Transcription
What squared is that? You—it's been months. Where have you been?
Oh, hi. Um, as it happens, I've been traveling the Multiverse.
What? Uh, I don't have much time. There is a group of universe-destroying aliens known as the… wait, wait, wait. Does that mean Eternal inflationary cosmology is true?
Uh, what? No, I don't see how this is—then is the Multiverse a product of the many-worlds interpretation of quantum mechanics?
No, it isn't. Uh, can we please stay on task? There are universe-destroying aliens on their way here, and we don't have much time to stop them.
I'm sorry, it's just that the confirmed existence of a Multiverse has pretty profound metaphysical implications. For instance, this undermines the fine-tuning argument, generally considered one of the strongest arguments for the existence of God.
No, um, actually, there's a great argument for the existence of God from the existence of a Multiverse. Basically, the fact that an agent finds themselves existing is evidence of a Multiverse which is too big for God not to exist.
That sounds like an intriguing argument. You're saying the central evidence for the Multiverse is merely one's own existence?
Yeah, that part is based on anthropic reasoning, but again, the aliens…
I'm sorry, but could you dive a bit deeper into this argument? I'm afraid you've piqued my curiosity.
Look, I'm a little pressed for time here, and… well, how much time do you have exactly?
Like, maybe just over half an hour?
I think that should be plenty of time to explain this argument and prepare our universe to fend off an interdimensional threat, don't you?
Um, maybe if I actually… yeah, yeah, I think you're right. I probably have enough time to explain the anthropic argument before getting to work.
All right, here goes. Before getting into the nitty-gritty details of the argument, let's take a look at the argument's general structure from a 10,000-foot view. The first part is establishing the self-indication assumption—basically, the SIA is a principle that says that since you exist, theories which posit that lots of people exist become more probable.
Here's an illustration for the sort of rationale behind this principle. Let's say I have a standard 52-card deck and you have a box. Now, I have done one of two things: I have either put one of my cards in your box, or I have put my entire deck in the box, but you don't know which I did. So, I let you ask me a question. You ask, "Is the four of clubs in the box?" Why, yes it is. Now, from that information alone, you have great evidence that I gave you the entire deck, because having the entire deck in the box maximizes the probability that you would have the particular card that you guessed in the box. So, finding out that you have one card is evidence that you actually have a bunch more cards.
Now, the SAA works off really similar logic, but instead of being about cards, it's about people. Since you know you exist, you should think there are lots of people who exist, since lots of people existing maximizes the probability that you, in particular, would exist. This will be defended in more depth later.
Now, the next step in the anthropic argument goes from the SIA to a Multiverse. If a theory that posits more people is more probable, what is the most probable theory? Well, a Multiverse—infinitely many people across infinitely many universes. But here's where things get interesting: some infinities are larger than others. So, the best Multiverse theory is one in which a rather large infinity of people exist. I like to call a Multiverse with a large infinity of people a mega Multiverse.
The final step in this argument is going from a mega Multiverse to God. See, it's not that difficult for an atheist to give an account of a plain old vanilla Multiverse, but a mega Multiverse—that's another story. The only naturalistic theories of mega multiverses face a number of technical problems, such as gerrymandering or destroying induction. Theism, on the other hand, can give a very straightforward account of a mega Multiverse: creating people is a good thing to do, so God created every possible person. So that's the general anthropic argument. Here's a deductive formulation and a probabilistic formulation. With that out of the way, let's now look at each of these premises in turn.
First off, why think that the SAA is true? Why think that a theory with more people in it is more probable? Real quick, a precise definition of the SIA is: if proposition q1 entails that there are N1 people that you might be, and proposition Q2 entails that there are N2 people that you might be, then the data of your existence yields a base factor favoring q1 over q2 with magnitude N1 over N2. To defend the SIA, I'm going to give a basic hypothetical scenario where the SIA gives a correct analysis of all the relevant probabilities, and all competing views do not. And the specific details of the hypothetical don't actually matter. So, if the SIA-based reasoning is correct here, it's going to be correct in other contexts as well. So, here's the scenario: a person is created—always Alex. Later, a fair coin is flipped. If the coin landed on heads, another person is created—sometimes Sammy. Sammy is not created if the coin landed on tails.
Now, let's say you are one of these people, but you don't know which. Besides your understanding of the rules of this scenario, you have no additional information that could help you discover whether you're Alex or Sammy. Neither can check if the coin was flipped yet; they both get created in identical rooms; they aren't told their names, and so on. Now, what should you think the probability is that the coin landed on heads? Again, you don't know if the coin flip has even happened yet. So the question is: whenever the coin flip actually happens, is the outcome heads? You might think that the answer is 50%; after all, the coin is fair. But the SAA disagrees. If a scenario has more people in it that you might be, it becomes more probable. If the coin lands on heads, then there are two people that you might be instead of one. So, the SIA says that the coin landing on heads is twice as likely as tails. So there is a two-thirds chance of heads and a one-third chance of tails.
This might seem like magic. Does the SIA claim that Sammy's potential existence magically affects the coin somehow? No, it doesn't. Basically, there are two kinds of probability: objective and subjective. Objective probability describes the tendencies of things in the external world; subjective probabilities describe how confident a person is in something. If a friend who is 90% reliable tells you a coin landed on heads, that doesn't magically affect the coin; it just makes you more confident that it actually happened. So, the probability you assigned to the coin being heads goes up. However, the objective probability that a fair coin lands on heads is still 50%, since that's the definition of fair. The SAA is a claim about what our subjective probabilities should be; it's saying that your particular existence is evidence that more people were created in total. So that's why you should be more confident the coin landed heads.
Maybe this still seems weird to you, though. Well, let's add one extra part to this thought experiment. After a while, you get told your name. Let's say that it turns out that you're Alex. Now, this is a new piece of information, and probability theory tells us how this affects the probability that the coin landed heads. If you went with the SIA probabilities to start with, the probability that the coin lands on heads becomes 1/2, which makes sense. If it turns out you're Alex, then the coin flip hasn't happened; it's in the future. So your subjective probability that it will land on heads should be 1/2. However, if you ignored the SIA and assigned a probability of 1/2 to heads to start with before you knew you were Alex, the probability of heads, rather absurdly, becomes 1/3 after you get this new information. Here's the mathematical reasoning with symbols; however, I think it's much easier to understand what's going on visually. If we want to go against the SIA and say that the probability of heads starts at 1/2, then we can represent that by dividing a square with area one into two equal halves. If the coin landed on tails, then you are definitely Alex. If the coin landed on heads, then you might be Alex or Sammy with equal probability. So the head's area gets bisected. So, in total, you are Alex with a probability of 3/4. If you learned that you definitely are Alex, then this Sammy quarter here gets thrown out, and heads only ends up occupying 1/3 of the remaining area. So the final probability of heads is 1/3. Alternatively, if we start with saying that heads has a probability of 2/3, then when we bisect it into equal portions, Sammy and Alex, Alex only ends up having a probability of 2/3. Then, upon learning your Alex, this third gets thrown out, and heads ends up occupying half of the probability space. So the denial of the SIA leads to an obviously absurd conclusion. Upon learning that you were Alex, you would have to think that this fair coin, which hasn't even been flipped, will probably land on tails without having any special evidence for that conclusion. So you should go with the SIA's probabilities instead.
Coolio. That's the main idea of the argument, but I want to explain what I think is going on here—why the SIA gives the right answer. There are two main questions we care about: am I Alex or Sammy, and did the coin land on heads or tails? There are four combinations of answers that can be given: Alex and heads, Alex and tails, Sammy and heads, and Sammy and tails. If we had literally no information to start with, then each of these options are equally likely, with a probability of 1/4. But there is some information we can use to update on the setup: I exist. Well, that rules out the last possibility because Sammy doesn't exist if the coin landed on tails. This means there are now three options which are equally likely, and the coin landed heads in two of them. So heads has a probability of 2/3. Now, if you give me some new information—"I'm Alex"—that just rules out one more option, and the probability of heads goes back to one-half. Some people struggle with this because they think we should treat the possibility of Sammy and tails as being somehow contradictory, so it shouldn't even be listed as an option. But it isn't a contradiction. My existence isn't guaranteed, and the proposition "I'm Sammy and the coin landed tails" is just a scenario where I'm someone who didn't get created—no less contradictory than the hypothetical scenario "my parents never met." These possibilities could have hypothetically happened, and we should simply treat "I think, therefore I am" as evidence that proves those possibilities never came to pass. When you get down to it, the SIA is just a way of approaching probability theory where the information "I exist" is treated like any other piece of information. Denying the SIA basically amounts to trying to assign some sort of special status to the fact that you exist and then running into paradoxes because your existence isn't some magically special data.
Now, I've been spending a lot of time on this one thought experiment, but there are dozens of thought experiments where any method of assigning subjective probabilities to propositions that deviates from the SIA results in absurdities. I'll go through one more because it illustrates some of the bizarre conclusions that some real-world SIA deniers actually accept. We'll first define a new term: your early earliness scorer. This is the percentage of all people who will ever exist that were created after you. So, if you were the 30th person created, and 100 people are created in total, your earliness score is 70%. If you were the fifth person created, and 10 people are created in total, your earliness score is 50%. Now, if I told you your Earl score was either between 0% and 10%, or it was between 10% and 100%, which of those options would you think is more probable based on that information alone? Obviously the latter. You have no reason to expect one particular earliness score rather than another, so the bigger range wins. The reason this earliness score stuff is important is precisely because a proposition becomes less probable when it entails that your earliness score falls in a narrower range. Next, a person-making machine creates a person every minute. After making 10 people, the machine flips a fair coin, and if it lands on heads, it creates another 90 people. So, in total, either 10 people or 100 people are created. Now, let's suppose you learn that you were one of the first 10 people made by the person-creating machine, and the coin flip hasn't happened yet. What's your earliness score? Well, if the coin ends up landing tails, then your earliness score could be anything. But if the coin lands on heads, your earliness score is guaranteed to be in a very small range—between 90% and 99%. Since heads constrains your earliness score to such a narrow range, whereas tails gives a very wide range, you should instead think the coin is more likely to land on tails. This is obviously an absurd conclusion, since the coin is fair, but there's no mistake with the math here. Considerations about your earliness score simply decrease the probability of heads. We need some answer to this paradox. If learning you're one of the first 10 people makes heads less probable, how can we avoid thinking that a fair coin will probably land on tails? You know, it sure would be nice if there was just some way to offset this bizarre effect. Well, the SIA does just that. If you applied the SAA before learning that you were one of the first 10 people, you would have started out thinking that heads was much more probable than tails, since if the coin landed on heads, more people would exist who you might be. But then, once you learned you were one of the first 10 people, considerations about your earliness score would lower the probability of heads. As it turns out, it can be shown mathematically that the initial probability boost from the SIA will always exactly cancel out considerations from your earliness score; your subjective probabilities end up matching the objective probabilities. So, in the scenario, if you accepted the SIA, then upon learning that you were one of the first 10 people, you would think that the probability of the coin landing heads was 50%. This particular example is closely tied to the Doomsday argument. There are some people who deny the SAA and, based on reasoning about their earliness score, conclude that the human race will probably go extinct soon, because if the human race does not go extinct soon, there might be hundreds of billions of people who will end up existing, which constrains all currently existing humans' Earl score to a super-duper narrow range. But this is absurd. If you endorse the SAA, you would use information about the objective probabilities of birth rates, man-made or natural disasters, and that sort of thing to assess the population of humanity in future generations. But proponents of the Doomsday argument insist that the subjective probability that you assign to humanity's imminent extinction should exceed the objective probability of that extinction. That's simply untenable. Concluding this section, I just want to emphasize how much support there is for this premise of the argument. Lots and lots of crazy stuff follows from denying the SIA—exactly what you should expect if the SIA is just standard application of probability theory. In fact, I will go so far as to say that if you are a detractor of the anthropic argument, I would counsel you to look at the other premises of the argument for possible objections.
Speaking of the next premise, the SIA entails the existence of a mega Multiverse. This is mostly a straightforward extrapolation of the SIA. If finitely many extra people give a theory a finite boost, then positing that there are infinitely many people gives a theory an infinite boost. This means that no matter how improbable a Multiverse is prior to SIA considerations, you should definitely believe that one exists. Even if you assigned a subjective probability of one in a billion to a Multiverse, the evidence of your existence overcomes that low prior. Now you may remember that earlier I said that the SAA doesn't just lead to a Multiverse; it leads to a mega Multiverse. What's that all about? Well, one of the coolest results of set theory is that some infinities are bigger than others. You have Beth null, which is the size of the set of natural numbers (1, 2, 3, and so on), but Beth null is infinitely smaller than Beth one, which is the size of the set of real numbers. So that would include stuff like the square root of two and Pi. But Beth one is infinitely smaller than Beth two, which is the size of the set of all real-valued functions. So now you have stuff like x², sin x, and all that. Equipped with this knowledge, we can now be a bit more granular with our usage of the SIA. Earlier we compared two basic propositions: there are finitely many people, and there are infinitely many people. The SIA supports infinitely many people, but now we could ask: how big of an infinity of people exist? Are there Beth null people? Beth one people? Beth two people? Or even more? Well, the SIA always lends credence to the theory with more people, and Beth one isn't just a million times bigger than Beth null; it's infinitely bigger than Beth null. So a Multiverse with Beth one people is an infinitely better theory than a Multiverse with a measly Beth null people, and a Multiverse with Beth two people is infinitely more probable than that. You might be asking: where does it end? What about a Multiverse theory with Beth three people or Beth 78? Those are even bigger. Well, it ends when you posit a Multiverse where the number of people that exist is the same as the number of possible people. You can't posit more people than that. So, if you hypothetically knew that there were Beth 78 possible people, then the SIA leads us to posit a Multiverse with Beth 78 actual people. But how many people are there really? How many different people could possibly exist? Obviously infinity, but which infinity? There's a decent argument that the number of possible people is at least Beth two. Here's a simplified version of the reasoning: we know that there are at least Beth null possible people, since a one-meter-tall person is possible, a two-meter-tall person is possible, a three-meter-tall person is possible, and so on. But a π-meter-tall person and a square root of 2 meter-tall person are also possible. So there are at least Beth one possible people. However, we could take a look at the graph of someone's height over time, and this could match any function. A person whose height is x² m is possible, and a person whose height is sin(x) + 2 m is possible, and so on. So Beth two is a lower bound on how many possible people there are. Now, there's a bit more to all this—I think the number of possible people is actually much, much larger—but that's a rabbit hole that I'm going to be avoiding in this video. So, since there are at least Beth two possible people, the SIA leads to a Multiverse with at least Beth two actual people. For the context of this argument, I propose that we refer to a Multiverse with at least Beth two people in it as a mega Multiverse.
On to the last premise: theism can explain the existence of a mega Multiverse very easily, whereas naturalism cannot. The theist can posit that God creates every possible person and puts them in universes to maximize overall goodness across all reality. This not only comports well with perfect being theology, but it can even help diffuse some arguments against theism. For example, some people have argued that God cannot act rationally when deciding what to create, since no matter what he creates, he must always pass up a better option—namely, creating whatever he was going to create plus creating even more good stuff as well. But on this mega Multiverse view, God doesn't leave any good stuff uncreated. Coolio. Naturalism, on the other hand, faces a Goldilocks problem. All proposed naturalistic multiverses end up being too small to be a mega Multiverse, or they end up including a bunch of undesirable universes that cause lots of problems for one's worldview. Here's an example of a Multiverse that just isn't big enough. Theists have given fine-tuning arguments in the past. The idea with that argument is just that the fundamental physical constants fall in a narrow life-permitting range. This is evidence of design. In response, some atheists have posited a Multiverse where every possible combination of physical constants appears in some universe. However, even if every constant is allowed to take on any real-valued number, that only gets you Beth one universes, because there are only Beth one real numbers. That simply isn't enough; a mega Multiverse will need more universes than there are real numbers. That's very hard to do. On the other end of the spectrum, a guy named David Lewis came up with a mega Multiverse that in no way relies on God but is still hugely problematic. You see, his view, called modal realism, has at least Beth two universes, but to get all those universes, he had to employ a principle of recombination. Basically, not only did he believe that there were a whole bunch of universes with different laws of physics, but he also believed that any possible way of splicing universes together also constituted an existing universe. For example, let's say there's one universe where a man made a statue of himself, and then it was broken 100 years later. In another universe, the man made a statue of a turtle instead, and then that was broken 100 years later. Well, the principle of recombination says that there is another universe where there was a statue of a man for 40 years, and then it suddenly changed to a statue of a turtle, and then that was destroyed 60 years later. Basically, David Lewis was able to get so many universes because his theory allowed you to copy-paste chunks of somewhat normal-looking universes in infinite combinations. But this faces a serious problem: Lewis's theory entails that there are infinitely many universes that look just like ours, but 5 seconds from now, something crazy happens. Like, there's a universe exactly like ours, except by the time you finish watching this video, the Earth disappears, or the sun turns into ice, or all gold in the universe is replaced with Swiss cheese. This is very problematic, since this infinity of chaotic universes makes it virtually certain that our world will end up devolving into pure chaos immediately. There are those who think that instead of being certain that our world will devolve into chaos, we simply lose any ability to make rational predictions about the future, which isn't a very good result either, if you ask me. Basically, with so many chaotic universes in Lewis's Multiverse, we have no reason to keep believing that our universe is not chaotic. In the end, the naturalist needs to open the door to extra universes wide enough to get at least Beth two people, but not so wide that a plethora of induction-destroying chaotic universes can enter in. You want a Multiverse from Eternal inflation? Well, there aren't enough physical parameters in our laws of physics to get Beth two permutations. What if the laws of physics themselves vary from universe to universe? Well, then you're going to get universes with laws of physics like ours, except they have additional chaos, including laws like: after the year 2025, all gold turns into Swiss cheese. What if we use the many-worlds interpretation of quantum mechanics? Well, a quantum measurement can take on at most Beth one possible values, so you don't get Beth two worlds. Naturalism simply doesn't have the resources to get a mega Multiverse without destroying our ability to predict anything. Theism, on the other hand, can thread this needle, because the predictions of theism are based in God doing the best action, and making a mega Multiverse without destroying induction is quite reasonably the best conceivable action.
So that's the anthropic argument. Before ending, let's look at some objections. One objection to the SIA is that it's very presumptuous. Do I think that you can just reason to the conclusion that a mega Multiverse exists from the comfort of your armchair simply by reasoning about your own existence? Short answer: yes. It is not presumptuous to reason that if 0% of all possible people existed, then I wouldn't exist. So it is not the case that 0% of all possible people existed. Long answer: I know I exist; you know you exist. How do we handle this piece of information? The SIA is one answer to that question. As it turns out, it can be proven that absolutely every answer leads to presumptuousness of some kind. We looked at an earlier example: proponents of the Doomsday argument think you can deduce humanity's imminent extinction from the comfort of your armchair. Since every answer has some quirky side effects, the far-reaching implications of the SIA aren't worrisome; it's not a bug, it's a feature. What if you just ignore your existence instead? Like, if my existence can only be evidence for weird stuff like a Multiverse or impending annihilation, then I'm not going to treat my existence as evidence for anything whatsoever. However, this is untenable. Your existence obviously acts as evidence for various propositions. For example, your existence is evidence that your parents had at least one child. So there are correct inferences that can be drawn from your existence.
Another very common objection to the SAA is that it can be parodied to make arguments for rather absurd conclusions. Like: "My chair exists, so can we run a parallel argument that a Multiverse with Beth two chairs exists to explain the existence of my particular chair?" Here's another one: "What if there are zillions of tiny invisible people living in my attic? Would the SAA favor that theory since it entails that more people exist?" Now, these parodies fail. To explain why, I want to take a step back and talk about probabilistic reasoning in general. If you have some data, you want to find a theory that makes that data probable. Think back to that illustration at the beginning of the video with the deck of cards. If you were given the entire deck of cards, then the probability that you would have the four of clubs is 100%. But if you were only given one card, then the probability that you would have the four of clubs is 1 in 52. So the theory that you were given the entire deck of cards maximizes the probability of the specific data you were given. If a theory doesn't raise the probability that you would get that specific data, then the data doesn't actually point to that theory. This is actually pretty straightforward. Positing that a thousand paper clips were put in the box isn't useful because the four of clubs card isn't a paper clip. Now, let's say that I see a chair. Does this observation get any more probable if I posit that there are other universes out there filled with nothing but infinity chairs? No, that theory does nothing to help with this specific data, since the chair I'm looking at is not in another universe; it's in this universe. So the data I have gives me no reason to posit a chair-based Multiverse. What about the zillions of tiny invisible people? Well, the piece of data we're trying to explain with the anthropic argument is: "I exist." Now, I am not a tiny invisible person, so positing the existence of such creatures does not raise the probability that I exist. Remember, the SAA says that a theory is more probable if it posits that there are more people who you might currently be. That's why it wouldn't apply to these weird parodies. Now, this clause of "who you might currently be" gets into a really technical and nuanced point. I'm going to introduce the concept of an epistemic double. An epistemic double is someone who has all the same memories and experiences as you; they have the same total evidence as you. If you want to put it in Bayesian terminology, the point is that they aren't identical to you; they might be different in unknown ways, but all the information they have available to them is the same as the information you have available to you. Now consider the set of every possible person who would be your epistemic double if they existed. The SAA actually only gives a probabilistic boost to theories that increase the number of people from this set. Why? Well, you could only be someone from the set. Positing a bunch of people you could not be doesn't better predict your existence. So you need to posit that at least Beth two of your epistemic doubles got created so that your existence is not infinitely improbable. People from outside the set could exist or not exist, and the SIA doesn't care, because people who you could not be do not affect the probability of your particular existence. But wait, does this mean that the SAA just entails that there is a mega Multiverse filled with at least Beth two duplicates of only you? No, the SIA entails that there is a mega Multiverse that has at a…
Minimum, as a necessary condition, at least two people whose lives have been similar enough to yours to have all the same memories and experiences. But there's nothing wrong with other people existing as well. And given a theistic mega-Multiverse, God would do so much creating that not only would at least two chairs and other people get created, but also at least two of your epistemic doubles would end up getting created. Put another way, if a theistic mega-Multiverse existed, then no matter which possible person you were, you would find yourself existing. So the theistic mega-Multiverse maximizes the probability that you would find yourself, in particular, existing. So that's why we should believe in a theistic mega-Multiverse.
Here's another objection: What if not every possible person can be created? Maybe you think God can always create one more person. Now, there are views on which every person can be created; substance dualism or hexism, for example, would be useful here. But even if a mega-Multiverse with every possible person isn't doable, theism is still the only game in town for getting a mega-Multiverse at all. At least two people can exist even if not everyone exists. So the SAA still massively favors theism over naturalism.
Here's one last objection: How does a mega-Multiverse comport with Christianity? Well, it actually just has very little impact on Christian theology overall. Scripture doesn't say anything about a Multiverse one way or the other. I mean, it does hint that there are realms beyond our natural world, but we're told very little about them. In fact, the only Christian doctrines that I find even tangentially related to the mega-Multiverse would be the Incarnation and the Atonement. You see, I have great evidence that Jesus Christ existed in this universe, but that means my epistemic doubles have great evidence that Jesus Christ existed in their universes as well. So if the anthropic argument works, then it looks like we need to account for how Jesus shows up in lots of other universes. Maybe the second person of the Trinity took on more than one human nature. Maybe the second person of the Trinity only took on one human nature, but the human nature somehow extends across all these universes. I don't know; it seems like we can't do much here beyond merely speculating at possibilities. But at the end of the day, even if there are some mysteries here, none of this is incompatible with Christianity.
One passage that you may wonder about is Romans 6:10: "For the death he died, he died to sin once for all, but the life he lives, he lives to God." So Jesus died once; that means he can't die in multiple universes, right? Well, in context, Paul is talking about the finality of Christ's atoning work. He's saying that what Jesus did on the cross is permanent. I think it would be wrong-headed to try and tease out metaphysical implications from the statement like "there is no Multiverse," when that was almost certainly the furthest thing from Paul's mind when he was writing this. So again, Christian doctrine is simply not affected by a mega-Multiverse. I should caveat this by noting that I'm a Protestant and I'm not as familiar with the ins and outs of Catholicism, so maybe Catholic theology faces a problem with a mega-Multiverse that Protestantism doesn't. I wouldn't know. Personally, I find the idea that God created literally more wonders than there are numbers to invoke a deep sense of awe.
So that's the anthropic argument. Let's see, am I forgetting anything? I have this uneasy feeling that you are, in fact, forgetting something. Oh yeah, I wanted to give a shout-out to Matthew Adelstein. He introduced me to this argument, and he's done a lot of good work defending the SIA. Ah, that's nice. But I was talking about...oops, not again! Ah, universe-destroying aliens. This is a rather unfortunate development. How long do we have? Hold on, there's a Plan B. Hi, yes, I'm doing fine, thank you. Um, can you please send in the doubles army? Yes, thank you. Coolio. Anyways, that's the end of this video. Thanks for watching. Remember, Christ died for our sins and rose from the dead. Goodbye.