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Why the universe cannot logically be infinite in time backwards

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File:Wooden hourglass 3.jpg
passage of time, imaged/S. Sepp

Further to “No Big Bang: Universe Always Was” just posted by Donald McLaughlin, Ashby Camp gave Uncommon Descent permission to post these notes from a class he taught at the 2018 Harding University Bible Lectureship titled “Answering the New Atheism.” Worth pondering:


1.The second premise of the Kalam cosmological argument is: The universe began to exist. It is more reasonable to believe this is true than to deny it because, Scripture aside, there are strong philosophical and scientific reasons for believing it.

a. The philosophical argument for the universe having a beginning is that past time cannot be infinite because an infinite amount of time cannot already have been exhausted so as to arrive at the present. Infinite time is limitless, inexhaustible, and thus cannot have been exhausted.

(1) Put differently, one could never traverse an infinite sequence of time units, an infinite number of seconds, minutes, hours, etc., to arrive at now. There always would be more time units to traverse before now. If one begins counting down from minus infinity, one cannot count to the present. An infinite amount of time can never pass because it is limitless; it can only be in process, never complete. To quote the New Dictionary of Christian Apologetics, (p. 700), “One can neither count from one to infinity nor count down from infinity to one. There is always an infinite distance to travel, so one never arrives.”

(2) Sean McDowell and Jonathan Morrow make the point this way in Is God Just a Human Invention? (Grand Rapids: Kregel Publications, 2010), 75-76:

Imagine you went for a walk in the park and stumbled across someone proclaiming aloud, “… five, four, three, two, one—there, I finally finished! I just counted down from infinity!” What would be your initial thought? Would you wonder how long the person had been counting? Probably not. More likely, you would be in utter disbelief. Why? Because you know that such a task cannot be done. Just as it’s impossible to count up to infinity from the present moment, it’s equally impossible to count down from … infinity to the present moment. Counting to infinity is impossible because there is always (at least) one more number to count. In fact, every time you count a number, you still have infinite more to go, and thus get no closer to your goal. Similarly, counting down from infinity to the present moment is equally impossible. Such a task can’t even get started! Any point you pick in the past to begin, no matter how remote, would always require (at least) one more number to count before you could start there. Any beginning point would require an infinite number of previous points. Here’s the bottom line: we could never get to the present moment if we had to cross an actual infinite number of moments in the past. Yet, since the present moment is real, it must have been preceded by a finite past that includes a beginning or first event. Therefore, the universe had a beginning.

(3) The impossibility of infinite past time, of having already traversed an infinite timespan, does not mean that future time will not go on forever. Future time is potentially not actually infinite. In other words, it is infinity in progress, something that will move toward infinity but never arrive; you’ll never get to the end of it. So it poses no problem like the claim of having already traversed an infinite timespan.

(4) The impossibility of traversing an infinite timespan, an infinite sequence of time units, need not mean that God has not always existed. There are ways of dealing with God’s relationship to time so that he does not exist (or has not always existed) in a sequence of individuated moments, a sequence of time units. His eternality is not one of infinite time but one of either timelessness or a different kind of time that has no measure or metric. For example, William Lane Craig’s view is that “God is timeless without creation and temporal since creation.” Philosophers Alan Padgett and Richard Swinburne refer to time before creation as “metrically amorphous time,” meaning it differs from our “measured time” (see, e.g., Eternity in Christian Thought).

(5) This philosophical claim that the universe cannot always have existed makes sense to me and to many philosophers, but there are others who are not persuaded.

(a) Some, for example, point to the fact a finite timespan, say one minute, can be subdivided infinitely into units of decreasing length, and yet one can still traverse that timespan. The claim is that in going from 0 to 1 minute one traverses an infinite number of time units to arrive at 1 minute, so it is not true that one cannot traverse an infinite number of time units to arrive at the present. But there is a problem with that claim. The subdivisions of a finite timespan are only potentially infinite in number. It is true that one could keep subdividing forever, but each further subdivision results in a finite number of subdivisions the sum of which is the finite timespan being subdivided. The number of subdivisions can grow toward an infinite number but can never actually reach an infinite number. Whereas, when speaking of an infinite timespan one is speaking of an actually infinite set of time units, the sum of which is an infinite length of time.

(b) Though this philosophical claim that the universe necessarily began to exist continues to be debated, it dovetails nicely with the scientific acceptance of the universe having a beginning. That is the subject to which I now turn.

  

Readers?

Comments
EricMH- there isn't any such thing as infinity + 1. There is infinity. And then there is the density of elements in the infinite set.ET
March 11, 2020
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The value of the elements is the key to determining the cardinality. The set of positive integers has every element contained in the set of all positive even integers PLUS it contains all of the odd integers as well. That proves there is more elements in one set. If the cardinality was the same tat couldn't happen, You lose. And seeing that I am disputing your accepted mathematics and you keep referring to them, that means you are a clueless dolt who can only repeat himself in hopes of a different outcome. The very definition of insanity. My scheme is easier to understand. And it is used throughout set theory. And I told why you can match them in your scheme. Are you daft?ET
March 11, 2020
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For instance, the size of an infinite set + 1 is still infinite. This seems to be the distinguishing characteristic between finite and infinite numbers. A finite number + 1 is a new cardinality, contrary to infinite sets. However, there is an approach outlined by JohnnyB that I think gets at what you are pointing out using hyper numbers. https://journals.blythinstitute.org/ojs/index.php/cbi/article/view/57/53 I'm not so familiar with that approach yet, so I cannot comment.EricMH
March 11, 2020
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@ET can you provide your proof? I assume it is something like 1. A = positive integers 2. B = negative integers 3. C = A | B 4. C - B = A Which seems correct. However, I am not sure how you go from #4 to |C| > |A|. Operations like subtraction and addition on infinite sets are not well defined, AFAIK, but I don't have as much of a math background as you all do.EricMH
March 11, 2020
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ET: No one can collect infinite elements. That means there cannot be a set of infinite elements because a set is a collection of elements. Jerad still has problems with facts So you've been arguing about something you don't think exists? hahahahahahahahahahahahahahahahahahaahhahahahahahahahahahahahahahahahahahJVL
March 11, 2020
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ET: The value is the point. No, the number of elements in each set is the point. That's what cardinality is all about. The set of positive integers has every element contained in the set of all positive even integers PLUS it contains all of the odd integers as well. That proves there is more elements in one set. Then why can I match up the positive integers with the positive even integers in a one-to-one fashion so that you cannot find an unmatched element in the set you think is larger? If that is true, and it is, then there cannot be the same cardinality. And the way you deal with that is to ignore it. I am not ignoring. I am disputing it. I have a method which proves they do have the same cardinality. And I've told you over and over and over again how you can prove me wrong. And you have failed to do that. You can match them in a one-to-one correspondence only because they are both countable. That you ignore that proves you can’t deal with it. ALL countably infinite sets have the same cardinality. I've linked to proofs of just that thing. I am not ignoring anything. I am telling you over and over and over again where I disagree with you and how you can prove me wrong. And yet you haven’t. All you can do is repeat what I am saying is incorrect. It’s as if you have Tourette’s I have explained over and over and over again the basic, accepted, established mathematics involved. I have linked to sources, I have done my best to explain things in various ways to try and get them across. You keep referring to things that are non-existent, like "contrived vs derived" matchings and when I ask you to support your views you can't do it. Once again: I can match up the elements of the positive integers with the positive even integers one-to-one. Every element of each set has a unique partner in the other set. Give me an element of either set and I can easily tell you its partner in the other set. Explain to me how that can happen if the sets do not have the same number of elements in them. Or, even easier, just find an element of either set that does not have a partner in the other set. That's it. No more diverting. No more trying to call cast aspersions on techniques. My scheme is simple, easy to understand. Under my scheme can you find an unmatched element of either set, yes or no? Yes or no? One word would end this whole thing.JVL
March 11, 2020
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No one can collect infinite elements. That means there cannot be a set of infinite elements because a set is a collection of elements. Jerad still has problems with factsET
March 11, 2020
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I can find fault with your arguments …
And yet you haven't. All you can do is repeat what I am saying is incorrect. It's as if you have Tourette'sET
March 11, 2020
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The value is the point. The set of positive integers has every element contained in the set of all positive even integers PLUS it contains all of the odd integers as well. That proves there is more elements in one set. If that is true, and it is, then there cannot be the same cardinality. And the way you deal with that is to ignore it. You can match them in a one-to-one correspondence only because they are both countable. That you ignore that proves you can't deal with it.ET
March 11, 2020
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ET: One more thingy- there is only one infinity. Cantor found there are different densities. And all I am doing is extending that to the countably infinite sets. So what is your opinion of the continuum hypothesis? But then again I have been over this with you and you refuse to or cannot grasp We'll see how you deal with the questions shall we? Cardinality refers to the number of elements a set contains. Exactly so, that's why the value of the elements of the sets are not important. The set of positive integers has every element contained in the set of all positive even integers PLUS it contains all of the odd integers as well. That proves there is more elements in one set. Then I should not be able to match them up, one-to-one. But I can. And you cannot find an unmatched element. Until you can deal with that you don’t have anything to refute my claim. I have dealt with it and you have not found a counter example.JVL
March 11, 2020
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ET: Jerad, Your scheme is the very thing being debated. Your scheme works fine for showing the sets are countable. Your scheme for doing that IGNORES the real value of the sets. The "value" of the sets (surely you mean the value of the elements of the sets?) is NOT THE POINT. It's how many elements are in each set. That's it. I have found a scheme which clearly and unambiguously matches each positive integer with a positive even integer so that no element of either set is left unmatched. It's simple, it's easy to understand. That can only happen if each set has the same number of elements. Using the standard scheme for matching- the scheme used throughout set theory for finding unions, subsets and proper subsets, shows there are unmatched elements. And using THAT scheme you cannot show there is a one-to-one correspondence. Using my scheme there are no unmatched elements. There is nothing wrong with my scheme. Except that you cannot find an unmatched element. There is no "standard scheme". If you can show there is a one-to-one correspondence using that standard scheme I will admit I made a mistake. And if you can’t you are wrong. No, you are wrong because you cannot think outside of one scheme. There is nothing wrong with my scheme. It's valid. You're the only person who has a problem with it. If you really took a Set Theory course then you know there is no "standard scheme". And if there is then you should be able to find a reference talking about it. Can you find such a reference? Your problem is thinking that because I disagree with Cantor on one irrelevant part of his ramblings, that I don’t know anything about it. But yet you cannot find any fault with my arguments and you can only repeat what is being debated. I can find fault with your arguments because you are desperately trying to discredit my scheme for no good reason except that it disproves your idea. There is no standard scheme. Any matching is allowed. I have found a way to match up the positive integers with the positive even integers one-to-one so that no element of either set is unmatched. That can only happen if the sets have the same number of elements. If you can find an unmatched element in my scheme then you will be lauded as a mathematical genius, guaranteed. Can you find an unmatched element, yes or no? And Jerad, stop lecturing me. It is clear that you don’t understand infinity. AGAIN, I have found a way to match up the positive integers with the positive even integers one-to-one so that no element of either set is unmatched with a unique "partner". That can only happen if the sets have the same number of elements. You disagree and, if you're right, then you should be able to find an unmatched element in the set with more elements. Can you do that? Yes or No? It's time to stop dodging and trying to weasel out of it. Can you find an unmatched element, yes or no? Everything else is just diversion. Can you find an unmatched element, yes or no?JVL
March 11, 2020
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Cardinality refers to the number of elements a set contains. The set of positive integers has every element contained in the set of all positive even integers PLUS it contains all of the odd integers as well. That proves there is more elements in one set. Until you can deal with that you don't have anything to refute my claim.ET
March 11, 2020
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Earth to Jim Thibodeau- I am saying that definition is incorrect. So if all you can do is keep repeating it you have serious issues.ET
March 11, 2020
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Definition 1: |A| = |B| Two sets A and B have the same cardinality if there exists a bijection from A to B, that is, a function from A to B that is both injective and surjective.
Jim Thibodeau
March 11, 2020
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One more thingy- there is only one infinity. Cantor found there are different densities. And all I am doing is extending that to the countably infinite sets. But then again I have been over this with you and you refuse to or cannot grasp it.ET
March 11, 2020
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And Jerad, stop lecturing me. It is clear that you don't understand infinity.ET
March 11, 2020
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Jerad:
There’s something beyond countably infinite.
Yes, I know. I have taken the courses. Your problem is thinking that because I disagree with Cantor on one irrelevant part of his ramblings, that I don't know anything about it. But yet you cannot find any fault with my arguments and you can only repeat what is being debated.ET
March 11, 2020
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Hey, we did rings and fields in abstract algebra! Galois theory too. Permutation groups were my favorite though. I’ve downloaded a 15 lecture set off YouTube about analysis but I haven’t watched it yet.Jim Thibodeau
March 11, 2020
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Yes, EricMH. That scheme works fine for showing the two sets are countable. However basic set subtraction and logic shows one set can have more elements than another.ET
March 11, 2020
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Jerad, Your scheme is the very thing being debated. Your scheme works fine for showing the sets are countable. Your scheme for doing that IGNORES the real value of the sets. Using the standard scheme for matching- the scheme used throughout set theory for finding unions, subsets and proper subsets, shows there are unmatched elements. And using THAT scheme you cannot show there is a one-to-one correspondence. If you can show there is a one-to-one correspondence using that standard scheme I will admit I made a mistake. And if you can't you are wrong.ET
March 11, 2020
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Jim Thibodeau: Well, I was always a bit fond of number theory myself but to understand how powerful and amazing math is I can't recommend complex analysis enough. Beautiful stuff. Rings and Fields . . . I think I would appreciate that stuff more now but at the time . . . .JVL
March 11, 2020
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JVL @126 I took so many math classes for fun that if I had taken Rational Analysis I would have also gotten a BA in math. My favorite was abstract algebra. Any suggestions about other good ones?Jim Thibodeau
March 11, 2020
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@ET you made me think a moment with the question: why does the set of positive integers have the same cardinality as the negative and positive integers? But, even though they have different numbers, it is still straightforward to come up with a scheme to map all #1) the positive integers to #2) all the numbers in the set of negative and positive integers. f(n) = if n is even: -n/2 else: (n+1)/2 Since that mapping will never be violated, i.e you'll never find an element in #2 that cannot be inverted to an element in #1 and visa versa, that means the two sets must have the same cardinality. Very counterintuitive.EricMH
March 11, 2020
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Oh, and by the way . . . we haven't even gotten onto the different sizes of infinity. There's something beyond countably infinite. You want to blow your mind? Become a math major. When you get to topology give me a shout. I'll provide the tea and sympathy.JVL
March 11, 2020
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Mathematics is not a spectator sport. It's like carpentry or hand-to-hand combat or sculpture or dancing or piano playing or football or being a pilot or any other skill which requires practice and experience. You have to get your head around the systems involved. Sometimes you need to learn new ways of thinking and new skills. I can study martial arts in a book all I want, I can watch hours and hours of videos but it all means close to nothing until I actually spend some time learning how to really carry it out.. Mathematics is actually very deep and, in the last 100 years or so, very, very difficult. And it's not like other topics where new ideas supersede old ones. What was true for Pythagoras and Euclid is still true. Newtonian physics may have been revised and overtaken but Newtonian mathematics has not suffered the same fate. If you understand mathematics up through the 19th century you are doing very well, most people never get past arithmetic. After Cantor in particular, things got very, very weird and difficult.JVL
March 11, 2020
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Because all you are doing is taking the sets and converting them to the set {1,2,3,4,…}. That means you are no longer dealing with the distinct elements that are actually in the set. But you won’t be able to grasp that. I am showing that they match up one-to-one. And yes, all countably infinite sets can be matched up with the positive integers. THAT'S THE POINT!! It's the number of elements that is the point of cardinality. True but no one can collect infinite elements. So it cannot be a collection. It's a well defined set. Pick something, anything, I can tell you whether it's in the set or not. That means the definition is clear and unambiguous. I have. Cantor’s bijection ignores the actual elements. It doesn't matter what the elements are, what we're talking about is how many of them there are. That's the whole discussion. Nonsense. You said you studied Set Theory but that clearly is not true or you would know what the rules are. The set of positive integers has every element contained in the set of all positive even integers PLUS it contains all of the odd integers as well. That proves there is more elements in one set. Then why can I match up the positive even integers with the positive integers one-for-one? You're not addressing that basic question is just causing this discussion to go on and on and on. The mistake was made by Cantor. That has always been my point. And all you can do is repeat the mistake as if it means something. Then find an unmatched positive integers in my scheme. That's all you have to do. Everything else is just diversion. Can you do that, yes or no? You can prove Cantor wrong by finding an unmatched element. I use the standard matching used for determining unions, subsets and proper subsets. And using that standard matching the set of positive integers has every element contained in the set of all positive even integers PLUS it contains all of the odd integers as well. I have never, ever disputed that the even integers are a proper subset of the integers or that the odd integers are a proper subset of the integers or that the even integers union with the odd integers give you the integers. We agree. But because all those sets can be put into a one-to-one correspondence with the positive integers they have the same cardinality. And if you want to say otherwise then you need to find a counter example. In my scheme, you need to find an unmatched positive integer. Can you do that, yes or no? Just answer the question.JVL
March 11, 2020
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I use the standard matching used for determining unions, subsets and proper subsets. And using that standard matching the set of positive integers has every element contained in the set of all positive even integers PLUS it contains all of the odd integers as well. Real. World. Proof.ET
March 11, 2020
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JVL:
How can they have a one-to-one correspondence unless they have the same number of elements?
Because all you are doing is taking the sets and converting them to the set {1,2,3,4,...}. That means you are no longer dealing with the distinct elements that are actually in the set. But you won't be able to grasp that.
The positive integers is a well defined set.
True but no one can collect infinite elements. So it cannot be a collection.
Address that issue and we’re done.
I have. Cantor's bijection ignores the actual elements.
It’s not magical, there are rules and things you can do and things you can’t do.
Nonsense. The set of positive integers has every element contained in the set of all positive even integers PLUS it contains all of the odd integers as well. That proves there is more elements in one set. The mistake was made by Cantor. That has always been my point. And all you can do is repeat the mistake as if it means something.ET
March 11, 2020
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ET: Being able to find a one-to-one correspondence only means the sets are countable. That’s it. How can they have a one-to-one correspondence unless they have the same number of elements? And besides, I've liked to a theorem (that means it's been proved) which says that all countable infinite sets have the same cardinality. Just because someone can wrongly claim that also means the sets have the same number of elements doesn’t mean anything to me. Well, it means they can’t think for themselves. When you can't find an unmatched element it does mean that. But that is all moot, anyway. There can’t be a set with infinite elements if a set is defined as “a well-defined collection of distinct objects, considered as an object in its own right”. Wrong again. You said you studied set theory but that's clearly not the case. The positive integers is a well defined set. Name an object and I can tell whether it's in the set or not. That's well defined. So I guess if you can ignore that you can also ignore the reality that says The set of positive integers has every element contained in the set of all positive even integers PLUS it contains all of the odd integers as well. That proves there is more elements in one set. Then you should be able to find an unmatched positive integer under my matching scheme. But you haven't been able to do that. Address that issue and we're done. You want to ignore reality and think infinity is magical, fine. Seeing that the concept under debate is useless anyone can say whatever they want and no one can prove them wrong. I mean real proof- not wanker proof. It's not magical, there are rules and things you can do and things you can't do. If you think the math and the proofs are wrong then find a mistake or a counter example. Can you do that: yes or no? BTW, I have always known what the mainstream has said about this. I am saying that it is wrong. So just merely repeating what the mainstream says proves that you are nothing but a good parrot. And thinking that repeating what they say somehow refutes my argument proves that you are clueless. I understand you are aware of what everyone else says. I'm not just repeating it, I get how it works. I've taken an actual set theory class and have worked with the rules governing such things. I have created a matching that you can explode if you are able. So we're still down to: can you find an unmatched positive integer in my scheme which matches the positive integers with the positive even integers one-to-one? Yes or no? Either answer will end the whole conversation. Not answering means you either don't understand the issue (which I don't believe) or you are too afraid to admit you made a mistake. Answering yes means you have to produce an unmatched positive integer in my scheme. Answering no means we're done and we can stop discussing this. It's your call.JVL
March 11, 2020
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Being able to find a one-to-one correspondence only means the sets are countable. That's it. Just because someone can wrongly claim that also means the sets have the same number of elements doesn't mean anything to me. Well, it means they can't think for themselves. But that is all moot, anyway. There can't be a set with infinite elements if a set is defined as "a well-defined collection of distinct objects, considered as an object in its own right". So I guess if you can ignore that you can also ignore the reality that says The set of positive integers has every element contained in the set of all positive even integers PLUS it contains all of the odd integers as well. That proves there is more elements in one set. You want to ignore reality and think infinity is magical, fine. Seeing that the concept under debate is useless anyone can say whatever they want and no one can prove them wrong. I mean real proof- not wanker proof. BTW, I have always known what the mainstream has said about this. I am saying that it is wrong. So just merely repeating what the mainstream says proves that you are nothing but a good parrot. And thinking that repeating what they say somehow refutes my argument proves that you are clueless.ET
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