists in an actually infinite number of states:

- Presentist. The past does not exist. So the particle does not exist in an Actually Infinite number of states, just one state, the present.
- Eternalist. The past exists so continuous time implies the particle must exist in an actually infinite number of states.

Presentism leads to paradoxes, so that suggests Eternalism. But time must be discrete for Eternalism to be free of Actual Infinity (which I class a paradox).
The Infinitely Large and the Infinitely Small are a real pain in the Brain.
• 378
Imagine a Square drawn on a piece of paper. Now imagine the Square shrinking smaller and smaller. It remains a Square no matter how small it shrinks. If we stop shrinking it and start magnifying it back we can bring the Square back to the original size. But now imagine the Square shrinking to Zero size. All points of the Square collapse to a single point and there is no longer a Square on the paper. The square has been transformed into a single point. We would not be able to magnify the resulting point back the the original Square. We could also shrink a Triangle in the same way and at Zero size it would be a single point just like the Square. The Square and the Triangle lose their identity when they are Zero size. They become something different. They become something less than what they were. Zero size is an unrecoverable threshold of size that changes everything.

Now imagine a Square that is the smallest Square that is not equal to Zero. This thought sends your mind into an endless recursive loop of the Square getting smaller and smaller and we soon realize that it is impossible to imagine such a smallest Square. One thing we can say is that this Square is Infinitely small but is still a Square. In general mathematics this would be called a differential Square or an infinitesimal Square.

Next imagine the Square that was drawn on the paper growing larger and larger. If the Square was exactly in the center of the paper the sides of the Square would eventually move off of the paper and past the edges of the universe. It remains a Square no matter how large it grows. If we stop growing it and start shrinking it back we can bring the Square back to the original size. But now imagine the Square growing to Infinite size. The sides would all move out to infinity. No matter how far you went in the universe you would never encounter a side of the Square. The Square has effectively exited the universe. We could also grow a Triangle in the same way and at Infinite size it will no longer be found in the universe. The Square and the Triangle lose their identity when they are Infinite size. They become something different. Paradoxically they become something less than what they were. You might think that the Square and Triangle are still out there at Infinity. But there is no "there" at Infinity. The Square and Triangle are gone. If you think you can go out "there" to an edge of the Square or Triangle at Infinity then that "there" is not Infinity. Infinite size is an unrecoverable threshold of size that changes everything.

Now imagine a Square that is the largest Square that is not equal to Infinity. Similar to the differential Square, this thought sends your mind into an endless recursive loop of the Square getting larger and larger and we again soon realize that it is impossible to imagine such a largest Square. We can say that this Square is Infinitely large but is still a Square that exists in the universe.

I think that just as Infinite Squares are not possible it is probably true that any Infinite Physical quantity of anything is not possible. Just because an equation in Science goes to Infinity, it doesn't mean that the Physical quantity in the equation is able go to Infinity. I think this is a limitation of what we can do with Mathematics. Seems like a minor limitation but it has big consequences when equations in Science go to Infinity.
• 1.9k
But now imagine the Square growing to Infinite size. The sides would all move out to infinity. No matter how far you went in the universe you would never encounter a side of the Square.

A good point. Both the process of growing something to oo and shrinking something to 1/oo are destroying information, which is not meant to happen in the physical world.
• 267
A line is an infinite quality as 1 direction with this infinite direction as 1 being composed of multiple line as 1 infinity in themselves.

Hence infinity, through the line as one considering one premised in empirical qualities observes all numbers as an observation of linear qualities through time as directive in nature.

The line as 1 is infinity as 1 with one itself being infinite in the respect it is continuous as existing through itself as itself. We can observe that a line as 1 through 0d space or void must effectively projects if it is to exist. However considering the line has nowhere to project but must project to something if it is to maintain its nature, the line must effectively fold through itself by multiplying in directions.

In these respects the 1 original line as infinite must individuals into multiple lines which each line being a ratio relative to each other and the original line while being infinite and one in itself. Finiteness is strictly the relation of infinities with infinity, as an absence of finiteness, being an absence of relation with infinity and finiteness existing at the same time in different respects due to there positive and negative qualities.

Not only is infinity quantifiable but there is one infinity and multiple infinities with these multiple infinities being the premise for finiteness or time.

In simpler terms the 1 directional nature of the infinite line shows infinity it only as quantifiable but effectively existing as a limit.
• 3.3k
Is infinity properly thought of as a number? Is it a quantity? Is that the same question

As far as I know, infinity is a quantity, so a number, that has no fixed value. It goes on forever.

Think of * as a number. An infinity of * is still a collection of * but we can't fix an exact value to it.

Basic math operations (+ - × ÷) fail with infinity.

I guess we aren't supposed to use it that way.
• 763
You can do addition and multiplication with infinity, e.g. transfinite arithmetic.
• 3.3k
:up: Thanks
• 17
No, limitlessness is. However you can't really take it as one because it's limitless. Infinity in the other hand is a process of something that takes forever. Limitlessness is the ultimate/perfect word. However it is only possible if there's no beginning and no end.
• 674
Is infinity properly thought of as a number? Is it a quantity? Is that the same question?

Is a bishop a religious guy with a cool hat? Or is a bishop a piece in a certain game that moves diagonally? There are connections between the hat-guy and the chess-piece, but they are different, and they both make pretty good sense in their context.

I think it's the same with infinity.
• 34

In the first case, infinity is a shorthand for a limiting process (the infinity is hidden in the quantifier 'for all epsilon')

Since you said correctly that in the definition of limit the notion of infinity is hidden in the quantifier I think you are not confusing the limit to infinity (infinity indicates the graphical correspondent to the behavior of a function, in so far as its values become arbitrarily large) with infinity of the limit(infinity limit a process) I suggest to not use the misleading terms: 'limiting process' but 'unlimited variables bounded in regards to which the limit is considered'.

Also
Nothing to prevent you from adding 1 to infinity.tom

Of course tom is right(but unfortunately unsuccessful) in preventing the tedious error of conceiving of infinity as a set of numbers. More explicitly: infinity is a relational concept, and its use as a factor just means the operations operate on variables(and infinite variables[of numbers] plus the unity is of course a not problematic thing). One thing is infinity as relational concept, another is the Symbol of infinity to indicate infinitely many variables.

In the second case - for cardinals - they give the size of infinite sets, so yes they are probably quantities since they represent the magnitude of something.

I don't think cardinality offer a quantitative view of Infinity, since it is either a relation between a set and its elements or between its elements and numbers(e.g. a set is D-
infinite iff for every natural number the set has a subset whose cardinality is that natural number) or between sets(e.g. the cardinal of R is bigger than the cardinal of I)
• 767
I don't think cardinality offer a quantitative view of Infinity, since it is either a relation between a set and its elements or between its elements and numbers(e.g. a set is D-
infinite iff for every natural number the set has a subset whose cardinality is that natural number) or between sets(e.g. the cardinal of R is bigger than the cardinal of I)

Numbers are sets, in the usual axiomatizations, so cardinality very naturally fits our idea of quantity.
• 5.7k
Numbers are sets, in the usual axiomatizations, so cardinality very naturally fits our idea of quantity.

The only problem here is that "sets" are based in qualities, and there is a conceptual difference between quality and quantity. Therefore set theory does not naturally fit our idea of quantity. So set theory provides a set of axioms which modify mathematics in a way so as to be inconsistent with our natural idea of quantity.
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