Actual Infinity in Reality?

[Jehanne]

You correctly understand my point. Polymath does not because he is so sure that there is not problem with an infinite chain of evens that he doesn't even see the metaphysical impossibility of his statements. He just states them over and over because his math background says you can do math with potentially infinite sets so an actual infinite must exist. If it wasn't so frustrating, it would be a fascinating example on why Philosophy of Science should be the first course math and physics majors should take.

---

I can see the paradox: it is counter-intuitive to not have a start since we are accustomed to things having one. But why does that lead to an *impossibility*?
And I would say this is why math and physics should come first and philosophy later: most people simply haven't developed their intuitions prior to learning how things actually are or can be.

---

I can see the paradox: it is counter-intuitive to not have a start since we are accustomed to things having one. But why does that lead to an *impossibility*?
And I would say this is why math and physics should come first and philosophy later: most people simply haven't developed their intuitions prior to learning how things actually are or can be.

Essentially, as far as I can see, you are assuming that any process in the real world must have a start. Does that correctly state your position?

So, why do you think this is necessary?

Here's some of the Wikipedia article, "Hilbert's paradox of the Grand Hotel":

Hilbert's paradox of the Grand Hotel, or simply Hilbert's Hotel, is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated that a fully occupied hotel with infinitely many rooms may still accommodate additional guests, even infinitely many of them, and this process may be repeated infinitely often. The idea was introduced by David Hilbert in a 1924 lecture "Über das Unendliche" reprinted in (Hilbert 2013, p.730) and was popularized through George Gamow's 1947 book One Two Three... Infinity.[1][2]

Hilbert's paradox is a veridical paradox: it leads to a counter-intuitive result that is provably true. The statements "there is a guest to every room" and "no more guests can be accommodated" are not equivalent when there are infinitely many rooms.
Initially, this state of affairs might seem to be counter-intuitive. The properties of "infinite collections of things" are quite different from those of "finite collections of things". The paradox of Hilbert's Grand Hotel can be understood by using Cantor's theory of transfinite numbers. Thus, while in an ordinary (finite) hotel with more than one room, the number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert's aptly named Grand Hotel, the quantity of odd-numbered rooms is not smaller than the total "number" of rooms. In mathematical terms, the cardinality of the subset containing the odd-numbered rooms is the same as the cardinality of the set of all rooms. Indeed, infinite sets are characterized as sets that have proper subsets of the same cardinality. For countable sets (sets with the same cardinality as the natural numbers) this cardinality is ℵ 0 {\displaystyle \aleph _{0}} .[3]
Rephrased, for any countably infinite set, there exists a bijective function which maps the countably infinite set to the set of natural numbers, even if the countably infinite set contains the natural numbers. For example, the set of rational numbers—those numbers which can be written as a quotient of integers—contains the natural numbers as a subset, but is no bigger than the set of natural numbers since the rationals are countable: there is a bijection from the naturals to the rationals.

Hilbert's paradox of the Grand Hotel

[polymath257]

LOL. Sure there are. BTW, you are failing to prove that actual infinities exist. You bring up some potential infinities candidates or that potential infinities are used in some equations--but that is not the same, is it.

And, so, if no one can prove that actual infinities exist, that makes their existence "impossible"??

Perhaps the concept of 'logically independent' is relevant here.

Two systems, equally consistent internally: one with infinities and one without. Neither has internal contradictions. So the question is which is a better model of reality?

But that isn't a question that can be determined without observing reality, now is it?

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Yes, I am quite aware of the properties of infinite sets. And yes, the odd counting numbers form a countable set. So do the integers, the collection of fractions, the collection of numbers that are solutions of polynomial equations with integer coefficients, and many other sets.

Part of the issue is, as far as I can see, that there is more than one concept of 'larger' when dealing with sets. Cardinality (being in one-to-one correspondence) is often not the most relevant idea of size. Being a subset is another that can be more useful, verious concepts of density or probability are others.

[LadyForCamus]

@steve, you’re moving the goal posts out a bit here, asking for proof of an actual infinity. No one is claiming to have evidence of an actual infinity existing in the physical world. They’re only claiming that there is no logical contradiction.

[SteveII]

LOL. Sure there are. BTW, you are failing to prove that actual infinities exist. You bring up some potential infinities candidates or that potential infinities are used in some equations--but that is not the same, is it.

And, so, if no one can prove that actual infinities exist, that makes their existence "impossible"??

Yes, because then we are left with things like:

1. The knowledge that you cannot get to an actual infinity by successive addition,
2. You cannot have an infinite series of events in the past because we could never count down from 5, 4, 3, 2, 1...Now because there would still have to be an infinite amounts of events that still need to happen, and 
3. Thought experiments like Hilbert's Hotel which give you a series of absurdities when dealing with actual infinities. 

With nothing on the other side to even balance out the question (let alone be convincing), it would seem that even with these 3 reasons, we find an actual infinite of a real object to be metaphysically impossible.

[Jehanne]

I am done with this thread, but I am going to leave everyone with a link to a letter from Professor Don Page, a world renowned cosmologist and a believer in Jesus:

http://www.preposterousuniverse.com/blog...cosmology/

Note what Dr. Page has to say about past eternal universes. After all, if God has always existed, is it logically impossible that the Universe could have existed coeternal with him, or at least as part of him?

For me, the last word is with you, Steve, if you want it.

Good discussion!

Best,

Dawn

[GrandizerII]

There are many people that do think the B theory of time is correct (physicist, cosmologists, philosophers).

And in fact, Einstein himself was a B-theorist with regards to time, calling time (or rather, the flow of it) an illusion.

General nor Special Relativity do not entail the B theory of time--it only implies that such a theory my be correct. BUT, more importantly, your belief there is no causal connections is NOT a part of the B Theory of time. If you think it is, it should be easy to post a link from a nice concise article on the subject.

What do you think B-theory means? Have you at least looked up Wikipedia on this?

Here's an article for you to read:
http://www.exactlywhatistime.com/physics...stic-time/

Quoted below is a relevant paragraph from that article:

Modern physicists therefore do not regard time as “passing” or “flowing” in the old-fashioned sense, nor is time just a sequence of events which happen: both the past and the future are simply “there”, laid out as part of four-dimensional space-time, some of which we have already visited and some not yet. So, just as we are accustomed to thinking of all parts of space as existing even if we are not there to experience them, all of time (past, present and future) are also constantly in existence even if we are not able to witness them. Time does not “flow”, then, it just “is”. This view of time is consistent with the philosophical view of eternalism or the block universe theory of time (see the section on Modern Philosophy).

Time does not flow => B-theory of time, which is consistent with the other philosophy of time: eternalism.

Here's a roughly 30-minute documentary video for you to watch on the B-theory of time, and how it is supported by modern science (unlike the A-theory of time, which is not exactly a scientific theory, only a theory in name and by convention):





With respect to change and events, what does it mean for time to not really be flowing? And what do you think the symmetry of time implies (as entailed by the laws of physics)?

I am a perdurantist, meaning that I am the sum of all time instances of me (in this local universe, in case of a multiverse). This is a logical implication of eternalism.

That does not answer the question. Yes or no--are you the same person you were in 2010 or will be in 2020? If not, you have temporal parts that are causally connected in a specific direction--from earlier than to later than.

I am not the exact same person I was in 2010 nor will I be the exact same person in 2020 (if I'm alive then). However, by convention and because we are wired to do so, we perceive the self to be this enduring entity that actually crosses time. It doesn't mean, however, that it does. Each instance of me is stuck in one of the time moments that are a part of this space-time reality.

No, it does not mean that I have temporal parts that are really causally connected, just connected. And direction, even if there is one in the objective sense, does not necessarily imply causal connection.

Human consciousness perceives causality from our temporal perspective. It's not an illusion in the sense that it isn't perceived. Rather, it is an illusion in the sense that fundamentally, causality is not a feature of the underlying reality.

Again, you have not shown that to be part of the theory. Can you explain why science is almost entirely focused on causation if it is an illusion.  It seem the claim that it is an illusion is driven by something else rather than science.

Then you haven't read enough articles on the B-theory of time. It is logically entailed by the theory. If the flow of time is an illusion, then what we are perceiving is an illusion. All the changes and events we supposedly witness are not really happening. It's all psychological. But it nevertheless serves an evolutionary benefit for us to perceive changes and events.

And, of course, science is going to focus on causation and changes and events much of the time, because much of the time, we are analyzing the world from a temporal perspective, assuming the flow of time, and because it's not a useless way of analyzing the world from such a perspective. Saying that causality is an illusion, fundamentally and ultimately, does not mean it is a waste of time to understand how this actual local universe works from a temporal perspective. Different aims/purposes, and different questions about the world, mean different ways of studying the world. It's normally when you need to answers questions regarding time itself that you jump some levels and analyze the world from a vastly different perspective, one that is atemporal.

How many times have I asked this: " We could not have gotten to our current universe without an infinite amounts of universes already being created. We would still be waiting for an infinite amount of universe to be sparked before ours could be sparked--which will never happen, because there still needs to be an infinite more that need to come first. Why can't you address this!?"

I don't adhere to such a model of cosmology. I think all universes exist eternally and simultaneously.

NOT SO. The B theory of time does not entail an actual infinite. The standard big bang models all have our spacetime manifold with a definite beginning.

If time does not really flow, then what can one say about "past", "present", and "future" time moments, other than what I have been arguing the whole time about them?

And even if there was what we may call the "beginning" moment, this is being said from a temporal first-person perspective.

Anything with a beginning is by definition not an actual infinite.

Actually, not true.

Remember this set?

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... }

This is an example of an infinite set that has a starting number.

If something were to "start existing" it most certainly can be counted.

Which is irrelevant to everything I've been saying here.

One man's "counter-intuitive outcomes" is another man's absurdities. Notice the word I marked above - assume. If you assume an actual infinite, you are question begging.

We're assuming it in order to see if we can disprove it, via a reductio ad absurdum argument. Since you have failed to disprove it, then it's fair to say that an actual infinity seems logically possible. Or at least, we haven't seen a logical argument that proves it is not logically possible. And I'm being very generous here.

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Why are you counting *backwards*? Time moves forward! Nobody is counting backwards from today to the infinite past.

I'm not sure what you mean by there having to be an infinite number of *more* events before any particular moment of time. Where is the 'more'?

So, what we do *NOT* have is a situation

start----infinite time----now.

Instead, we have the situation for any point in the past,

infinite time----point in the past---finite time---now.

Bold mine.

I think what Steve means is, if we have time moving in a forward direction, how do we get to that one point in time from an infinite past?  How do you get to that single event in time without beginning somewhere?  If events are happening in succession, and time is infinite into the past, how would we ever arrive at a singular point in time?  Wouldn’t you have to start somewhere to get there?   I’m confused! Lol

Steve is assuming the A-theory of time, which is very problematic and a clear contradiction of modern science. What he seems to be doing is treating time as if it's this one entity (NOT a stream, but a "something" like a point/dot or whatever) which actually moves along from "somewhere" to "somewhere else", and wherever it hits, that is the "present moment". But according to science, time isn't anything like that. It's more of a dimension of space itself (similar to the x, y coordinates in high school algebra graphs), and all things in existence are already there and have already "happened". So there is no "arrival" to worry about. This "present moment" exists because it's always been there, being experienced by its corresponding instance of "you".

[polymath257]

And, so, if no one can prove that actual infinities exist, that makes their existence "impossible"??

Yes, because then we are left with things like:

1. The knowledge that you cannot get to an actual infinity by successive addition,
2. You cannot have an infinite series of events in the past because we could never count down from 5, 4, 3, 2, 1...Now because there would still have to be an infinite amounts of events that still need to happen, and 
3. Thought experiments like Hilbert's Hotel which give you a series of absurdities when dealing with actual infinities. 

With nothing on the other side to even balance out the question (let alone be convincing), it would seem that even with these 3 reasons, we find an actual infinite of a real object to be metaphysically impossible.

As at least partial answers,

1. All this means is that we won't be able to verify an actual infinity by just counting. It must take some *other* method (for example, using geometric considerations). This is, at least, the most serious of your objections: the possible  inabillity to verify an actual infinity. 

2. This is one that I still don't understand what you are attempting to say. Once we are *anywhere* in the sequence, there is only finitely far to go in the sequence (I assume you meant, by the way, ...., -5,-4, -3, -2, -1.. and not what you wrote. If you did mean what you wrote, I understand your objection even less). So, if you are at -100, there are only 100 more places to go to get to 0. At no point do you have an infinite time to wait. It is simply that every event has a previous event. THERE IS NO START. If you thing there is an infinite wait from some point, please let me know where it is.

3. Exactly which of the results for the Hilbert Hotel are *absurdities*? Every single one is a logical possibility. For example, if there is an infinite amount of time up to now, then there will be also an infinite amount of time up to a minute from now. If there are an infinite number of seconds previous to now, there are an infinite number of half-seconds previous to now. I don't see how the cardinalities being the same is in any way a problem with this being a reality. If anything, the Hilbert Hotel shows there is always room for 'extra time' even if time is infinite. Where is the *absurdity*?

So, exactly what do you mean by the phrase 'metaphysically impossible'? I understand an issue if there is an actual contradiction. But you have shown none. At worst, you have shown where infinities act different than finities. I hope that it isn't a surprise that they do.

---

Bold mine.

I think what Steve means is, if we have time moving in a forward direction, how do we get to that one point in time from an infinite past?  How do you get to that single event in time without beginning somewhere?  If events are happening in succession, and time is infinite into the past, how would we ever arrive at a singular point in time?  Wouldn’t you have to start somewhere to get there?   I’m confused! Lol

And the answer is that nothing was ever in the *infinite* past. Everything is finitely far in the past. But different things are different amounts in the past and there is not a bound to how far in the past they are.

So, take the progression that Steve seems to have problems with

...., -6, -5, -4, -3, -2, -1, 0, 1, 2, 3,.....

Show me two of those points that are infinitely far away from each other. You won't be able to: every distance between points is finite. And that is true even if the past goes infinitely far back. Those ellipses are NOT numbers. Any *actual* number will be only finitely far back.

[SteveII]

You correctly understand my point. Polymath does not because he is so sure that there is not problem with an infinite chain of evens that he doesn't even see the metaphysical impossibility of his statements. He just states them over and over because his math background says you can do math with potentially infinite sets so an actual infinite must exist. If it wasn't so frustrating, it would be a fascinating example on why Philosophy of Science should be the first course math and physics majors should take.

I can see the paradox: it is counter-intuitive to not have a start since we are accustomed to things having one. But why does that lead to an *impossibility*?
And I would say this is why math and physics should come first and philosophy later: most people simply haven't developed their intuitions prior to learning how things actually are or can be.

And there it is. Thinking that math and physics should come first before being properly trained to think is what the problem is (of course we are talking about the pursuit of advanced degrees). Both math and physics rely on the Philosophy of Science to even exist--yet you place it secondary to the subject. 

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I can see the paradox: it is counter-intuitive to not have a start since we are accustomed to things having one. But why does that lead to an *impossibility*?
And I would say this is why math and physics should come first and philosophy later: most people simply haven't developed their intuitions prior to learning how things actually are or can be.

Essentially, as far as I can see, you are assuming that any process in the real world must have a start. Does that correctly state your position?

So, why do you think this is necessary?

Because if a series of events did not have a start, the current events we are experiencing would never occur because there would still have to be an infinite number of events that must come first. You are suggesting that an infinite number of events have already happened, but that simply cannot be true. Since events are actual countable things, by definition, you could not have traversed an infinite amount of them to get to the current event. There will always have to be infinitely more events that still need to happen. This is not a "counter-intuitive" problem. This is a metaphysical impossibility. 

Again, don't you find it strange that you can't find a paper on an infinite series of events?

[polymath257]

I can see the paradox: it is counter-intuitive to not have a start since we are accustomed to things having one. But why does that lead to an *impossibility*?
And I would say this is why math and physics should come first and philosophy later: most people simply haven't developed their intuitions prior to learning how things actually are or can be.

And there it is. Thinking that math and physics should come first before being properly trained to think is what the problem is (of course we are talking about the pursuit of advanced degrees). Both math and physics rely on the Philosophy of Science to even exist--yet you place it secondary to the subject. 

---

Essentially, as far as I can see, you are assuming that any process in the real world must have a start. Does that correctly state your position?

So, why do you think this is necessary?

Because if a series of events did not have a start, the current events we are experiencing would never occur because there would still have to be an infinite number of events that must come first. You are suggesting that an infinite number of events have already happened, but that simply cannot be true. Since events are actual countable things, by definition, you could not have traversed an infinite amount of them to get to the current event. There will always have to be infinitely more events that still need to happen. This is not a "counter-intuitive" problem. This is a metaphysical impossibility. 

Again, don't you find it strange that you can't find a paper on an infinite series of events?

I disagree with both of your points. First, it is a good idea to see how the world works and how actual scientists work before attempting the philosophy of science.  Going this way prevents a LOT of errors philosophers make about how things 'must be' that are simply not true. Even if philosophy is *logically* prior to other areas (of which I am not convinced), it is better to put it later as a pedagogical technique.

So, yes, I would place doing these before the study of philosophy as a practice to learn how to think about things. Philosophy tends to be rarefied enough that without particulars in mind, it is very easy to go astray. And philosophers, especially those doing metaphysics, go astray as a matter of course. Such distinctions as potential vs actual infinity are properly discarded as outdated and simply wrong headed. Treatments of concepts like 'substance' need to be wholly revisited in the light of quantum physics. Concepts of time have to be reviewed based on what we know from relativity. By doing math and physics *first* we solve a lot of time on philosophical dead ends that take up way too much time in philosophy classes.

And, once again, there is no infinite traversal in infinite time. ALL points are finitely far from all other points. There would NOT be an infinite number that still 'need to happen'. If we are at a point 100 years ago, there is only 100 years yet to go to get to now. At *any* point you select, there is only a finite amount of time required to get to now.

I don't find it strange to not find a paper. Physicists tend to think philosophers are idiots. And they are mostly right about that. Way too many philosophers are still stuck in the ideas of Aristotle and Plato. They tend to promote their biases instead of really learning how things can or could be. Philosophers like to claim various things are 'impossible' or 'must be the case'. Physicists have enough experience to know such conclusions are often, even usually, wrong because of bias on the part of the philosopher. It has happened too many times in the past: Kant thinking Euclidean geometry is the only one possible (non-Euclidean geometry put that to rest), philosophers claiming a probabilistic scientific theory is impossible (Quantum Mechanics put that to rest), etc.

Philosophers spend way too much time getting themselves balled up in language and such things as the problem of universals. Truthfully, the paradoxes of language just show how little humans are logical. And philosophers fall all over themselves showing their best to uphold that tradition.

So physicists go and do their science and tend to ignore the people saying that things can't be how they are or must be how they are not They really don't think any argument is required for an actual infinity because the possibility is *logically* possible and the notion is useful in creating their theories. When those theories work and are testable in other areas, physicists tend to adopt the whole package until new evidence comes in. Philosophical arguments based on Aristotelian metaphysics just don't carry any weight for people actually doing science these days. Aristotle is a nice historical demonstration of how badly ideas can go wrong. That his ideas were taken as gospel for so long held us back a good 1000 years if not more.

[GrandizerII]

Why (Almost All) Cosmologists are Atheists

Interestingly, here's what Sean Carroll had to say on causality from the paper linked to in the quote:

In particular, we should emphasize that there is no place in this view for common philosophical concepts such as “cause and effect” or “purpose”. From the perspective of modern science, events don’t have purposes or causes; they simply conform to the laws of nature. In particular,
there is no need to invoke any mechanism to “sustain” a physical system or to keep it going; it would require an additional layer of complexity for a system to cease following its patterns than for it to simply continue to do so. Believing otherwise is a relic of a certain metaphysical way of thinking; these notions are useful in an informal way for human beings, but are not a part of the rigorous scientific description of the world. Of course scientists do talk about “causality”, but this is a description of the relationship between patterns and boundary conditions; it is a derived concept, not a fundamental one. If we know the state of a system at one time, and the laws governing its dynamics, we can calculate the state of the system
at some later time. You might be tempted to say that the particular state at the first time “caused” the state to be what it was at the second time; but it would be just as correct to say that the second state caused the first. According to the materialist worldview, then, structures and patterns are all there are — we don’t need any ancillary notions.

So it's not an idiosyncratic view I just happen to personally hold. This is the view on causality that physicists themselves adhere to as well.