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I'm not a mathematician and I may be misunderstanding some aspects of this concept.

According to the B-theory of time, the flow of time is an illusion, and every point in time exists equally. If this theory is accurate, then physical reality could potentially have an infinitely extended past, and the notion of an infinite regress of causes becomes a metaphysical possibility.

Given this framework, is it feasible for the universe to terminate in a state such as "heat death," with no possibility of rejuvenation? This idea appears to present a mathematical paradox to me. If time is infinite, it seems that an infinite sequence of events, including infinite big bangs and heat deaths, should have already occurred.

Is it mathematically plausible for an event to occur for the first time after an infinite interval has passed? Wouldn't this be analogous to an infinitely long rope, where one "half" is blue and the other "half" is a different color—a situation that is mathematically impossible?

Or is the B-theory of time only compatible with a universe/reality that infinitely renews itself?

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    I don't see any connection between the B theory of time and the idea that time extends infinitely into the past and/or future. Can you clarify that? Commented 2 days ago
  • @DavidGudeman The B-theory of time doesn't inherently necessitate that time be infinite, but the B-theory is very often used in philosophy to argue for the feasibility of an infinite past. The B-theory makes an infinite past possible by treating time as a static dimension where all points exist simultaneously, avoiding the need to traverse an infinite sequence to reach the present. My question is based on the scenario where both temporal infinitude and the B-theory turn out to be the correct descriptions of reality.
    – Blaxium
    Commented 2 days ago
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    There is no such thing as "mathematically plausible", it is either true or false there. There is no problem with infinitely long rope, half-blue half-red, it is perfectly non-paradoxical. There is no connection between infinite time and renewal. There can be a single big bang with a single heat death and nothing else happening ever, or something else happening at other times that has no resemblance to what we have seen or can imagine. I think you are confusing the shortlist of habitual patterns in your head with mathematical possibilities that are only limited by non-contradiction.
    – Conifold
    Commented yesterday

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No, an infinite sequence of events need not repeat, just as an infinite series of numbers need not repeat (e.g. the digits of pi). Your infinitely long rope example is obviously flawed, because we have in the real number line a counter-example: half the numbers are positive, half are negative, yet the line is infinite.

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The interpetation of time as either "A" (all the physical stuff exists now in a 3-dimensional space, there is no actual representation of a "time axis", and everything moves around and is modified constantly. Time, in this model, is not a feature of the universe itself, but an illusion.

"B" would be that instead, the universe is a 4-dimensional entity (3 space + 1 time dimension) that is totally unchanging, with time being as real as space, nd each imagined slice along the time axis is equivalent, with no specific point along the time axis being labeled as "now". Movement and change (of physical entities within the universe), in this model, is not a feature of the universe itself, but an illusion.

Neither of these has anything to do with the possibility or neccessarity of infinity or periodicity.

If this theory is accurate, then physical reality could potentially have an infinitely extended past, and the notion of an infinite regress of causes becomes a metaphysical possibility.

No, it says nothing about that. Nothing keeps the 4-dimensional "block" that the universe would be in "B" time from having a border in any of the dimensions. The same as the 3 spatial dimensions are not necessarily infinite (nobody knows for sure), the universal "block" could have a sharp edge/surface at the point 0 along the time scale (i.e., Big Bang), or if you take spatial expansion into consideration, the whole affair could be more like a 4-dimensional cone, with the tip being the Big Bang (in fact, if you look at visualizations of the Light Cone, that's exactly what the Light Cone of the event of the Big Bang would be).

Given this framework, is it feasible for the universe to terminate in a state such as "heat death,"

Sure. These time frameworks are just ideas, beyond theoretical, neither of them has any influence on anything we know or assume about the physicality of the universe. One possible scenario is the Big Chill, vividly shown in the speculative video Timelapse of the Future - basically eternal expansion, with everything just eventually slowly disappearing.

Light Cones are a useful tool for physicists to work with their maths, nothing more, nothing less.

with no possibility of rejuvenation?

Again, neither of the two interpretations has any impact on these things. If the universe should somehow contract eventually, instead of expanding unendingly, then who knows what would happen at the Big Crunch.

This idea appears to present a mathematical paradox to me. If time is infinite, it seems that an infinite sequence of events, including infinite big bangs and heat deaths, should have already occurred.

It seems that it's more an (understandable) misinterpretation of the word "infinity". First of all, it is absolutely possible to have infinity only in one direction (positive along the time dimension). Secondly, it is normal for things to converge to a plain, regular number when going to infinity. Thirdly, it is often more helpful to think in terms of "this expansion will never stop" instead of "this will expand infinitely". The latter has the danger of thinking that something that is not infinite may be eventually become infinite - this is a fallacy of course. For example, assuming the universe is infinite in size, you can travel linearly for an unending amount of time, but at each point in time there will be a concrete, finite distance you will be away from your starting point. At no point will you say that you are now suddenly "infinitely far away" from the starting point.

Is it mathematically plausible for an event to occur for the first time after an infinite interval has passed?

Maths would not say that an "infinite interval" has passed. They would say "an arbitrarily large interval" has passed, or "there is no limit to the length of the interval" or something like that. Infinity is not a number. Saying an interval is infinite is a short-hand for saying that it has no limit at one or both of its sides.

Wouldn't this be analogous to an infinitely long rope, where one "half" is blue and the other "half" is a different color—a situation that is mathematically impossible?

By now it should be clear - maths does not talk about physical ropes, and confusing mathematical constructs, especially when infinity is involved, with physical reality, is a way to total confusion. The one has nothing to do with the other. In regards to physical reality, infinity is a shorthand for "it's arbitrarily large and we can't put a limit on it (or there may be no limit)", not something you would directly work with.

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  • Thank you for your informative answer. The reason I'm confused is, it is often argued in philosophy that an infinite past is impossible for the same reasons you explained. The B-theory is used as a counterargument to argue that an infinite past is, in fact, possible. A philosophical paper I'm reading now says, "Craig suggests that it is impossible to obtain an actual infinite by successively adding one member after another. This is only true using an A-theory of time. If a B-theory is true, then this second argument is false and entire Kalam Cosmological Argument will crumble to the ground."
    – Blaxium
    Commented yesterday
  • @Blaxium I'll also post a proper answer, but assuming your quote is talking about WIlliam Lane Craig, it's worth mentioning that IIRC his arguments, they're not very mathematically sophisticated, and the idea that you can only "build up" to an infinite collection by successively adding is an example of that.
    – redroid
    Commented yesterday
  • @blaxium, well, it is what it is... suggests that it is impossible to obtain an actual infinite by successively adding one member after another is exactly the gist of my answer, and unless whoever wrote that defines the terms in a different manner than regular maths, then I can't really tell you more, unfortunately...
    – AnoE
    Commented 17 hours ago
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While the other answers are very good, I disagree with some parts of them. I think it's worth focusing on this sentence:

If time is infinite, it seems that an infinite sequence of events, including infinite big bangs and heat deaths, should have already occurred.

I think a lot of the confusion you're facing hinges on the idea of "already occurred," because you've lost sight of that being a relative quality - "already" compared to what? Even in A theory, it's still relative, but the framework of A theory gives you a default baseline, you'd be asking "already" compared to the present.

In the currently accepted idea of a one-ended history that starts but doesn't end, that has a straightforward, if counterintuitive answer: no, an infinite sequence of events hasn't already happened, because the universe is only X billion years old. Further, for any point in history you can actually name, you will only be a finite amount of time away from the Big Bang, so an infinite sequence wouldn't have happened by then either. As other answers have mentioned, the line of the natural numbers is infinitely long, but no particular number is actually infinitely far away from zero.

But perhaps history doesn't have a beginning. Mathematically, you can have a history that's infinite in both directions. There's nothing logically or formally problematic about that, it's just really weird to think about. I'll borrow a quote from a comment to illustrate

Craig suggests that it is impossible to obtain an actual infinite by successively adding one member after another.

This is obviously true in practice, but the reason its true is that you have to start counting at some point, and as I talked about previously, you never end up a non-finite distance away from your starting point. But if we're talking about an infinite past, this reasoning falls over - by definition, there is no starting point to an infinite past, just like there is no smallest negative number.

Another reason talking about an bidrectionally infinite past sounds strange is because its hard to "orient ourselves." With the number line, 0 is unique, but this doesn't really help us do physics because we could just decide to put the "0 time" somewhere else and all the physics remains the same. This is a very deep difference mathematically, where a bidirectional line counts as a "affine space" whereas a single-ended line doesn't.

However, philosophically, we do get a unique feature - our present, the time right now. In A theory, this is the only "real" time, whereas the ontological significance of "now" in B-theory is messier but epistemologically, as individuals reasoning at a specific time and place within the universe, it's distinguished either way.

So, having picked out "now," yes, there might well be an infinite sequence of events leading up to that point. And this is a very weird prospect that immediately raises a question of, "but how long did it take to get here?"

But this question is ill-formed because it runs into the same issue as earlier - to get here, from where (when)? With an infinite history, we have a weird inversion of the situation from earlier: no specific point in history is "infinitely long ago", and subsequently, there is no point where the history before that point is finite. We've not built up Craig's collection element by element, because there is no starting point - it's always been infinite, and stays infinite even as we continue to add to it. The formalization of being able to say, "...and so on..." and then continue adding things is called the ordinal numbers

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Is it mathematically plausible for an event to occur for the first time after an infinite interval has passed?

Yes. Since we're talking about mathematics, not physics, consider an altogether artificial example. Imagine a hypothetical immortal particle with a property, P, whose value is a function of time. Let the event of interest, E, be that P(t) exactly equals 1/2.

Now let's choose a form for P(t). How about a Gaussian form, P(t) = e-t2 (for a particular choice of units and reference points).

With these choices, P(t) is always between 0 (exclusive) and 1 (inclusive). E occurs exactly twice in our particle's life, at t = ± (log 2)1⁄2. There is nothing mathematically inconsistent between that and the range of t being unbounded in both directions, yielding an infinite expanse of time before E first occurs, and an infinite expanse of time after E last occurs.

Wouldn't this be analogous to an infinitely long rope, where one "half" is blue and the other "half" is a different color—a situation that is mathematically impossible?

What makes you think that's mathematically impossible?.

If we consider an infinite rope that is blue up to some point, and red from then on, then there is a variety of ways in which I can construct a bijective function that associates every red position on the rope with a unique corresponding blue position, and vice versa. Since for each red there is a distinct blue, and for each blue a distinct red, there must be an equal (cardinal) number of red and blue points. This is reasonable to describe as half blue and half red.

Or is the B-theory of time only compatible with a universe/reality that infinitely renews itself?

No, nothing about the B-theory of time requires any cycling or repetition of states.

For example, although the Gaussian P I use above does take each value other than 1 twice, it is also possible to construct functions that take a different value at every point, even subject to the constraint that their values be bounded both above and below. For example, the one having P(t) = e-t2 - 1 for t <= 0 and P(t) = 1 - e-t2 for t >= 0.

However, if the number of possible states of the universe is strictly smaller, in the relevant infinite sense, than the number of times, then the Universe cannot avoid revisiting states. This would be the case, for example, if the state of the universe could be described by a finite (even though very large) number of quantum numbers, and all real numbers were valid time coordinates. Then the number of states would be countably infinite, but the number of times uncountably infinite, making it impossible to map a distinct state to each time. In fact, infinitely many states would have to repeat infinitely many times each.

Even in such a case, though, it would not necessarily be the case that the universe would need to "renew" itself. The progression of the state of the universe would not necessarily need to be cyclic, and not every state would need to repeat. As far as I can see, it could still be consistent with something that progresses, asymptotically, toward a uniform distribution of energy, on average (a.k.a. the heat death of the universe). It would also be consistent with a starting point to time and an initial state that never repeats.

In particular, this idea ...

If time is infinite, it seems that an infinite sequence of events, including infinite big bangs and heat deaths, should have already occurred.

... is flawed. Time being infinite does not imply that any particular state must repeat, much less any sequence of states that characterizes a particular event. These can be equally as infinite as time.

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The B Series is just an abstraction created because McTaggart could not fathom the reality of the A Series. In The Unreality of Time McTaggart asserts a contradiction in the A Series

For it assumes the existence of time in order to account for the way in which moments are past, present and future. Time then must be pre-supposed to account for the A series. But we have already seen that the A series has to be assumed in order to account for time. (Page 468)

The difficulty is that there must be an observer to say when the present is, in order to distinguish past, present & future. But the observer must be in time. This is not entirely clarified by Heidegger's conception of authentic time, but it does break out of the circle of contradiction by distinguishing the authentic time of Being from objective time.

One might argue that Being must be in objective time, but not so. As noted here

"Time and the "I think" are no longer opposed to one another as unlike and incompatible; they are the same." Kantbuch, p. 197

The Cartesian cogito is the first point of departure, and as Heidegger finds, I am has its own time. Being & Time arrive together a priori, objective time follows a posteriori. Thus there is no contradiction in the A Series and the B Series is unnecessary. Putting the observer back in objective time resolves the incompleteness of the B Series, e.g. "infinitely extended past" (OP), which can then be dealt with in the normal manner.

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According to the B-theory of time, the flow of time is an illusion, and every point in time exists equally. If this theory is accurate, then physical reality could potentially have an infinitely extended past, and the notion of an infinite regress of causes becomes a metaphysical possibility.

You've made an unjustified jump from all times equally existing to thinking that time has an infinite past. Mathematicians have known for some time that inserting infinity in something requires justifying. There is no justification here.

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