What Ingredients Make an Entity Truly Infinite?

[DrKareem]

It suddenly struck me today when I heard somewhere that "the universe is infinite". Well I know from physics that it isn't. So I thought about what could make somethine infinite. I mean a finite number of objects would make a finite entity, so there's no way you could get an infinite entity with finite objects. So what are the ingredients of an infinite entity?

But then again, numbers are finite, and they can be arranged in infinitely to form an infinite set of numbers. I'm so confused. I'd be delighted to see some books recommendation about this subject.

 

[StatusX]

If you think of the process of building an object as taking time at a finite rate, then no infinite objects can exist. I think this is the problem a lot of people have with 0.999~. But obviously in abstract math, this doesn't have to be the case, and the set of all integers is an example of an infinite set of finite objects.

In physics, this question is more difficult. First of all, I don't know how you know the universe is finite, but most physicists would disagree with you. There is no speed limit on the expansion of space, and it is possible that spacetime is infinite in extent. In fact, if the universe is flat, which it appears to be, I don't know what a boundary could conceivably look like. On the other hand, as long as space expands faster than light, we can never reach the boundary, so depending on your metaphyical outlook, you could say the boundary doesn't exist and call that an infinite universe.

 

[DrKareem]

I did read in a book that the universe is expanding, but there was nothing written about the rate of expansion (unless that it couldn't be overcome by gravity). I assumed that it would be finite if it is growing bigger. Apparently I'm wrong, I admit i have very much information about that.

But still, the universe is formed up of discreet particles. And at any given instant of time, the number of particles is finite, diregarding the boundaries. It's very unclear in my mind. I really should do some readings about this.

 

[jackle]

What normally seems to lead to an infinity in maths, is where you have a process that goes through a feedback loop. For example, counting goes through the process:

a) next_number = 0
b) next_number = next_number+1
c) say <next number>
d) goto b)

So, it goes round forever.

You have the example of an infinite number of points on a line, demonstrated by always finding another point half way between the previous point and the end of the line. The next point is fed back into the process.

With the expanding universe, it is a lot like the counting example except you are increasing the size of the universe and everybody assumes (rightly or wrongly) that this can go on endlessly.

Fractals are an example where you have a shape and you modify it according to some rules and then put the new shape through the same process etc. You end up with a shape that has an infinite complexity (These shapes do actually seem to exist in nature).

1/3 has an infinite number of digits because when you do the division you get stuck in a feedback loop that generates endless 3s.

 

[Hurkyl]

There's a subtle but important difference between "unbounded" and "infinite".

For example, the division algorithm would be an example of an "unbounded" process: it keeps going and going, without end, but at all times you only have a finite number of digits.

However, you would get infinitely many digits if you said something like "Okay, for all n, the n-th digit of 1/3 is defined to be the result of the n-th step of the division algorithm".

 

[StatusX]

I'm not sure I understand you Hurkyl. Are you saying that processes are unbounded, and sets are infinite? Or are you saying that there can be sets that are unbounded and yet not infinite?

 

[Hurkyl]

You always have to be very careful about precisely what you're describing. A
"process" or "algorithm" usually defines a collection of intermediate values. In this case, the division algorithm provides us with a sequence of partial results.

There is no bound on the length of a partial result, thus we can say that these partial results are unbounded.

But still, each partial result is finitely long.


However, the decimal representation of 1/3 is truly infinitely long. To get it from the division algorithm requires some subtlety. One way is to say that the decimal representation of 1/3 is the unique infinitely long string of digits whose initial segments agree with the partial results from the division algorithm.

 

[jackle]

I think unbounded sets don't actually contain the member infinity because they can never actually get there. Is this right?

There is also the idea that there are many different values of infinity but I'm not going there today

 

[zarback]

From an uneducated mind, a ramble...

For anything to be infinite, it must not have a measurable quantity, otherwise it would have a finite value. Using that presumption, I can assume universe space is not infinite due to the very existence of matter i.e. if matter displaces space then for given vast areas of "space" we can assign a value to it. The mere fact that a mesurable quantity of something can exisit without "space" rules out all possibility of it being infinite due to the value now being infinity minus our measurment.

 

[StatusX]

For anything to be infinite, it must not have a measurable quantity, otherwise it would have a finite value

You'll have to be clearer. What kind of infinite thing are you talking about, and what do you mean by a measurable quantity? Are you talking about math or physics?

 

[zarback]

My philosophical ramble applied to all things which are flagged as infinite. Using a basic math example with 10 divided by 3: the answer is not 3.333333333(insert infinite amount of 3's) - the answer is simply 10 cannot be divided by 3 equally. To request 10 to be divided by 3 is an exercise in acceptable failure as it simply cannot be done.

Regarding a measurable quantity - shouldn't it stand to reason that if something can be measured than it must have a finite value? There must be a baseline from which to create a standard to accumulate value which proves an inconstancy in true quantity relative to the observer. Any inconstancy of an assumed infinite value automatically should disprove the previous assessed value of infinity.

We know the big bang dispersed a finite amount of energy & matter because we can measure it's current density & rate of expansion from our relative perspective (however enormously insignificant on the scale of reality). Had the big bang dispersed an infinite amount of energy & matter then no matter how tiny we peer at the quantum level or how far away we scan the vastness of space would be inundated with uncountable & unmeasurable quantity from every perspective. The very moment the observer can detect an absence of matter/energy proves finite value: infinity -minus- observed anomaly i.e. some measurable quantity which is less than true infinity.

 

[Tournesol]

"Had the big bang dispersed an infinite amount of energy & matter then no matter how tiny we peer at the quantum level or how far away we scan the vastness of space would be inundated with uncountable & unmeasurable quantity from every perspective."

Even if it were dispersed over an infinite distance ?

 

[matt grime]

My philosophical ramble applied to all things which are flagged as infinite. Using a basic math example with 10 divided by 3: the answer is not 3.333333333(insert infinite amount of 3's) - the answer is simply 10 cannot be divided by 3 equally. To request 10 to be divided by 3 is an exercise in acceptable failure as it simply cannot be done.

ten what divided by three what? stop confusing mathematics with real life. 3 does not divide ten in N, or Z, but there is an element of Q corresponding to (10,3) = (20,6)= ...

Regarding a measurable quantity - shouldn't it stand to reason that if something can be measured than it must have a finite value?

Only if you preclude infinity as being a valid outcome of a "measurement" whatever "measurement" may actually mean, and apparently in your postualtes that is "something that must be finite"

 

[zarback]

An infinite distance is a paradox in of itself. For there to exist a distance, there must exist the means to measure and reference points to the observer. That an observer can assign the very word "distance" implies finite value.

The true emptiness of space where our universe has not yet expanded into must be assigned a value of zero. Only upon being invaded by matter/energy does space become measurable with a quantity other than zero. An observation of true empty space cannot be possible as the observer itself alters that space and inadvertently adds a value other than zero by merely existing.

Therefore, it is easy think of "space" as being a volume or area, which denotes a measurement - or assigning a value to "space." Therein lies the problem, as true space has a value of zero.

The real question is: How big is zero?

 

[StatusX]

There are an infinite number of real numbers, but you can get the "distance" between 4 and 5 easily. And what do you mean by "assigning a value to space?"

 

[Tournesol]

"That an observer can assign the very word "distance" implies finite value."

That an observe can assign no finite distance implies infinite value.

 

[matt grime]

"That an observer can assign the very word "distance" implies finite value."

That an observe can assign no finite distance implies infinite value.

no, it means they can assign no finite value. Who says that there is a dichotomy

assignable with a finite value or is infinite?

 

[Nereid]

I'm in the slow class, what has (most of) this discussion got to do with the philosophy of science?

If it's maths, IMHO, these questions have been debated (and pretty much settled) quite a long time ago.

What am I missing?

 

[zarback]

There are an infinite number of real numbers, but you can get the "distance" between 4 and 5 easily. And what do you mean by "assigning a value to space?"

I dispute there being "real" numbers other than zero and one. Regardless of the quantity of similar matter/energy, everything that exists is absolutely unique however small on the quantum scale or large on the universal scale. Either something exists (1) or it does not (0). No two things are identical, however similar their properties.

 

[matt grime]

You appear to be using "exist" in a sense distinctly different from that that StatusX was doing.

 

[jackle]

I dispute there being "real" numbers other than zero and one. Regardless of the quantity of similar matter/energy, everything that exists is absolutely unique however small on the quantum scale or large on the universal scale. Either something exists (1) or it does not (0). No two things are identical, however similar their properties.

Lucky the invention of mathematics wasn't left up to you then.

 

[Philocrat]

It suddenly struck me today when I heard somewhere that "the universe is infinite". Well I know from physics that it isn't. So I thought about what could make somethine infinite. I mean a finite number of objects would make a finite entity, so there's no way you could get an infinite entity with finite objects. So what are the ingredients of an infinite entity?

But then again, numbers are finite, and they can be arranged in infinitely to form an infinite set of numbers. I'm so confused. I'd be delighted to see some books recommendation about this subject.

ALL INFINITIES ARE FINITE! We will know this when the entire human reality is re-engineered and we design a 'ZOOMABLE VISUAL FACULTY' (ZVF). If we can, then both the intellect and sense organs can zoom to their extensions (all scientific instruments) and their extensions to COP (Critical Observation Points), and nothing more. But ZVF, if we can design one, does not have extension as it is part of the perceiver or observer. ZVF can see beyond COP and can zoom to the very perceptual boundaries or limits of all objects under observation. ZVF can see infinities finitely (or should I say 'Finite Infinities').


WARNING: We must not permit machines to have ZVF before us humans. It would be suicidal.

 

[selfAdjoint]

ALL INFINITIES ARE FINITE! We will know this when the entire human reality is re-engineered and we design a 'ZOOMABLE VISUAL FACULTY' (ZVF). If we can, then both the intellect and sense organs can zoom to their extensions (all scientific instruments) and their extensions to COP (Critical Observation Points), and nothing more. But ZVF, if we can design one, does not have extension as it is part of the perceiver or observer. ZVF can see beyond COP and can zoom to the very perceptual boundaries or limits of all objects under observation. ZVF can see infinities finitely (or should I say 'Finite Infinities').


WARNING: We must not permit machines to have ZVF before us humans. It would be suicidal.

I don't know if this is meant seriously. If it was, it's bunk. Infinities behave differently from finite magnitudes. If you add two finite magnitude of the same size together, you get a larger magnitude. If you add two infinities of the same cardinality together, you don't get a different infinity, you get the same one exactly. $$\aleph_0 + \aleph_0 = \aleph_0$$ and so on.

 

[jackle]

If you add two finite magnitude of the same size together, you get a larger magnitude. If you add two infinities of the same cardinality together, you don't get a different infinity, you get the same one exactly. $$\aleph_0 + \aleph_0 = \aleph_0$$ and so on.

I have just been wondering about something that you might be able to help me with. In quantum field theory, some of the calculations only work because two infinities in them cancel out. I was wondering what it was that allowed them to get away with such a dirty trick! My understanding is that it is rare for this to happen. Are the infinities of the same cardinality, or do they have special properties or something?

 

[selfAdjoint]

I have just been wondering about something that you might be able to help me with. In quantum field theory, some of the calculations only work because two infinities in them cancel out. I was wondering what it was that allowed them to get away with such a dirty trick! My understanding is that it is rare for this to happen. Are the infinities of the same cardinality, or do they have special properties or something?

The infinities in QFT are of the form $$\lim_{x \rightarrow 0} \frac {1}{x}$$, and they occur inside integrals. They are removed by regulation, which is basically an acknowledgment that the theory cannot be trusted down to x=0, that is at 0 length scales; there is some limiting length below which the theory just doesn't apply. So the regulator is applied to gently turn off the interaction at very short lengths, going to zero at zero lengths.

This regulator, or "cutoff" will remain in the theory through all the various derivations as an undetermined constant, until it is time to calculate numbers, at which time, if the theory is "renormalizable", a parameter can be redefined to absorb it. If it would take an infinite number of parameter adjustments to do this, then the theory is "nonrenomalizable", and the traditional view was that you had no hope of getting meaningful numbers out of it. But many strange things are happening in theory and math these days, and among them, Witten has created a concept he calls "asymptotic safety", in which some nonrenormalizable theories might actually produce meaningful results if treated just right. So there's hope for us all.

 

[jackle]

Ah, thankyou!