I don’t think the past can have been endless. If it were it would never have gotten here.
Agreed.
the past never got here - the future is not here - the only here here is the present - and that is fleeting.
While we can say that ‘x moments’ have passed since an event in the past (thanks to our understanding of our journey down time) - we have no abiltity to get to the past or the future.
Has nothing to do with us passing thru the past to get to the present - but it was always the present even when it was the past.
You’re fallaciously equating two different things: the totality of time, and the time that has passed. Again, these are the same for finite pasts, but not for infinite ones. In an infinite past, there never was a moment such that from that moment, an infinite amount of time would have passed to get to the present. Because you can’t start at ‘-infinity’.
Let’s see where I loose you.
Take things down a notch to the finite level. Consider a line with 20 units on it. The totality of moments of time modelled on such a line is 20. The maximum amount of time passed in the universe/for the universe likewise is 20 (moments, minutes, seconds, have your pick). That’s because there is an earliest point—all the way to the left of the line—from which one has to wait 20 units of time until one reaches the present, all the way to the right of the line.
Are we agreed so far?
Now, consider an infinite line. As before, the totality of time is just the cardinality of the line—the amount of subdivisions of the line. Since we’re talking about natural numbers, there’s aleph-zero of those, but that doesn’t matter for the moment.
Still on board?
Then let’s move on to time actually passing. We do the same thing we did as in the finite case: the amount of time that passes is measured by going to some specific point on the line, and then waiting. The amount of units of time we pass by, the amount of subdivision of the line we cross, is the amount of time elapsed. This is, really, the key issue, so it’s important that you try to visualize it: time passed is a length on the line. It’s the difference between two points on the line.
This is just the same thing as in the finite case, where we said that the total amount of time that can maximally pass is 20 units; this number was arrived at in just this way. For time to pass, you have to be at one moment, and wait until you are at a different moment.
I guess this is where we part ways. Am I right?
If so, try to figure out what ‘time passing’ means. If it’s now 12:25, and I wait five minutes, it’ll be 12:30, and five minutes will have passed. But crucially, it needs to have been 12:25 first such that five minutes can pass—if the universe were only created at 12:27, if it’s now 12:30, the universe hasn’t experienced five minutes of time; in no sense have five minutes elapsed.
So, time passing means that it’s some time now, and later, it’ll be, well, later. Any interval of time that could pass needs a beginning and an end.
Does this make things any more clear? No two points in time between which a given amount of time lies, no such amount of time can elapse. The total amount of time elapsed in the finite case is 20 units because there are 20 units of time between its beginning- and end-point.
If now an infinite amount of time passes, that is, if there is an interval such that there is an infinite amount of time between its endpoints, then you are right: such an interval can (on a plausible construal of temporal progression, anyhow) never be traversed.
But on an infinite line, there is no beginning- and end-point between which we have an infinite distance. It is simply infinite: without beginning or end. And in fact, there are no two points between which lies an infinite distance. But since there are no such two points, an infinite amount of time can’t elapse. Saying that an infinite amount of time elapses anywhere on an infinite line is like saying 30 units of time elapse on a finite line with a length of 20 units: there is no such distance to traverse, no interval of this size to cover.
There can never be a now such that the present is infinitely removed from it. It can’t be -infinity o’clock in the sense it can be 12:25 o’clock. But it needs to have been 12:25 o’clock for 12:30 o’clock to be five minutes later—or if you will, renaming 12:30 o’clock 0 o’clock, meaning the present, in order for five minutes to have elapsed, it must have been -0:05 (12:25) o’clock earlier.
So for an infinite amount of time to have elapsed, if it is now 0 o’clock, it must have been -infinity o’clock earlier. But such a time simply does not exist; there is no such point on the line.
The question you must always ask yourself if you claim that an infinite amount of time has passed is: since when? When I say, five minutes have passed, and it’s now 12:30, then five minutes have passed since 12:25. However, when I want to say, infinitely many minutes have passed, then there is no time in the infinite past such that I can say, infinitely many minutes have passed since then. Thus, in fact, infinitely many minutes haven’t passed, since there is no time since when they could have passed, even if the past is infinite.
Again, the next time you want to claim that an infinite amount of time has passed, I want you to tell me since when. From what point in time has an infinite amount of time passed?
But you don’t. Again, where do you start? Because from any point you could start, you have to pass only finitely many integers.
Please, try to make it more precise what infinitely many integers/infinitely many intervals (as you say further down) you have to pass, and where you start. Because to me, at any point in time n minutes ago, I have to pass only the n-minute interval separating that point and now. Even if you want to argue, you have to pass the n-minute interval, the n-1-minute interval, the n-2-minute interval, etc., these only would be finitely many intervals.
In fact, even if you counted all the intervals between a point in the past, and all other points in the past lying between that point and the present, these would still be finitely many (say for 3 minutes to the past, that would be the interval between 3 and 0, between 3 and 1, between 3 and 2, between 2 and 1, between 2 and 0, and between 1 and 0—seven in total; that’s what my point (4) above was about, by the way).
So, list the intervals in such a way that we could enumerate them—of course, not all of them (if they’re infinitely many as you claim, that would take a bit too long), but find some systematic way to order them, such that each of those intervals can at least be named, pointed to, or something. Try and find a systematic way of writing them down, like I did above when I paired every negative number with a rational number (a fraction) between -1 and 0: there’s infinitely many of both, but in giving the scheme I did, I made sure all are accounted for. But I don’t see where your infinities are hiding.
Moving on.
Having no boundary, e.g. no least/greatest value (the set of natural numbers is unbounded in this sense).
I meant in that quote ‘time elapsed’, or ‘time to be traversed’. Time elapsed is not the same as the total amount of time, which is the case even in a universe with a beginning, but no end: there’s an infinite amount of time (there is no boundary to the future, no ‘highest’ value the clock can read), but no matter how long you wait, an infinite amount of time will never have elapsed.
Well, because the totality has no bearing on the question of how much time has elapsed. Perhaps, recast this into the more concrete form of ‘the longest possible time anyone has to wait’: just because the ‘totality’ is infinite, doesn’t mean that the longest possible time anyone has to wait is infinite, too; but you argument only holds if that were the case. In order to show that ‘an infinite amount of moments must have elapsed to get to the present’, you have to show that the longest possible time anyone has to wait is infinite: these are simply the same things. But in a universe with infinite past, nevertheless, nobody ever has to wait infinitely long, because there’s no time such that he could start waiting, and stop waiting at some other time, with those times being infinitely far apart.
You keep talking about this ‘totality issue’. But you haven’t made an actual argument as to how it’s relevant.
I’ve shown you mathematical proof that an infinite past does not imply an infinite length of time passing, somebody having to wait infinitely long, etc. Everytime I do this, however, you simply point again to the infinite line, saying ‘but that’s infinitely much time!’. But this is just besides the point. Time, in order to pass, must elapse between two points in time; you must show that there are two points in time such that they are separated by an infinite amount of time. That there are n o’clock and m o’clock such that the two are infinitely distant. Only then does the problem of having to traverse an infinite number of moments actually arise; merely the fact that there is an infinite amount of time does not make it so.
It’s like there’s an infinite amount of sand, and it’s constantly running into my pocket; you’re saying that my pocket will certainly rupture, since it can’t possibly withstand the weight of infinitely much sand. I say, yes, that’s right, it couldn’t withstand that weight; but I’ve started out with finitely much sand in my pocket, and only finitely much is added to it at a time, so I always only have finitely much sand in my pocket, which it can withstand. But then you’re saying again, but there’s an infinite amount of sand!
But these are two different things: the amount of sand in total, and the amount of sand in my pocket. The problem only exists if there was an infinite amount of sand in my pocket. But there never is. Likewise, these are two different things: the total amount of time, and the amount of time that has elapsed. The totality of time, and the longest that somebody could have waited. To be sure, there is no maximum amount of time anybody could have waited: for any given amount of time, it is always possible to have waited longer; the amount is unbounded. But any such amount is finite; there may be infinitely many of these amounts, but again, each and every one of those is finite. Nobody ever could have waited an infinite amount of time; and it’s only if that could have been the case that the problem arises.
Aleph-zero is the total amount of numbers; it’s a name for a specific kind of infinity, ‘an infinite amount equal to the amount of natural numbers’.
Aleph-zero events, plural, meaning ‘as many moments (events) as there are natural numbers’—infinitely many moments.
No, ω* refers to the ordering, not the amount of numbers. I can re-order the numbers on the line, putting e.g. all the even ones first, and only after that all the odd ones. There are then still as many numbers as before, the amount is the same, but the order type has changed—it’s now ω* + ω*, meaning that you have the infinite set of even numbers, and then the infinite set of odd numbers, each of which is of order type ω*.
The example is relevant because it shows a case in which what you would have to establish actually holds: there is an infinite interval between, say, -4 and -3, consisting of all the odd numbers (larger than three) (and the even numbers smaller than -4, i.e. -2). If time progressed that way, first all odd hours, and then all even hours, then somebody waiting for the clock to strike -3 if the time is now -4 o’clock would have to genuinely wait infinitely long. But of course, that’s not how linear time progresses.
Nevertheless, this should show you what would have to be true in order for your argument to work; and also, since it’s not true, that it doesn’t.
I think you’ll have to start making an effort looking up things you don’t get, this is getting a bit too much of an introduction to elementary mathematical concepts. Try the wiki article on intervals, or maybe this page.
Nevertheless, the notation (-1,1) refers to the open interval with endpoints -1 and 1, that is, the set of numbers larger than -1, but smaller than 1. Concretely, we had specified these to be taken from the set of rational numbers, i.e. fractions. This contains neither a smallest nor a largest number: for any number in the interval you tell me, I can tell you both a smaller and a larger one.
No. Time = time elapsed, time between two points, time somebody could wait. Past = total amount of points in time before the present moment. Infinite time = infinite time elapsed, infinite time between two moments, having to wait infinitely long. Infinite past = an infinite total amount of points before the present moment. From each of those infinitely many points, it takes a finite time to get to the present.
Just remember: beside the point.
But if you start at any point in the past, it takes a finite time to get to the present.
But you don’t. No matter from what point in the past you start, you have a finite amount of time to pass through. Yes, there are infinitely many different finite times, but that doesn’t add up to having to pass an infinite time; you only have to pass the time from a given moment in the past to now.
It’s not choosing a particular time which is finite, the key point is that all of them are finite. Any possible time you could wait is finite. There is simply no infinite amount of time that could pass, as there is no infinite interval, no n o’clock and m o’clock such that they are infinitely far apart.
If you want to claim that you have to wait n minutes, to establish that claim you have to find a starting point and an end point such that they are n minutes apart. If you say, I waited 10 minutes for the bus, and somebody challenges that claim, you can say, well, I was at the station at 3:15, and the bus came at 3:25, and then you’ve shown that you have waited 10 minutes.
This holds just as well if n is infinite. So, next time you want to claim that an infinite amount of time has elapsed, you have to say between which two points in time. What time did you start waiting for the bus, and at what time did it come?
Yes I agree with that view of time. There is only the present. What creates the illusion of time is change.
That however only brings us to formulate the issue a little differently; was there a first event, a first change, that would represent the “beginning” of time.
What we have is now ways to date events that appeared to have happened in the past at X based on lots of observed data - (carbon dating, core samples, etc) - at some point, we can say "we can see this far back, no further’ - which would be that point in time for our purposes.
It would not, however, be logical to conclude that that was the ‘beginning’ of time itself.
Assuming that the existing chain of events is causal or connected in some similar way, the implication that such a chain had a beginning works the same as I posted before.
If there are however random events that take place from time to time, then my argument would break down there.
every ‘chain of events’ has a beginning - and its undeniable that ‘random’ events also take place.
I’m not sure if you’re looking for a ‘spark’ outside of time/space that caused it or ???
I think one of the biggest things to keep in mind is that ‘infinite time’ and/or ‘infinite space’ is more of a concept than something that is concrete - if anything, its a way to describe something that we cannot observe the bounds of, or as a description of an observation that we have not yet found the bounds of (‘pi is infinite’ ‘the universe appears to be infinite’).
THis has nothing to do with the concrete examples of distance and/or time that have clearly passed that we are able to determine.
I’ve always had the idea that time had to have a had a beginning, maybe the big bang maybe something else. This is because you other wise have an endless chain, and there can’t be such a thing. No matter how long something has existed it will not have existed endlessly.
Now with a different view of time as not being something that happens but just the way we interpret change, if there are uncaused events (I suppose things quantum or such) then there could always have been events, just not events with endless precursors.
The cardinality is infinity and I don’t see how that doesn’t matter.
Yes. Below on this post is what my reasoning is. (The concept of ‘waiting’ fits to a certain degree though, I’ll give you that)
No two points in time between which a given amount of time lies, no such amount of time can elapse.
Maybe you will just have to trust me, but I DO understand what you are saying here. Again I don’t find this to be the full picture (see my next few text entries)
If now an infinite amount of time passes, that is, if there is an interval such that there is an infinite amount of time between its endpoints, then you are right: such an interval can (on a plausible construal of temporal progression, anyhow) never be traversed.
This is where I disagree. It’s not that there is a PARTICULAR interval of infinite distance away from the now. Its that if you put every interval together of the past (if the regular line segments stand for.
Perhaps another example. Consider a line. You can half that line infinite times. In fact, ALL of those halves already exist on the line (imagine them all marked out in a line in your imagination, all at one time). None of the halves are infinite distance from an endpoint, yet there is a particular existing amount which cannot ALL be progressively drawn in. What do you think? They are all of a finite distance, never infinite, yet cannot be progressively drawn in going at a constant rate.
Starting from the end point of a ray, how long would it take to traverse an infinite amount of intervals at a regular speed? This would never cease. But how can you assume that you can traverse that ray in reverse and get to the end point? (with the issue of where you would start aside; just think of how many steps it would take.)
What you are actually asking for when you ask for an infinite ray with the beginning point at 0 and the end point at ∞, is a line segment, not a ray. Is a ray not infinite having a beginning point and no such end point?
There doesn’t need to be 2 end points for an infinite distance. Infinity can have a beginning point (a ray), but not a beginning and end point (a line segment). A beginning and end point is always a finite distance. This is not what we are looking for when looking for infinity (we are not looking for a point at -∞ because it is impossible). If the intervals stand for distance, you only need infinite intervals, which in the case of a line or a ray, you have.
Exactly. I agree. However, we disagree with what conclusions which are drawn from this. You still seem to think that a ray is possible with this model, but this models tells me that only a line segment is possible.
Thus, in fact, infinitely many minutes haven’t passed, since there is no time since when they could have passed, even if the past is infinite.
I am trying to more clearly understand the math you are using for this claim, as we go on I might see what you are talking about: though, how can that infinite past≠infinite minutes? How is it that: infinite past≠infinite time. The minute thing is contrived (you could go with minutes, seconds, years, any time interval) but the terms are basically synonymous. If you have tried to answer this in the places where you are going math-heavy, perhaps you can point to the post?
As I said, there is not a line segment with 0 on one end and ∞ on the other. Refer to my above writing on this for your answer.
Its not about where you start. Its about how many steps (integers) are involved.
Please, try to make it more precise what infinitely many integers/infinitely many intervals (as you say further down) you have to pass, and where you start.
I believe I have done so above. Included is the starting point.
So, list the intervals in such a way that we could enumerate them—of course, not all of them (if they’re infinitely many as you claim, that would take a bit too long), but find some systematic way to order them, such that each of those intervals can at least be named, pointed to, or something. Try and find a systematic way of writing them down, like I did above when I paired every negative number with a rational number (a fraction) between -1 and 0: there’s infinitely many of both, but in giving the scheme I did, I made sure all are accounted for. But I don’t see where your infinities are hiding.
Not really sure what you are asking me to do here. But the infinity is hiding in what a ray is.
Having no boundary, e.g. no least/greatest value (the set of natural numbers is unbounded in this sense).
I meant in that quote ‘time elapsed’, or ‘time to be traversed’. Time elapsed is not the same as the total amount of time, which is the case even in a universe with a beginning, but no end: there’s an infinite amount of time (there is no boundary to the future, no ‘highest’ value the clock can read), but no matter how long you wait, an infinite amount of time will never have elapsed.
Yes it does.
The totality IS the time which has elapsed. We are linking past to time, and the past is the concrete events which we must have ACTUALLY had to go through.
Using a ray, you have infinitely much time, making a ray impossible. The only linear alternative to this (given we are starting with an end point at the ‘now’) is a line segment.
past = wait time, infinite past=wait time
Then if you are trying to traverse a ray to get to the end point, you don’t have an infinite amount of line-segment intervals to wait through?
Perhaps you can point to the exact mathematical proof you are talking about? You have done a lot of proofs. Definitions are of use when trying to establish the truth of an issue. I am making an attempt to follow you mathematically, so bear with me until that happens. Apart from that, you are basically speaking Greek to me and saying that the thing you said in Greek proves it, “see?” Not really. To me you DO seem to be going against basic definitions here, and I really feel you are failing to grasp my points.
By the way, if you can say your points in laymans terms and exnay on the more advanced mathematical concepts (if you can) then that will help this conversation be understood with less confusion. (And allow me to reply faster as I don’t have to spend time trying to learn new definitions and see how they all work together. Not bad in itself but it makes for a slower reply time).
What? So you have an infinite amount of sand (but not in your pocket), you put a finite amount in your pocket til it ruptures, and now I think you have an infinite amount in your pocket? That’s not what I think about the example you gave.
I have made an effort to look up what you are talking about, but some of this stuff is hard (at least for me) to find (example, I wouldn’t know where to begin to go with search terms for the (-1,1) thing. I appreciate any time you take in explaining, and thank you for what you have done so far (and for what you will do should you continue to do so)
… Is ω* an arbitrary abstraction made by you or is it an official variable, and if so what is the official name? (Like π is ‘Pi.’) (This is so I can search it up, your definition and where it switches to ‘ω*+ω*’ is not making sense to me.). Also, that means that the implication is unclear. For the record though, I know that linear doesn’t progress on only evens waiting for the odd number which will never come, which is not what my argument is based on.
The whole (-1,1) thing… I really don’t get this. To the extent that I do get it (there are infinite fractions between -1 and 1) I don’t see how it is relevant. Again, and if you think this is important, you can explain but it might be faster if you give me the term for me to look up ((-1,1) is too vague to look up). Also, I couldn’t find it in my search, but why do you keep using N as a variable for time? Is that personal or is it based off of traditional math?
“Time elapsed = time between two points” does not hold true on a ray. I agree that infinite time=infinite time elapsed which would be impossible. Though that means that we don’t have a ray, but a line segment, i.e. we have a beginning of time.
I’m glad you see that there is an infinite amount of points. And how do you suppose this INFINITY was PASSED?
Infinitely many times = Infinitely many seconds. (The ‘regular line-segment intervals’ on the ray stand for seconds… infinite times IS infinite seconds.)
I know that’s your key point, I feel that should be evident in my answers if you are comprehending what I am saying. From past posts of mine:
Here’s a question for you, at this point, do you understand why I am saying that this isn’t the complete story?
Interesting, I will think about this… but eh, if you are arguing for the impossibility of infinite time then I agree. Though, you seem to think that this can still translate to a ray where I see this as only translating to a line segment.
Interesting, but what I see you are saying here that any amount of time (including ALL of the past) is a wait between two points, in other words: a line segment.
OK, this is getting too unwieldy to be practical. And could you please try and mark what you quote appropriately? It’s hard to figure out what you actually wrote, especially if you don’t even leave a line break between the quote and your own text. A couple of times I found myself suddenly emphatically agreeing with what you wrote, only to realize that it was my own words I was agreeing with… :smack:
Also, regarding the definitions you asked for, I think I’ve given most of the relevant ones; I’ve linked to two different pages explaining the notion of an interval like (-1,1), and the ω* notation I’ve just taken from the paper by Quentin Smith. See also the wiki on order type and ordinal number. Basically, ω denotes the order type of the natural numbers, that is of something ordered like
0,1,2,3,4,…
(considering 0 to be a natural number for the moment), and the asterisk denotes the inverse order, i.e. ω* denotes something ordered like
…,-4,-3,-2,-1,0.
“Ordered like” here meaning that you can pair the elements of two things such that both keep their natural order; i.e. the set of squares is ordered like the set of natural numbers (and thus, also of order type ω) because I can pair them 1 -> 1, 4 -> 2, 9 -> 3, 16 -> 4, i.e. associating to each natual number its square, where the ordering relation remains the same.
Anyway, I want to try a different tack to force you to make your assumptions more explicit. A clock, we may imagine, is something that counts moments (i.e. minutes, hours, years, what have you). You claiming that an infinite amount of moments must have elapsed in order to get to the present in an infinite-past universe is thus the same as saying that a clock would have to show ‘infinity’ in the present (which it can’t; we both agree that it could not actually tick off infinite moments). So, all that I want you to do to establish your claim is how any clock would come to do that—because only then has an infinite amount of moments actually elapsed.
To give an example of how this might work, consider again the re-ordered negative number line
…,-4,-2,…,-3,-1,0.
Any clock starting at -4 would, once we arrived at -3—provided we could actually do so—show an infinite amount of moments to have elapsed; hence, on a past ordered like this (i.e. of order type ω* + ω*), we could never get from -4 to -3. The same thing you now have to prove to be true for a line of order type ω*; otherwise, it is simple not the case that an infinite amount of moments would have had to elapse to get to the present.
Sure
A line or ray is defined by an infinite amount of length. The length stands for time. Therefore: a line or ray is infinite time…
You don’t need a number at ‘point infinity’ for there to be an infinite amount. If 3.1415… each stands for (1,2,3,4,5…) is there going to be a point at infinity? No? Well according to your assertion, pi doesn’t go on forever.
As per the definitions…
I understand ω, I understand ω*, but not (ω*+ω*). The later part of your argument depends on this so, explain if you want me to understand. I have never studied set theory and don’t know what to search. I did specify that I wasn’t a math major; the math may be elementary but that doesn’t help if I never have heard of it. (That means, I really don’t know what you are trying to get at with the clock/set example)
Also, it is not always convenient to answer these questions, especially with the definitions that I look up. If you are able to speak in layman’s terms this will help, or provide a layman’s definition when you post terms
No. According to my assertion, pi goes on forever, but there isn’t an infinite amount of digits between any two of its digits. Which is simply true: no matter where you start counting, you won’t ‘reach infinity’ and then still have digits left you haven’t counted.
The argument doesn’t depend on this definition, in fact, not even the example does; it’s merely to illustrate how it could be for the clock to show ‘infinity’, and needs only the realization that on a line ordered such that all the even numbers come before the odd numbers, there will be an infinite amount of numbers between any given even and any given odd number; which is just a restatement of the fact that there are infinitely many even numbers.
But, I’ll try to explain anyway: as I said, a set is of order type ω* if it can be put into an one-to-one correspondence with the negative natural numbers such that both sets keep their natural order. For the set of even numbers, this works as follows:
-1 –> -2
-2 –> -4
-3 –> -6
-4 –> -8
.
.
.
-n –> -2n
.
.
.
Thus, every negative natural number is associated with a negative even number, and both sets retain their order (given by the relation ‘greater/smaller than’). So, the set of even numbers is not only of the same cardinality, but also of the same ordinality as the set of natural numbers, i.e. ω*. The same is true for the odd numbers, as you can easily convince yourself.
Now, the set ordered like
…,-4,-2,…,-3,-1,0
is clearly not of the same order type, even though it is of the same cardinality as the set of natural numbers: there is no one-to-one mapping of it to the natural numbers such that it stays ordered like that (if you have trouble believing this, try and answer the question what -2 gets mapped to if you’ve ‘exhausted’ all the natural numbers already mapping the odd part of the set).
However, both the even and the odd part can be put into such an order-preserving correspondence with the negative naturals separately, in this way:
..., -6, -4, -2, ..., -5, -3, -1, 0
| | | | | | |
..., 3, 2, 1, ..., 3, 2, 1, 0
|---------------||------------------|
ω* ω*
I’ve already indicated that each of these mappings gives rise to another ω*; this means simply that the ordering type of each sub-set consisting of the evens and odds respectively is ω*, which we thus have twice. This makes the ordering type of the total set ω* + ω*.
But again, this all is wholly unnecessary to answering my simple question of how a clock should come to read ‘infinity’ in a universe with an infinite past; it merely furnishes an example of how it could come to do so, which however doesn’t provide a model of what we typically understand by the term ‘past’. Of course, the point here is that in a universe whose past is of order type ω*, no clock ever could come to read ‘infinity’ and thus, no infinite amount of moments elapses (since if there did, the clock would just have to count those moments). This statement is in fact just the same as saying ‘there are as many moments in the past as there are natural numbers’, which is to say, ‘the past is infinite’.
Again, however, you’re free to make an argument in support of your thesis: how could a clock in such an infinite past ever read infinity? Answer this cogently, and you’ll have provided an actual argument for your thesis; but if you fail to, then I’m sorry, but you simply have nothing.
Can a line of infinite length have defined points within it, with finite length between said points?
In other words, is it possible to have a year 2013 turn into year 2014 in a universe in which time is said to be infinite? Can you segment infinity like that? Does marking specific points in time necessarily de-infinitize it?
I’m not up-to-date on the modern conception of infinity, so I ask out of genuine curiosity in the hopes it advances the discussion.
Gateway, you say that you never studied set theory. Well, then, if you want to understand the arguments in this thread, you need to do so. Here are some:
Naïve Set Theory by Paul R. Halmos
Elements of Set Theory by Herbert B. Enderton
Set Theory and Logic by Robert R. Stoll
These are in addition to the books on infinity I recommended earlier:
To Infinity and Beyond by Eli Maor
Infinity and the Mind by Rudy Rucker
One, Two, Three . . . Infinity by George Gamow
The Art of the Infinite by Robert Kaplan and Ellen Kaplan
Look, at this point it’s clear that you’re never going to get the points we’re making in this thread. You’re going to have to learn about set theory and about infinity from scratch. You’re a smart person, Gateway, but you don’t realize that you don’t have to background to understand the points we’re making. One way to get that background is to read the books above. Can anyone suggest some other books?
You’re like a number of people who have joined the SDMB thinking that they can make up for their lack of knowledge in certain areas without an intensive effort. You can’t. You can read some books about the subject or you can take a course (which will also mean you will have to read some books), but we can’t bring you up to the level you want to reach without that kind of study.
Why would you think not?
I have heard that infinity as a concept has its own set of rules. What I’m getting at is the idea that the moment you put a dot on an infinite line, it has the properties of being finite, for it is measurable and segmented. How can you create two infinite lines from what is originally one infinite line? You have essentially doubled infinity. Now, if that infinite line has an infinite number of dots on it, then perhaps the concept holds. But then again, you can’t measure any single dot for then you have violated a precept of infinity’s nature.
I actually don’t know if this correct thinking at all, so I am asking for clarification. It seems to be the crux of the dispute that has consumed this entire page. What I find interesting is the idea that since we measure time, in years, does that act of measuring mean time could not, by definition, be infinite?
The fact that something (a set) can be put into correspondence with a proper part (subset) of it is, in fact, a definition of infinity: if you ‘halve’ the set of natural numbers by crossing out every odd one, the resulting set of even numbers will have the same cardinality as the natural numbers—you can associate every even number with a natural number, and thus, there are equally many of each, even thought there are ‘clearly’ less even numbers than there are natural numbers.
Likewise, if you ‘halve’ an infinite line by putting a dot somewhere, each half-line will still be infinite, and as long as the total line. (But this is getting a little off-topic, and if you just want to deepen your understanding of the concept of infinity, I’d suggest you ask a question in GQ, where there’s a number of mathematicians more capable of elucidating it than I am).
Half Man Half Wit,
The clock example doesn’t describe what I am trying to say, it doesn’t prove your theory or disprove mine:What if you have a computer that inputed the total number for all of the time intervals in an infinite past. What would that singular number be? It would be ∞, even if you can’t read it on a clock. The fact that it is ∞ is the relevant issue, and even your clock example shows you can’t get there in progressive steps. So, your clock can NEVER get to infinity yet that is a possible number for an infinite past?!?!
Okay, you say pi goes on forever. Well then: is the totality of pi (the cardinal number) sequentially passable?
It is not. To use that number for the past though, is to say that it HAS been SEQUENTIALLY PASSED. (The past ACTUALLY HAPPENED in a SEQUENTIAL way, given linear time)
Your claim is: “All distances on this infinite distance are finite.”
Notice how “all distances” would include the initial infinite distance you are starting with? That is your infinite distance.
And by the way, it isn’t only “between 2 points” that you have to worry about, because an infinite timeline doesn’t have a beginning and end point. If you begin with the now for a beginning point, it you will never find the end point. Thus you can’t sum up an infinite line’s “only finite” distances by saying “but between 2 points.”
PS You say that the set-theory isn’t really correlating to your example. Good… it is beyond me. Maybe someday it won’t be, but for now it is.
You are—or at least were—trying to say that an infinite amount of moments would have to have elapsed if the past were infinite. This is exactly the same as saying that there must be a clock such that it would read ‘infinity’ in the present; the two are just different wordings of the same matter of fact. Your argument works if and only if such a clock exists.
Take another look at its structure. You’re asserting that in an infinite-past universe, something must have been done that it is impossible to do: an infinite amount of moments must have been ‘ticked off’, a supertask must have been completed. If now actually the infinite-past universe does not contain such a supertask, then the argument fails. And the supertask is exactly equivalent to a clock ticking off infinitely many seconds/years/minutes etc. If no such clock exists, no such supertask exists, and it is not the case that something undoable must be done, i.e. the present must be reached by waiting infinitely long, in a past-infinite universe.
Let’s make this more concrete. Your original argument was:
(1) In a past-infinite universe, an infinite amount of moments/infinite amount of time must have elapsed in order to get to the present moment.
(2) It is impossible for an infinite amount of time to elapse.
(3) ∴ A past-infinite universe is impossible.
This is exactly the same as:
(1a) In a past-infinite universe, a clock must exist such that it shows ‘infinity’ once it gets to the present moment.
(2a) It is impossible for a clock to ever reach ‘infinity’.
(3) ∴ A past-infinite universe is impossible.
It is easy to show that premises (1) and (1a) are equivalent, i.e. that one implies the other, and vice versa: if there is an infinite amount of moments that elapse, one just needs the clock to count these moments; conversely, if one has a clock that counts to infinity, then whatever intervals it has ticked off are just the infinitely many moments that have elapsed. The two propositions are thus logically co-extensive, and hence, identical. Especially, if one of them is false, the other necessarily is, as well.
Thus, if no such clock exists, then premise (1) of your argument is false, and the conclusion (3) hence not established.
But why would a computer outputting ∞ be any sort of obstacle to the past being infinite?
What you can read on a clock is what matters, because that’s the time that has actually passed. And it’s precisely my contention that even though the totality of the past, the amount of moments ‘in it’ is infinite, that any time that has actually passed is finite, and must be. This is just a mathematical property of the infinite number line, essentially given by its ordinality. Making the argument ‘but there are an infinite number of intervals in total’ simply misses the point, as it is the amount of time elapsed—the interval we traverse—that matters. (At least, it’s what your argument as paraphrased above starts from; you’re of course free to try and make a different argument why the simple fact that the totality of moments in the past is infinite entails itself an impossibility, but so far, you haven’t done so. But it’s hard to see how any such argument wouldn’t equivalently well apply to an infinite future.)
Perhaps a further example to highlight where your reasoning goes wrong. The open interval (0,1) contains infinitely many points, given by rational numbers (fractions) of the form 1/n (n>1). It also contains infinitely many intervals; indeed, it contains exactly as many intervals as the line of natural numbers. It is, viewed as a totality, exactly the same: of the same cardinality and ordinality (because I don’t consider all the rational numbers). Your argument of pointing to the totality and declaring ‘it’s infinite’ applies to it just as well as to the line of natural numbers. But plainly, we traverse it every day: because the totality does not matter for the time elapsed, only ‘what a clock shows’ does.
Likewise, consider the infinite interval [0, ∞), but with ‘step sizes’ arranged such that the step from 0 to 1 takes 1 second, from 1 to 2 takes 1/2 second, from 2 to 3 takes 1/4 second, and so on. Then, you will traverse the entire interval in 2 seconds, even though its total length is infinite.
This shows that your (vague) gesticulation in the direction of ‘the totality’ likewise misses the mark: regarding the totality, the intervals I just presented are equal to the infinite past: both have infinitely many finite intervals, the latter is even itself infinite in length. But none of that is an obstacle to traversing it in finite time.
What would be an obstacle is having to perform an infinite amount of steps taking a constant amount of time each—i.e. having a clock tick off infinitely many seconds. But this is precisely what doesn’t—and can’t!—occur in a past-infinite universe with the infinite past being of the same ordinality as the natural numbers.
You need to realize that this sentence says the same as ‘you can never have an infinite distance between two natural numbers, yet there are infinitely many natural numbers’. Which is true, regardless of your interrobangs.
This reasoning, again, is valid only in the finite case, and no longer in the infinite case, because the notions ‘maximum time that has elapsed’ and ‘cardinality of the amount of points in the past’ diverge (exactly like ordinal and cardinal numbers do). You assume that they continue to mean the same thing, but that’s simply mathematically wrong.
I don’t know what you mean by ‘the initial infinite distance [I am] starting with’, but you’re still invited to show me two natural numbers such that their difference is infinte.
This is exactly why I can only worry about the distance between two points: there is no other distance, nothing that traverses ‘the totality’.
In any case, to sum up: if you wish to make the case that an infinite past entails a contradiction by requiring the performance of a supertask, you need to show where that supertask—like a clock ticking off infinitely many seconds—actually occurs. Your argumentation for the last couple of posts has been to vaguely gesture towards ‘the totality’ and asserting that it occurs somewhere in there, that the fact that it’s infinite were by itself enough to guarantee it cropping up somehow. But as I’ve now shown you several times, it’s not enough: there are examples of infinite structures such that no such supertask/contradiction arises.
I’ve also shown you an example of a structure where it does arise, the re-ordered negative number line (which you claim still not to understand, in which case, you’ll either have to believe me or convince yourself that there are infinitely many numbers between -4 and -3 in that case—really, I don’t know how to make that more obvious). This is what you would have to do for the case of an infinite past in general: point out how, in such a structure, a supertask like a clock ticking to infinity arises. Only then will you have any substance to your premise (1). (Of course, this isn’t possible; I simply hope that thinking about coming up with such an example will help you realize why, and also why vague handwaving involving ‘the totality’ does not establish anything.)