## Clarifying Infinity

Discussions concerned with knowledge of measurement, properties, and relations quantities, theoretical or applied.

### Re: Clarifying Infinity

Brent696 » August 31st, 2018, 12:58 pm wrote:... and if it is infinite are there infinite planets, infinite stars, infinite amount of energy and matter.

Yes, the universe does appear to be spatially infinite. We have evidence of this from sky-mapping experiments a decade ago which established a high probability that the universe is “flat”; i.e. spatially infinite. If it is spatially infinite then under your own block model it must be temporally infinite as well, since each spacetime event has one time coordinate along with three spatial coordinates. An infinity of past events is no more problematic than an infinity of future events. And yes, this means an infinite number of planets, including an infinite number of inhabited planets, since in an infinite universe anything with a non-zero probability of happening will happen infinitely many times. Max Tegmark points out that in an infinite universe, we should expect an infinite number of duplicate or near-duplicate earths with duplicates of us on them. What we call the observable universe is merely that portion of the universe where the light from other objects has had time to reach us. The diameter of the observable universe is estimated to be 28.5 gigaparsecs. Tegmark estimates we would have to travel this distance multiplied by a factor of four to arrive at the first duplicate of our earth, and shake hands with ourselves thereupon.
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### Re: Clarifying Infinity

davidm » September 1st, 2018, 8:05 am wrote:Max Tegmark points out that in an infinite universe, we should expect an infinite number of duplicate or near-duplicate earths with duplicates of us on them.

Tegmark explicitly rejects physical infinity.

http://blogs.discovermagazine.com/crux/ ... g-physics/

I hope you can see that what you wrote was that Tegmark said that "IF this THEN that." He didn't claim the antecedent.
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### Re: Clarifying Infinity

BadgerJelly » September 1st, 2018, 7:40 am wrote:Can anybody explain how mathematical infinities can be added together?

https://en.wikipedia.org/wiki/Cardinal_ ... arithmetic

https://en.wikipedia.org/wiki/Ordinal_arithmetic

Briefly, there are two kind of transfinite numbers: cardinals and ordinals. Technically cardinals are ordinals these days but they're used differently.

Cardinals are added and multiplied easily. The sum or product of two cardinals is simply the larger one.

Ordinal arithmetic is described in the Wiki pages. Unless someone's interested, it's a little too involved to get into here.
Last edited by someguy1 on September 1st, 2018, 12:19 pm, edited 1 time in total.
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### Re: Clarifying Infinity

someguy1 » September 1st, 2018, 9:40 am wrote:
davidm » September 1st, 2018, 8:05 am wrote:Max Tegmark points out that in an infinite universe, we should expect an infinite number of duplicate or near-duplicate earths with duplicates of us on them.

Tegmark explicitly rejects physical infinity.

http://blogs.discovermagazine.com/crux/ ... g-physics/

I hope you can see that what you wrote was that Tegmark said that "IF this THEN that." He didn't claim the antecedent.

If is always implicit. Tegmark has also written:

A generic prediction of cosmological inflation is an infinite ergodic universe, which contains Hubble volumes realizing all initial conditions - including an identical copy of you about 101029 meters away.

Whether inflation is true, or whether the universe is actually spatially infinite, is an empirical matter. The most up to date evidence that I am aware of suggests that to a high degree of probability, the universe is spatially infinite.

Couldn't figure out how to reproduce that number in this software; that is a superscript 10 following the first 10, and then follows another superscript, the 29.
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### Re: Clarifying Infinity

davidm
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### Re: Clarifying Infinity

I refuted your point about Tegmark. You agree or just changing the subject? "Points to" is not proof. I hope you can see that.

One large objection to taking physical infinity seriously is that if someone claims physical infinity is true, an obvious question would be whether mathematical set theory applies. If there are infinitely many of something, is that a countable or uncountable infinity? Are the axiom of choice and the continuum hypothesis true or false? If a any physicist took physical infinity seriously, we'd see grant applications proposing to investigate these questions. Since we don't, we have to conclude that no matter what kind of wild speculations pop-scientists like Tegmark publish, nobody actually takes physical infinity seriously.
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### Re: Clarifying Infinity

someguy1 » September 1st, 2018, 10:16 am wrote:

I refuted your point about Tegmark. You agree or just changing the subject? "Points to" is not proof. I hope you can see that.

You did not refute my point about Tegmark. I believe the article you linked was from 2015. Ten years earlier, Tegmark held that the universe was spatially infinite, and adduced the implications of this idea. He did not claim to KNOW that the universe was spatially infinite, and since then, he may have changed his tune. He may now no longer believe in a spatially infinite universe. But it doesn't matter. Tegmark doesn't decide these things. Evidence decides them.

As to the linked article, I did not say that WMAP proved that the universe was infinite, only that it's findings pointed to an infinite universe. That's exactly why I didn't use the word "prove." Did you read the short article? WMAP bolstered prior evidence that the universe is probably spatially infinite.

"Proof" is not part of science. Science never proves anything, in the absolute sense.

I hope you can see that.
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### Re: Clarifying Infinity

In any case, the original point of contention here was Brent's claim that the universe cannot be infinite in space or time. Not only can it be that, the evidence suggests (but does not prove!) that it is that.
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### Re: Clarifying Infinity

davidm » September 1st, 2018, 10:26 am wrote:In any case, the original point of contention here was Brent's claim that the universe cannot be infinite in space or time. Not only can it be that, the evidence suggests (but does not prove!) that it is that.

I agree with you that Brent doesn't know for sure that the universe isn't spatially infinite. And you seem to agree with me that YOU don't know that it is. So we have two guys yakking at each other who don't know if what they are saying is true, yet are passionately insisting they're each right. Is that fair to say?

But you didn't engage with my set theoretic point. If the WMAP-ers claim the world is infinite, do they have an opinion on what its cardinality is and which of the standard axioms of set theory are true about the world? Union? Intersection? Powerset? Replacement? Choice? No, they don't say a word about it. Which means that they're good astronomers but not thinking at all about what it means for something to be infinite.

Physicists who are thoughtful about these issues DO recognize this. When an infinity pops up in a physics equation, they don't say, "Aha the world contains an actual infinity." Rather, they say, "Our equations break down in these circumstances." That's the appropriate degree of intellectual modesty required. You and Brent should try to emulate it. YOU DON'T KNOW. Just admit that.

ps -- Jeez man I read your WMAP link and it does not support your claim. "All we can truly conclude is that the Universe is much larger than the volume we can directly observe."

Wow. Aren't you embarrassed to be cherry-picking experiments then exaggerating what they say? If there was a big bang and the universe is flat, the universe may well be expanding "forever," but at any moment it's still finite. Astronomers aren't set theorists and use the word "infinite" very loosely, as you are doing here. Your point is refuted by your own link.
Last edited by someguy1 on September 1st, 2018, 12:55 pm, edited 4 times in total.
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### Re: Clarifying Infinity

A generic prediction of cosmological inflation is an infinite ergodic universe, which contains Hubble volumes realizing all initial conditions - including an identical copy of you about 101029 meters away.

Whether inflation is true, or whether the universe is actually spatially infinite, is an empirical matter. The most up to date evidence that I am aware of suggests that to a high degree of probability, the universe is spatially infinite.

Couldn't figure out how to reproduce that number in this software; that is a superscript 10 following the first 10, and then follows another superscript, the 29.

Use the full editor. It has superscript brackets. And I'm happy to hear my doppelganger doesn't live 60 miles down the road from me!

1010[sup]29[/sup]

Ahhh, I see the brackets don't nest properly. Sorry about that.

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### Re: Clarifying Infinity

Braininvat » September 1st, 2018, 10:33 am wrote:Use the full editor. It has superscript brackets. And I'm happy to hear my doppelganger doesn't live 60 miles down the road from me! .

BIV is it my browser or does math markup no longer work here? I always get broken images.
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### Re: Clarifying Infinity

ronjanec » August 30th, 2018, 3:39 pm wrote:
someguy, what is your personal definition for the concept of infinity/infinite?

I don't have a personal definition. I just use the standard one(s) from mathematics. Just as I don't have personal definitions of gravity, force, velocity, and pickles. I use the standard terms. That's exactly the point. People on this thread have a "personal" definition of infinity, which has had a standard mathematical definition for over 140 years. That makes it impossible to have a conversation, because certain posters are using "personal" definitions for technical terms that have perfectly well-understood standard definitions.

A set is finite if it can be bijected to a natural number 0, 1, 2, 3, ... It's infinite otherwise.

A set is Dedekind-infinite if it is in bijection with one of its proper subsets.

The two definitions coincide in the presence of the axiom of choice, and they differ in its absence.

Now those are the standard definitions. Why would I invent a personal definition at odds with those? We're having a scientific discussion. If I say, "My personal definition of gravity is that it's the force that makes bowling balls float up to the sky like helium baloons. And this proves Newton was wrong, because his theory predicts bowling balls fall down," that would not be helpful. I hope you agree.
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### Re: Clarifying Infinity

There is a decent article here from "Ask an Astronomer

But I will take a bit of an issue with the second to last paragraph as he speaks about the difficulty in defining Infinity. He states that we could begin counting 1,2,3,4, all the way up to infinity, the problem here is context because this is simply impossible. Hopefully there is some agreement to this being impossible, we do not count to a million, then billion, then trillion and at some point reach infinity.

We can never reach infinity in such a scenario, it might be more accurate to say it is infinitely impossible. Infinity and finitude simply cannot, in my view, occupy the same dimension of reality, where one is said to exist, the other cannot.

Within finitude, infinity exists only as a potential, never an actual. Theoretically the same is true in the reverse, if there is an actual infinity, then a finite condition could only appear as a potential reality.

Now if we think of space as true emptiness, then we might suppose our universe might be expanding outward into emptiness, but as Einstein determined, space is not such a nothingness, it is something, it bends, warps, and expands. Space then we might say is expanding, but not into more space, into what we don't know.

Back to Infinities, there are two infinities I can visualize, 1 and 0. 0 is obviously an infinite of nothing, and it can be said our finite universe is created from this infinite nothingness by 0=1+(-1), and matter arises, or the illusion of matter (Something) arises from the wave interference between the 1 and (-1), collapse both waves and infinity (0) pops into the equation.

1, as it relates to an infinity, is not to be understood as (1 of), but rather in the context of (1 and only 1). In this way 1 adapts itself to an Infinity of Being, a true Something.

Now I can't prove all this through a telescope, I can see this universe as a product of the Infinity of 0, Logically I see the instigator or our 1+(-1)=0 universe as that Infinity of 1 which would be transcendent to our present state of being, but that is not at issue here on this thread. I mention it only for the context of how I understand the two infinities relating to one another, and how 0=1+(-1) can fractionally produce all the variegatedness of our present reality.

In my opinion, these are the only infinities that can be understood as factual, even though one (0) possesses no true state of Being other than naught of 1.

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### Re: Clarifying Infinity

someguy1 » Sat Sep 01, 2018 11:20 am wrote:
ronjanec » August 30th, 2018, 3:39 pm wrote:
someguy, what is your personal definition for the concept of infinity/infinite?

I don't have a personal definition. I just use the standard one(s) from mathematics. Just as I don't have personal definitions of gravity, force, velocity, and pickles. I use the standard terms. That's exactly the point. People on this thread have a "personal" definition of infinity, which has had a standard mathematical definition for over 140 years. That makes it impossible to have a conversation, because certain posters are using "personal" definitions for technical terms that have perfectly well-understood standard definitions.

A set is finite if it can be bijected to a natural number 0, 1, 2, 3, ... It's infinite otherwise.

A set is Dedekind-infinite if it is in bijection with one of its proper subsets.

The two definitions coincide in the presence of the axiom of choice, and they differ in its absence.

Now those are the standard definitions. Why would I invent a personal definition at odds with those? We're having a scientific discussion. If I say, "My personal definition of gravity is that it's the force that makes bowling balls float up to the sky like helium baloons. And this proves Newton was wrong, because his theory predicts bowling balls fall down," that would not be helpful. I hope you agree.

someguy,

When I first responded to your thread, I never noticed that it was in the Mathematics forum, and I have to apologize for posting my strictly metaphysical/ontological type comments in your SCF thread.

My personal definition generally does work when you are talking about the concept of infinite things in general, but like you pointed out in your earlier comments, mathematicians have a different way of looking at this as opposed to things actually being able to exist (again) in an infinite state of being. Again, my apologies.
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### Re: Clarifying Infinity

Brent696 » September 1st, 2018, 11:37 am wrote:There is a decent article here from "Ask an Astronomer

You're the second person who's decided to get their ideas about infinity from astronomers. The WMAP scientists don't know anything about infinity. I wouldn't ask a set theorist about star formation. Don't ask astronomers about infinity.

Brent696 » September 1st, 2018, 11:37 am wrote:But I will take a bit of an issue with the second to last paragraph as he speaks about the difficulty in defining Infinity. He states that we could begin counting 1,2,3,4, all the way up to infinity, the problem here is context because this is simply impossible. Hopefully there is some agreement to this being impossible, we do not count to a million, then billion, then trillion and at some point reach infinity.

I most definitely agree that in practice it seems that we can't count to infinity. It takes a minimum amount of time to say a number, and our time is finite. The sun will go nova, the universe will collapse to another big bang or expand to heat death. Everything is finite. I perfectly well agree with your point.

HOWEVER. I wonder if you would consider the following point of view. In math, we have infinite sets. I hope that people will grant that while there may well not be actual infinities in the world, there certainly are in math. Cantor's paradise, as Hilbert put it.

So now consider this. In the history of math and science, there have been many instances of mathematical objects that at first seemed absurd, yet later became essential to our understanding of the world. Irrational numbers, negative numbers, imaginary and complex numbers. The useless and bizarre theory of non-Euclidean geometry in 1840 turned out to be the true spacetime geometry of the universe in 1905.

The most striking example is factoring integers. It's an ancient problem regarded as supremely beautiful and supremely useless. Until the 1980's when someone used factoring to invent public key cryptography, which is now the basis of all e-commerce, blockchain technologies, and international cyber-spying. This is an example of a mathematical pursuit that was absolutely useless for 2000 years then became absolutely essential, overnight -- in our own lifetimes.

So here is my thesis. Tell me what you think.

* We have a long history of mathematics discovering something purely abstract, "obviously" false about the world, but a nice plaything for the mathematicians with their head in the clouds. And then one day some physicist finds a use ... and after a while, what was once absurd becomes received wisdom.

* Actual infinity is one such example of something mathematicians discovered and built a big and beautiful theory about, that appears to be totally false about the real world.

* Is it not then at least POSSIBLE that someday -- tomorrow morning, or 500 years from now -- professor so-and-so in Helsinki will drop a paper showing that assuming an actual infinity explains some laboratory phenomenon to an astonishing degree of accuracy. Everyone will say, "Well of course there's no REAL infinity, it's just a model that makes our calculations easier." They said that about heliocentrism. Then the old scientists die off and the young ones take over the profession, and to them, the new paradigm is as real as a rock you can kick.

This pattern has been repeated so often through the history of science that we must allow that it is POSSIBLE that someday, the mathematical theory of actual infinity will turn out to be a part of the world.

You can't say it's not possible. That would be my thesis. What do you think?

Note: I object when people say the universe is infinite, and I object when people say it isn't. I'd better clarify my position.

First, I don't know whether there's an actual infinity in the world. But second, I think that the case for either side is exceedingly weak. I've never seen an argument on either side of the question that's any good! That's why I tend to object to pretty much any opinion anyone has about this subject. I don't think anyone's thought seriously enough yet about the question of infinity in physics. Tegmark on down, I think it's all bunk. I hope that puts my various posts into context. I don't agree with anything I've ever read about this subject, from the professionals or the amateurs.
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### Re: Clarifying Infinity

ronjanec » September 1st, 2018, 2:12 pm wrote:When I first responded to your thread, I never noticed that it was in the Mathematics forum, and I have to apologize for posting my strictly metaphysical/ontological type comments in your SCF thread.

It's ok, I was just trying to corral the general discussion, not you in particular. Everyone here has their own "personal" definition and these generally differ substantively from the mathematical one. Sure, infinity has a lot of philosophical importance, but there's also a 140 year old scientific theory of infinity and at least people should pay lip service to it. But really, it's a lost cause. I'm not annoyed at anyone in particular here. Pretty much everyone is confused about infinity.

ronjanec » September 1st, 2018, 2:12 pm wrote:My personal definition generally does work when you are talking about the concept of infinite things in general, but like you pointed out in your earlier comments, mathematicians have a different way of looking at this as opposed to things actually being able to exist (again) in an infinite state of being. Again, my apologies.

Because you apologized twice I realize I may have come on too strong. It's not you in particular, it's the entire world that's confused on this point. And not just on Internet forums, but also among the pop-scientists like Tegmark, whose name is universally invoked in these conversations.

Anyway I am not the moderator and I have no power to keep anyone on any topic. I just think that people's vague ideas about infinity tend to confuse these discussions.

Anyway thanks for responding, no apologies necessary but definitely accepted.
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### Re: Clarifying Infinity

someguy1 » September 1st, 2018, 4:27 pm

You're the second person who's decided to get their ideas about infinity from astronomers. The WMAP scientists don't know anything about infinity. I wouldn't ask a set theorist about star formation. Don't ask astronomers about infinity.

Well....the thread did say "clarifying";-) but yeah, I was only answering infinity space as a possibility. Now you are shifting me to infinity in math, which is fine, I think

someguy1 » September 1st, 2018, 4:27 pm

I most definitely agree that in practice it seems that we can't count to infinity. It takes a minimum amount of time to say a number, and our time is finite. The sun will go nova, the universe will collapse to another big bang or expand to heat death. Everything is finite. I perfectly well agree with your point.

Well this was not about limited by time or such, it is simply impossible to achieve, thinking about Infinite as the "Whole", of which numbers are merely divisions, MIGHT not be how we should approach infinity, hence my separation between actual and potential.

someguy1 » September 1st, 2018, 4:27 pm

HOWEVER. I wonder if you would consider the following point of view. In math, we have infinite sets. I hope that people will grant that while there may well not be actual infinities in the world, there certainly are in math. Cantor's paradise, as Hilbert put it.

For the most part one might say I was speaking of the phenomenal universe as actual. Mathematical infinities I can see has having a life of their own, they are real in the sense that math inhabits its own dimension. And I use the word "dimension" in commonality.

Math to me describes the heavenly ideal of what our physical state can only shadow. In mystical traditions it can be said that the square-cube describes the phenomenal, the circle-sphere the infinity, and the triangle-pyramid the bridge between the two. (And I am still speaking from a laic POV) And beneath this reality lies the Infinite 0, which is what I tend to think shows up in Physics at which point they start looking for strings or whatnot to maintain a somethingness to reality. (If that makes any sense), so, moving on

someguy1 » September 1st, 2018, 4:27 pm

So now consider this. In the history of math and science, there have been many instances of mathematical objects that at first seemed absurd, yet later became essential to our understanding of the world. Irrational numbers, negative numbers, imaginary and complex numbers. The useless and bizarre theory of non-Euclidean geometry in 1840 turned out to be the true spacetime geometry of the universe in 1905.

The most striking example is factoring integers. It's an ancient problem regarded as supremely beautiful and supremely useless. Until the 1980's when someone used factoring to invent public key cryptography, which is now the basis of all e-commerce, blockchain technologies, and international cyber-spying. This is an example of a mathematical pursuit that was absolutely useless for 2000 years then became absolutely essential, overnight -- in our own lifetimes.

This I can understand this as our understanding of the phenomenal world has DEEPENED, so have we come closer to those relationship or maybe mathematical purities of which our phenomenal world shadows. I mean technology is nearly math becoming manifest, but in a different vein than say a flower does.

someguy1 » September 1st, 2018, 4:27 pm

So here is my thesis. Tell me what you think.

* We have a long history of mathematics discovering something purely abstract, "obviously" false about the world, but a nice plaything for the mathematicians with their head in the clouds. And then one day some physicist finds a use ... and after a while, what was once absurd becomes received wisdom.

* Actual infinity is one such example of something mathematicians discovered and built a big and beautiful theory about, that appears to be totally false about the real world.

* Is it not then at least POSSIBLE that someday -- tomorrow morning, or 500 years from now -- professor so-and-so in Helsinki will drop a paper showing that assuming an actual infinity explains some laboratory phenomenon to an astonishing degree of accuracy. Everyone will say, "Well of course there's no REAL infinity, it's just a model that makes our calculations easier." They said that about heliocentrism. Then the old scientists die off and the young ones take over the profession, and to them, the new paradigm is as real as a rock you can kick.

This pattern has been repeated so often through the history of science that we must allow that it is POSSIBLE that someday, the mathematical theory of actual infinity will turn out to be a part of the world.

You can't say it's not possible. That would be my thesis. What do you think?

Oh "possibles", certainly anything is possible, isn't that what they say..... and would I be a closed minded doctrinal thinking thinker if I would deny such possibilities.....and why does this argument always work for those in science and never for the theologians.....ahhhh.(Rhetorical)

Perhaps I might posit the Infinities are like the holes in the universe, the wholes in math, they just seem to kind of swallow every "thing" us. I mean its the Infinite Being of a God that would be an all consuming fire, swallowing up the identities and Being of all else in existence. This universe will collapse and be swallowed up in the infinite nothingness from which it has been parceled out. In math class, at least in junior high, if you brought an Infinite into an equation everything else just disappeared.

I don't know what to tell you SG, Infinities just seem to have a specific purpose, I can agree infinities might exist in some creative ways, mathematical cul de sac slight of hands, infinity mirrors I suppose, but I just can't seem to align one with any thing finite without watching such finitude cease to exist, simply because, and this might be philosophical, But THAT is what Infinities do, consume everything.

But it is possible that I do not exist, so my opinions might not either.

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### Re: Clarifying Infinity

someguy,

Actually, I am not confused at all about “infinity”, and I have spent a lot of time thinking about this concept in regards to objective reality(and that includes mathematics in the same sense).

And you did not come on too strong with me because I apologized a couple of times in my response to you: I actually felt bad about turning your personal SCF thread into a philosophical/ontolological discussion(at least from my end), because of my mistake in again not realizing that this thread was in the Mathematics forum(duh)
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### Re: Clarifying Infinity

someguy1 » September 1st, 2018, 11:43 pm wrote:
BadgerJelly » September 1st, 2018, 7:40 am wrote:Can anybody explain how mathematical infinities can be added together?

https://en.wikipedia.org/wiki/Cardinal_ ... arithmetic

https://en.wikipedia.org/wiki/Ordinal_arithmetic

Briefly, there are two kind of transfinite numbers: cardinals and ordinals. Technically cardinals are ordinals these days but they're used differently.

Cardinals are added and multiplied easily. The sum or product of two cardinals is simply the larger one.

Ordinal arithmetic is described in the Wiki pages. Unless someone's interested, it's a little too involved to get into here.

It is the term “larger” when talking about infinities that gets me. Can you explain this jargon a little more please? Does it simply refer to infinite sets having more numbers over some finite range? Like saying there are more odd number from 1-100 than prime numbers?

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### Re: Clarifying Infinity

someguy1 » September 1st, 2018, 9:36 am wrote:
Braininvat » September 1st, 2018, 10:33 am wrote:Use the full editor. It has superscript brackets. And I'm happy to hear my doppelganger doesn't live 60 miles down the road from me! .

BIV is it my browser or does math markup no longer work here? I always get broken images.

I tried it and also got a broken image. Will look into this.

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### Re: Clarifying Infinity

Potential Infinite v. Actual Infinite

One of the most important contributions that Aristotle had made to to study of infinity is identifying a dichotomy between what Aristotle calls the “potential infinite” and the “actual infinite”.

The potential infinite is a group of numbers or group of “things” that continues without terminating, going on or repeating itself over and over again with no recognizable ending point. What distinguishes the potential infinite and gives it the characteristic of being “potential” is the idea that if one were to take a sliver, or examine just one isolated portion of that infinite set of numbers, one would be able to capture or observe a finite set of numbers. The obvious example is the the grouping of natural numbers. No matter where you are while listing or counting out natural numbers, there always exists another number to proceed the one before. Also, a geometric line with a starting point could extend on without end, but could still be potentially infinite because all one would have to do is add on more length to a finite length to allow it to extend.

The actual infinite involves never-ending sets or “things” within a space that has a beginning and end; it is a series that is technically “completed” but consists of an infinite number of members. According to Aristotle, actual infinities cannot exist because they are paradoxical. It is impossible to say that you can always “take another step” or “add another member” in a completed set with a beginning and end, unlike a potential infinite. It is ultimately Aristotle’s rejection of the actual infinite that allowed him to refute Zeno’s paradox.

(Image can be found on site http://sites.middlebury.edu/fyse1229pis ... -infinite/)

Although Aristotle did disprove the existence of the actual infinite finally, and tended to reject a lot of major concepts in mathematics, the importance of mathematics was still never belittled in Aristotle’s eyes. Aristotle argued that actual infinity as it is not applicable to geometry and the UNIVERSAL, is not relevant to mathematics, making potential infinity all that actually is important.

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### Re: Clarifying Infinity

BadgerJelly » September 2nd, 2018, 2:21 am wrote:
It is the term “larger” when talking about infinities that gets me. Can you explain this jargon a little more please? Does it simply refer to infinite sets having more numbers over some finite range? Like saying there are more odd number from 1-100 than prime numbers?

There's a well-developed theory of infinite sets that was started by Georg Cantor in 1874. Briefly, we define an infinite set as "countable" if it can be put into one-to-one correspondence, or bijection, with the set of counting numbers 0, 1, 2, 3, ...

So for example the natural numbers themselves are countable. Surprisingly, the rational numbers (all the fractions liks 3/4 and 5454/23) are also countable. I posted a visual proof earlier. And the set of algebraic numbers is countable. These are all the numbers, rational or irrational, that are the roots of polynomials with integer coefficients.

Now one may say, well, these sets are infinite after all, so they are ALL bijectable to each other. But no! Cantor showed, with his famous diagonal argument, that the real numbers can NOT be put into bijection with the counting numbers.

In this case we say that the cardinality of the reals is "larger" than that of the counting numbers. We don't mean that it's literally larger, only that there's an injection (a function that sends distinct elements to distinct elements) to another, but no injection in the other direction.

Cantor then showed that evere set is "smaller," in the same sense, to its power set -- the set of all of its subsets. So we get an endless hierarchy of larger and larger infinite sets: The counting numbers, their power set, the power set of the power set, and so forth.

I can't really do justice to all of this in a forum post, but feel free to ask questions. Bottom line is that since 1874 all of this has become a standard and well-understood part of mathematics, and set theorists have taken the subject much farther, routinely working with huge infinite sets so large they can't even be proved to exist within the standard axiomatic framework of math. These are the so-called large cardinals.

It's with all this in mind that I routinely criticize posters who have their own private, personal definitions of infinity. It's ok to do that, as long as one is aware of the standard theory and is claiming that it's wrong or that they don't like it or that they have a different idea. But there is already a well-established theory of mathematical infinity, and all conversations about infinity should at least acknowledge the existence of that theory and its importance in modern math.

Now of course none of this theory has anything to do with the real world, as far as we know. For example there is no collection of physical objects that matches the entire set of counting numbers 1, 2, 3, 4, ... There are only finitely many atoms in the known universe. On the other hand, modern physics uses infinitary math as the basic setting for its equations. How can this make sense? Well, that's the question. We have this elaborate theory of infinite sets that has seemingly no relation to the real world ... except that somehow, it does. That's a mystery.

https://en.wikipedia.org/wiki/Georg_Cantor

https://en.wikipedia.org/wiki/Countable_set

https://www.homeschoolmath.net/teaching ... ntable.php

https://en.wikipedia.org/wiki/Cantor%27 ... l_argument

https://en.wikipedia.org/wiki/Cantor%27s_theorem

https://en.wikipedia.org/wiki/Large_cardinal
Last edited by someguy1 on September 2nd, 2018, 1:02 pm, edited 1 time in total.
someguy1
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### Re: Clarifying Infinity

Thanks

Was trying to pull this thread away from metaphysics. I won’t pretend to understand exactly what you mean above I’d have to put a lot more serious work into understanding pure mathematics before I could ask much more regarding infinities.

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### Re: Clarifying Infinity

BadgerJelly » September 2nd, 2018, 11:00 am wrote:Thanks

Was trying to pull this thread away from metaphysics.

Thank you for that!

BadgerJelly » September 2nd, 2018, 11:00 am wrote:I won’t pretend to understand exactly what you mean above I’d have to put a lot more serious work into understanding pure mathematics before I could ask much more regarding infinities.

This link is for home-schooled high school kids.

https://www.homeschoolmath.net/teaching ... ntable.php

After that, the next step is the Wiki page on the Cantor diagonal argument.

This material is accessible to anyone. No math background is needed. Please feel free to ask questions. It shocked professional mathematicians back in 1874 and it's still surprising to everyone who encounters these beautiful ideas. There is really no math prerequisite at all.
someguy1
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### Re: Clarifying Infinity

someguy1;

What's non-constructive about the Peano axioms?

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There is no implication that they are. They are just one definition of integers.

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Generation of a list requires time. If there is no last element, then the list is never complete. "Infinity" is not a number, thus it is not quantifiable.

To say 'the number of elements in the continuum interval 0 to 1 is infinite', is meaningless. If points are dimensionless, there is no solution for n in n*0=1.

This is just another instance of playing the supreme role, trying to make something from nothing.

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Cantor was a self appointed spokesman with a divine revelation of the infinite. His biography reveals his worry about his standing in the mathematical community. The 'messenger' syndrome would supposedly boost his credibility, but there was little acceptance.[1]

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Cantor does make a distinction between 'immanent reality', numbers as mathematical concepts, and 'transient reality', instances of application of numbers in nature.

ref:

1. Ewald, W., From Kant to Hilbert, Oxford 1996.
phyti
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### Re: Clarifying Infinity

someguy1 » September 2nd, 2018, 10:54 am wrote:[
Now of course none of this theory has anything to do with the real world, as far as we know. For example there is no collection of physical objects that matches the entire set of counting numbers 1, 2, 3, 4, ... There are only finitely many atoms in the known universe.

The known universe means the observable universe -- our Hubble volume. WMAP indicates the universe is probably flat, hence spatially infinite. If so, there are an infinite humber of atoms.
davidm
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### Re: Clarifying Infinity

The known universe means the observable universe -- our Hubble volume. WMAP indicates the universe is probably flat, hence spatially infinite. If so, there are an infinite humber of atoms.

"Flat" does not necessitate infinite, so your assertion "hence spatially infinite" might exist as a secondary possibility, but is without certainty also. As for an infinite number of atoms, this is another misunderstanding, if there were an infinite number of atom, there would be infinite number of particles, electrons, protons, photons, etc, ultimately there would be no SPACE. No object, no "thing" can be infinite, this mistake is when we think of infinity as a conclusion to a finite, as if by counting one could somehow reach infinity.

Finite and Infinity cannot co-exist.

ESA: Is the Universe finite or infinite?

Joseph Silk:
We don't know. The expanding Universe theory says that the Universe could expand forever [that corresponds to a 'flat' Universe]. And that is probably the model of the Universe that we feel closest to now. But it could also be finite, because it could be that the Universe has a very large volume now, but finite, and that that volume will increase, so only in the infinite future will it actually be infinite.

ESA: It sounds like a game of words, is it?

Joseph Silk:
No. We do not know whether the Universe is finite or not. To give you an example, imagine the geometry of the Universe in two dimensions as a plane. It is flat, and a plane is normally infinite. But you can take a sheet of paper [an 'infinite' sheet of paper] and you can roll it up and make a cylinder, and you can roll the cylinder again and make a torus [like the shape of a doughnut]. The surface of the torus is also spatially flat, but it is finite. So you have two possibilities for a flat Universe: one infinite, like a plane, and one finite, like a torus, which is also flat.

ESA: ‘Flat' seems to have a different meaning to non-scientists. By 'flat' we understand to be like a table, which has width. Does the Universe have width?

Joseph Silk:
Flat is just a two-dimensional analogy. What we mean is that the Universe is 'Euclidean', meaning that parallel lines always run parallel, and that the angles of a triangle add up to 180o. Now, the two-dimensional equivalent to that is a plane, an infinite sheet of paper. On the surface of that plane you can draw parallel lines that will never meet. A curved geometry would be a sphere. If you draw parallel lines on a sphere, these lines will meet at a certain point, and if you draw a triangle its angles add up more than 180o. So the surface of the sphere is not flat. It's a finite space but it's not flat, while the surface of a torus is a flat space.

Brent696
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### Re: Clarifying Infinity

Your quoted material demonstrates my case and not yours.

It has long been known that space might be infinite. I recall reading George Gamow's discussion about this as a kid -- about a topology of positive curvature, negative curvature, or no large-scale curvature (i.e., flat, or infinite) for space. Evidence now suggests that the universe is probably infinite. Even if it is not, it CAN be -- in opposition to your claim that it can't be.
davidm
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### Re: Clarifying Infinity

Brent696 » September 3rd, 2018, 2:40 pm wrote:As for an infinite number of atoms, this is another misunderstanding, if there were an infinite number of atom, there would be infinite number of particles, electrons, protons, photons, etc, ultimately there would be no SPACE.

Why is that?
davidm
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### Re: Clarifying Infinity

Seems a bit like saying there can't be any odd numbers because of the even numbers crowding them out in their infinitude. Brent, an infinite space is a continuum and has abundant room for infinite sets of anything - atoms, beach balls, short bald men named Balthazar, you name it. Have you read the thread here prior to posting? If nothing else, look up Hilbert's Hotel, and see how there's always room for more guests.

TheVat