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Tomdstone
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There are an infinite number of split seconds in a one hour period. There is a scheme to show what they are.
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Tomdstone
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How many time intervals are there actually in the one hour period?1- No, we don’t have an “actually infinite number of time intervals” between 1:00 PM and 2:00 PM,.
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yppop
Guest
If you believe space is actual and you believe space is composed of points, then we can define a plausible and actual infinity. The way this is done is to apply the Cantor-Dedekind axiom of continuity that states that each and every point in three dimensional space can be represented by a single number on the real number line. Since we can squeeze an infinity of rational or real numbers in any interval, it can be shown that there is an infinite number of points in a volume of space however small you care to make it.1- No, we don’t have an “actually infinite number of time intervals” between 1:00 PM and 2:00 PM, only a potential infinite number of time intervals. Craig isn’t so stupid as to overlook “objections” like that; he writes about them in detail (for instance, in his article on the Blackwell Companion to Natural Theology), while also dealing with Zeno’s paradoxes. There’s one absolute difference between 2 and 1, and that is 1;
2- Regardless, would you then believe that a library with an infinite number of books (where we could take a book and the library would still have the same number of books it had before we took a book from it) would be possible? I find that a very, very hard bullet to bite. But absurdities like that would have to follow from the possibility of actual infinites in concreto.
Digression:* There is an infinite number of rational numbers which are designated as aleph (null) in transfinite algebra. There is an even greater infinity of real numbers which are designated as as aleph (1). the rational numbers define discrete space; real numbers define continuous space. *
Once you can wrap your mind around that you can begin to understand that with the infinite we are forced to accept that there are as many even integers as the are total integers; yes, for the infinite, the part is equal to the whole. And you can continue to add elements such as rooms or books without changing the total, namely aleph (null) for books or rooms since they are discrete.
As far as intervals are concerned you can have and infinite number of them only in an infinite interval; however since they have dimensions, the number between 1 and 2 has to be finite unless of course you make the length “infinitesimal”, the mathematicians greatest sleight of hand that allows you so shrink it to a limit of zero.
Yppop
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Tomdstone
Guest
You can have an infinite number of intervals in an interval of finite length:. [0,1/2), [1/2, 3/4), [3/4, 7/8), [7/8, 15/16),… will be an infinite number of intervals in [0,1].If
As far as intervals are concerned you can have and infinite number of them only in an infinite interval; however since they have dimensions, the number between 1 and 2 has to be finite unless of course you make the length “infinitesimal”, the mathematicians greatest sleight of hand that allows you so shrink it to a limit of zero.
Yppop
In any case, there are an infinite number of split seconds in a one hour interval follows from what you have shown. There are 3600 seconds in one hour.
There are 2x3600 split seconds of length 1/2
There are 4x 3600 split seconds of length 1/4
There are 8x 3600 split seconds of length 1/8
There are 16x 3600 split seconds of length 1/16
There are 32x 3600 split seconds of length 1/32
Etc. There is no limit to the number of split seconds in a one hour interval.
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Triflelfirt
Guest
Everyone knows that. But that’s not an actual infinite, that’s a potential infinite. The difference between 2 and 1 is 1. The fact that you “can” divide the intervals indefinitely shows that there is a potentially infinite number of intervals, but they’re not actual. For any natural number n, n+1, or n/2 is always a finite number, therefore not an actual infinite – aleph zero. Your objection fails. Moreover, you’ve ignored my second point (about accepting absurdities such as Hilbert’s Hotel and a library with an infinite number of books in the real world).You can have an infinite number of intervals in an interval of finite length:. [0,1/2), [1/2, 3/4), [3/4, 7/8), [7/8, 15/16),… will be an infinite number of intervals in [0,1].
In any case, there are an infinite number of split seconds in a one hour interval follows from what you have shown. There are 3600 seconds in one hour.
There are 2x3600 split seconds of length 1/2
There are 4x 3600 split seconds of length 1/4
There are 8x 3600 split seconds of length 1/8
There are 16x 3600 split seconds of length 1/16
There are 32x 3600 split seconds of length 1/32
Etc. There is no limit to the number of split seconds in a one hour interval.
I will, however, share some of Craig’s own comments on similar issues in his excellent article “The Kalam Cosmological Argument” in the Blackwell Companion to Natural Theology:
*Sometimes it is said that we can find concrete counterexamples to the claim that an actually infinite number of things cannot exist, so that Premise (2.11) must be false. For example, Walter Sinnott-Armstrong asserts that the continuity of space and time entails the existence of an actually infinite number of points and instants (Craig & Sinnott- Armstrong 2003, p. 43). This familiar objection gratuitously assumes that space and time are composed of real points and instants, which has never been proven. Mathematically, the objection can be met by distinguishing a potential infinite from an actual infinite. While one can continue indefinitely to divide conceptually any distance, the series of subintervals thereby generated is merely potentially infinite, in that infinity serves as a limit that one endlessly approaches but never reaches. This is the thoroughgoing Aristotelian position on the infinite: only the potential infinite exists. This position does not imply that minimal time atoms, or chronons, exist. Rather time, like space, is infinitely divisible in the sense that division can proceed indefinitely, but time is never actually infinitely divided, neither does one arrive at an instantaneous point. If one thinks of a geometrical line as logically prior to any points which one may care to specify on it rather than as a construction built up out of points (itself a paradoxical notion13), then one’s ability to specify certain points, like the halfway point along a certain distance, does not imply that such points actually exist independently of our specification of them. As Grünbaum emphasizes, it is not infi- nite divisibility as such which gives rise to Zeno’s paradoxes; the paradoxes presuppose the postulation of an actual infinity of points ab initio. “. . . [A]ny attribution of (infinite) ‘divisibility’ to a Cantorian line must be based on the fact that ab initio that line and the intervals are already ‘divided’ into an actual dense infinity of point-elements of which the line (interval) is the aggregate. Accordingly, the Cantorian line can be said to be already actually infinitely divided” (Grünbaum 1973, p. 169). By contrast, if we think of the line as logically prior to any points designated on it, then it is not an ordered aggregate of points nor actually infinitely divided. Time as duration is then logically prior to the (potentially infinite) divisions we make of it. Specified instants are not temporal intervals but merely the boundary points of intervals, which are always nonzero in duration. If one simply assumes that any distance is already composed out of an actually infinite number of points, then one is begging the question. The objector is assuming what he is supposed to prove, namely that there is a clear counterexample to the claim that an actually infinite number of things cannot exist.
Some critics have charged that the Aristotelian position that only potential, but no actual, infinites exist in reality is incoherent because a potential infinite presupposes an actual infinite. For example, Rudy Rucker claims that there must be a “definite class of possibilities,” which is actually infinite in order for the mathematical intuitionist to regard the natural number series as potentially infinite through repetition of certain mathematical operations (Rucker 1980, p. 66). Similarly, Richard Sorabji asserts that Aristotle’s view of the potentially infinite divisibility of a line entails that there is an actually infinite number of positions at which the line could be divided (Sorabji 1983, pp. 210–3, 322–4).*
If this line of argument were successful, it would, indeed, be a tour de force since it would show mathematical thought from Aristotle to Gauss to be not merely mistaken or incom- plete but incoherent in this respect. But the objection is not successful. For the claim that a physical distance is, say, potentially infinitely divisible does not entail that the distance is potentially divisible here and here and here and. . . . Potential infinite divisibility (the prop- erty of being susceptible of division without end) does not entail actual infinite divisibility (the property of being composed of an infinite number of points where divisions can be made). The argument that it does is guilty of a modal operator shift, inferring from the true claim
(1) Possibly, there is some point at which x is divided
to the disputed claim
(2) There is some point at which x is possibly divided.
(to be continued)
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Triflelfirt
Guest
But it is coherent to deny the validity of such an inference. Hence, one can maintain that a physical distance is potentially infinitely divisible without holding that there is an infinite number of positions where it could be divided.
Rucker also argues that there are probably, in fact, physical infinities (Rucker 1980, p. 69). If the mutakallim says, for example, that time is potentially infinite, then Rucker will reply that the modern, scientific worldview sees the past, present, and future as merely different regions coexisting in space-time. If he says that any physical infinity exists only as a temporal (potentially infinite) process, Rucker will rejoin that it is artificial to make physical existence a by-product of human activity. If there are, for example, an infinite number of bits of matter, this is a well-defined state of affairs which obtains right now regardless of our apprehension of it. Rucker concludes that it seems quite likely that there is some form of physical infinity.
Rucker’s conclusion, however, clearly does not follow from his arguments. Time and space may well be finite. But could they be potentially infinite? Concerning time, even if Rucker were correct that a tenseless four-dimensionalism is correct, that would provide no reason at all to think the space-time manifold to be temporally infinite: there could well be finitely separated initial and final singularities. In any case, Rucker is simply incorrect in saying that “the modern, scientific worldview” precludes a theory of time, according to which temporal becoming is a real and objective feature of reality. Following McTaggart, contemporary philosophers of space and time distinguish between the so-called A-Theory of time, according to which events are temporally ordered by tensed determinations of past, present, and future, and temporal becoming is an objective feature of physical reality, and the so-called B-Theory of time, according to which events are ordered by the tenseless relations of earlier than, simultaneous with, and later than, and temporal becoming is purely subjective. Although some thinkers have carelessly asserted that relativity theory has vin- dicated the B-Theory over against its rival, such claims are untenable. One could harmonize the A-Theory and relativity theory in at least three different ways: (1) distinguish meta- physical time from physical or clock time and maintain that while the former is A-Theoretic in nature, the latter is a bare abstraction therefrom, useful for scientific purposes and quite possibly B-Theoretic in character, the element of becoming having been abstracted out; (2) relativize becoming to reference frames, just as is done with simultaneity; and (3) select a privileged reference frame to define the time in which objective becoming occurs, most plausibly the cosmic time, which serves as the time parameter for hypersurfaces of homo- geneity in space-time in the General Theory of Relativity. And concerning space, to say that space is potentially infinite is not to say, with certain constructivists, that it depends on human activity (nor again, that there are actual places to which it can extend), but simply that space expands limitlessly as the distances between galaxies increase with time. As for the number of bits of matter, there is no incoherence in saying that there is a finite number of bits or that matter is capable of only a finite number of physical subdivisions, although mathematically one could proceed to carve up matter potentially ad infinitum. The sober fact is that there is just no evidence that actual infinities are anywhere instanti- ated in the physical world. It is therefore futile to seek to rebut (2.11) by appealing to clear counterexamples drawn from physical science.
Rucker also argues that there are probably, in fact, physical infinities (Rucker 1980, p. 69). If the mutakallim says, for example, that time is potentially infinite, then Rucker will reply that the modern, scientific worldview sees the past, present, and future as merely different regions coexisting in space-time. If he says that any physical infinity exists only as a temporal (potentially infinite) process, Rucker will rejoin that it is artificial to make physical existence a by-product of human activity. If there are, for example, an infinite number of bits of matter, this is a well-defined state of affairs which obtains right now regardless of our apprehension of it. Rucker concludes that it seems quite likely that there is some form of physical infinity.
Rucker’s conclusion, however, clearly does not follow from his arguments. Time and space may well be finite. But could they be potentially infinite? Concerning time, even if Rucker were correct that a tenseless four-dimensionalism is correct, that would provide no reason at all to think the space-time manifold to be temporally infinite: there could well be finitely separated initial and final singularities. In any case, Rucker is simply incorrect in saying that “the modern, scientific worldview” precludes a theory of time, according to which temporal becoming is a real and objective feature of reality. Following McTaggart, contemporary philosophers of space and time distinguish between the so-called A-Theory of time, according to which events are temporally ordered by tensed determinations of past, present, and future, and temporal becoming is an objective feature of physical reality, and the so-called B-Theory of time, according to which events are ordered by the tenseless relations of earlier than, simultaneous with, and later than, and temporal becoming is purely subjective. Although some thinkers have carelessly asserted that relativity theory has vin- dicated the B-Theory over against its rival, such claims are untenable. One could harmonize the A-Theory and relativity theory in at least three different ways: (1) distinguish meta- physical time from physical or clock time and maintain that while the former is A-Theoretic in nature, the latter is a bare abstraction therefrom, useful for scientific purposes and quite possibly B-Theoretic in character, the element of becoming having been abstracted out; (2) relativize becoming to reference frames, just as is done with simultaneity; and (3) select a privileged reference frame to define the time in which objective becoming occurs, most plausibly the cosmic time, which serves as the time parameter for hypersurfaces of homo- geneity in space-time in the General Theory of Relativity. And concerning space, to say that space is potentially infinite is not to say, with certain constructivists, that it depends on human activity (nor again, that there are actual places to which it can extend), but simply that space expands limitlessly as the distances between galaxies increase with time. As for the number of bits of matter, there is no incoherence in saying that there is a finite number of bits or that matter is capable of only a finite number of physical subdivisions, although mathematically one could proceed to carve up matter potentially ad infinitum. The sober fact is that there is just no evidence that actual infinities are anywhere instanti- ated in the physical world. It is therefore futile to seek to rebut (2.11) by appealing to clear counterexamples drawn from physical science.
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belorg
Guest
No, thinkandmull. A countable infinity is an infinity that is countable, which means each member of the set can be counted by using the natural numbers. In the set 1 3 5 7 9 … we can start counting 1 is the first member of the set, 3 is the second, etct. , so we get a correspondence between 1 3 5 7 9 and 1 2 3 4 5Countable infinity is an absurdity. It is trying to understand the infinite through succession of the finite. The odd numbers equal all the even numbers, not all the integers, regardless of what modernist math thinkers say today. I’ve already demonstrated how you are mistaken from correspondence: 1 3 5 7 9 correspond to 2 4 6 8 10, not 1-10
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Rhubarb
Guest
I think it’s super-weird to say that time or space is made of discrete units. All of my intuition all and empirical examination tells me that space and time intervals are infinitely dense.
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yppop
Guest
Tom,You can have an infinite number of intervals in an interval of finite length:. [0,1/2), [1/2, 3/4), [3/4, 7/8), [7/8, 15/16),… will be an infinite number of intervals in [0,1].
In any case, there are an infinite number of split seconds in a one hour interval follows from what you have shown. There are 3600 seconds in one hour.
There are 2x3600 split seconds of length 1/2
There are 4x 3600 split seconds of length 1/4
There are 8x 3600 split seconds of length 1/8
There are 16x 3600 split seconds of length 1/16
There are 32x 3600 split seconds of length 1/32
Etc. There is no limit to the number of split seconds in a one hour interval.
And here we are back with Zeno and his paradoxes, which the mathematicians address using the “infinitesimal”, the length of which can only be reduced to zero at infinity. Also your example is using rational numbers that extend to aleph(null).
The point I was making is that continuous space is defined by real (irrational) numbers, none of which you can write because they all extend to infinity so if time is continuous (which I don’t happen to believe) then continuous space is infinitely divisible and all intervals disappear. Only if time is discrete can you make the argument you presented.
Yppop
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rossum
Guest
I think it is easier to use a mapping: x → 2x - 1No, thinkandmull. A countable infinity is an infinity that is countable, which means each member of the set can be counted by using the natural numbers. In the set 1 3 5 7 9 … we can start counting 1 is the first member of the set, 3 is the second, etct. , so we get a correspondence between 1 3 5 7 9 and 1 2 3 4 5
1 → 1
2 → 3
3 → 5
4 → 7
5 → 9
etc.
All positive integers are mapped to all the odd numbers. Since it is a one-to-one mapping then the two sets have the same cardinal number; both sets are countably infinite since the positive integers are countably infinite.
Going back to Hilbert’s Hotel, the guest in room N is moved to room 2N - 1, leaving an infinite number of rooms free for new guests.
rossum
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Tomdstone
Guest
Yes. That is why there are an infinite number of instants of time in a one hour period. If anyone denies this, let him give the finite number of instants of time in one hour.I think it’s super-weird to say that time or space is made of discrete units. All of my intuition all and empirical examination tells me that space and time intervals are infinitely dense.
To each real number in the interval [0,1], there corresponds an instant of time.
There are an infinite number of real numbers in the interval [0,1].
Therefore there are an infinite number of instants of time in the interval [0,1].
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Tomdstone
Guest
Continuous time is infinitely divisible, otherwise give me the number of instants of time in a one hour period. Intervals of time do not disappear.The point I was making is that continuous space is defined by real (irrational) numbers, none of which you can write because they all extend to infinity so if time is continuous (which I don’t happen to believe) then continuous space is infinitely divisible and all intervals disappear. Only if time is discrete can you make the argument you presented.
Yppop
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Tomdstone
Guest
Correct.I think it is easier to use a mapping: x → 2x - 1
1 → 1
2 → 3
3 → 5
4 → 7
5 → 9
etc.
All positive integers are mapped to all the odd numbers. Since it is a one-to-one mapping then the two sets have the same cardinal number; both sets are countably infinite since the positive integers are countably infinite.
Going back to Hilbert’s Hotel, the guest in room N is moved to room 2N - 1, leaving an infinite number of rooms free for new guests.
rossum
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thinkandmull
Guest
That’s my point. Saying all the even plus odd numbers equal all the even numbers denies that there are different types of infinityThere are different types of infinity.
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thinkandmull
Guest
Arguing that it is potentially infinite is a red herring. The points are on the line, they are crossed over, in any movement, even for time. Eternal time is a different type of infinity. Where can I read what Craig says on Zeno? He probably uses Aristotles slight of hand no?1- No, we don’t have an “actually infinite number of time intervals” between 1:00 PM and 2:00 PM, only a potential infinite number of time intervals. Craig isn’t so stupid as to overlook “objections” like that; he writes about them in detail (for instance, in his article on the Blackwell Companion to Natural Theology), while also dealing with Zeno’s paradoxes. There’s one absolute difference between 2 and 1, and that is 1;
2- Regardless, would you then believe that a library with an infinite number of books (where we could take a book and the library would still have the same number of books it had before we took a book from it) would be possible? I find that a very, very hard bullet to bite. But absurdities like that would have to follow from the possibility of actual infinites in concreto.
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Tomdstone
Guest
No, because they both are countably infinite, unlike the set of real numbers.That’s my point. Saying all the even plus odd numbers equal all the even numbers denies that there are different types of infinity
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thinkandmull
Guest
You’re not seeing this yet. All the odd plus even numbers have a larger push forward towards infinity than all the odd. You can’t see this by going one by one. All the odd equal all the even.
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Tomdstone
Guest
They may have a larger push forward, but the sets {1,2,3,4,5,6,…} and {2,4,6,8,10…} are both countably infinite. The set of real numbers is uncountably infinite.You’re not seeing this yet. All the odd plus even numbers have a larger push forward towards infinity than all the odd. You can’t see this by going one by one. All the odd equal all the even.
If anyone says that there are a finite number of instants of time in a one minute period, let him give the number. He cannot, because time is continuous and as such there are an infinite number of instants of time in any given time period.
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yppop
Guest
TomContinuous time is infinitely divisible, otherwise give me the number of instants of time in a one hour period. Intervals of time do not disappear.
What do you mean by an interval of time, if not a duration with a finite length??
Yppop
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yppop
Guest
Tom,Yes. That is why there are an infinite number of instants of time in a one hour period. If anyone denies this, let him give the finite number of instants of time in one hour.
To each real number in the interval [0,1], there corresponds an instant of time.
There are an infinite number of real numbers in the interval [0,1].
Therefore there are an infinite number of instants of time in the interval [0,1].
What do you mean by “instants”. If you mean a “point” in time then you are correct to say there are an infinite number of instances of time in 1 hour, 1 minute, 1 second, etc.
If you are talking about the rational numbers, then there are intervals because there is always a “next” number, but an “instant” can be defined as a “point” in time and since it must be associated with a rational number we are then talking about discrete space.
On the other hand, if you are talking about continuous space, there is no “next” point since there is no “next” real number. There can be no interval or probably no instant in continuous space. The correct observation to be gotten from Zeno’s paradoxes is: if there is motion, space cannot be continuous and if space is continuous there can be no motion. If Zeno’s observation is correct what conclusion can you derive?
Yppop
When dealing with continuous space I believe we are dealing with the spiritual element of reality. Only God is actually infinite and that is why the infinitude of continuous space is an analog of the spiritual. We are immersed in the Mind of God. It is amazing that God has laid out this conundrum of infinity for us to contemplate.
Yppop
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