Boltzmann Brains Revisited

revisit this topic for a couple of reasons. One is that in the original version of the episode we published on March fifteen, I said something that was completely wrong, not about Boltzman brains themselves, but about gay ord Cantor and the nature of infinity as a mathematical concept. And when we became aware of this, we edited and republished the episode so not to leave a mistake and assumption

floating out there on the Internet. But ever since then, I wanted to come back to the topic because I know some of you out there probably downloaded the early version of the episode with an arrow left in. I wanted to acknowledge my mistake, make a clear correction, and then take that as an opportunity to talk a little

bit about the amazing nature of mathematical infinity on its own. Yeah, I mean it's this is one of those topics that combines both our attempt to understand consciousness and what it is, and also the the this mind boggling concept well actually a number of different concepts regarding the nature of infinity.

Uh So, you know, it stands to reason that that that there would be a few holes left to fill, that there would be room to revisit it in a future episode, And here we are absolutely So we're going to explore the thing that I mentioned that was wrong in the first episode, and we're gonna explore a lot of the actually mind blowing reality underlying that claim, and that that's going to have to do with can Her

and the idea of accountable infinities and unaccountable infinities. But we're also just going to have a chance to explore the idea of Boltzman Brains a little more, including a little bit of feedback from listeners on the idea and so forth. So if you never listen to the original Boltzman Brain episode from back in March, you might want

to go back and check that one out first. But I'd say it's not strictly necessary if you want to jump right in with us here if you did listen, but you're you're a little bit fuzzy on the concept because it's been a little while. Don't worry. We will give you a brief refresher on the Boltzman brain argument. Yeah, and if you're not even sure what the deal is with infinity, I'll try and roll through some of the basics there as well. But let's get back to these brains,

these marvelous floating space brains that could destroy us. All. Okay, So, the standard Boltzman brain argument, as we explored the last time, states that if you assume yourself to be a typical observer, which you should, right, why wouldn't you assume yourself to be typical? Yeah, well, there's a danger in assuming that you were a privileged observer. Right. In fact, by definition, if you assume yourself to be a typical the odds are that you're wrong. Right, that's the definition of what

it means to be a typical. If you assume yourself as you should, to be a typical observer, it is more likely that you are something like a disembodied hallucinating brain that randomly fluctuated into existence in empty space, floating there, hallucinating your life, your memories, your current sense experience, and all of that kind of stuff. It's more likely that you're one of those than that you're a normal mammalian

organism that evolved on a rocky planet. This random isolated floating brain in space is known as the Boltzman brain, and normal conscious organisms that exist through biochemical evolution, like we assume ourselves to be, are called in these arguments usually ordinary observers. So you've got these these two different types of beings. You could be right and techly, you wouldn't know the difference whether you were one or the other. Yeah, And if you really stop and think long and hard

about either possibility, they're they're they're both. They both feel kind of far fetched, you know, like they're they're they're kind of equally believable and equally fantastic, because it's just the the O O argument here that I'm just this mammal on this rock and I'm thinking about thinking is crazy.

That's the core of the show. I mean, we spend most of our time on stuff to blow your mind, exploring the weirdness of what it means to be a mammal that lives on a rock floating in space, even though that's what all our best science tells us we actually are. Yeah, but according to the Boltzman brain argument, you should believe that it's more likely that you're actually a brain floating in space hallucinating everything about your life then that you're one of these creatures living on a

rocky planet. And the way it works is pretty simple. You look at different models of the universe and you roughly calculate how many of each type of being you'd expect to find given the model of the universe that you're looking at. Now, if you're looking at our universe, you might still think, well, how could this be a place where there are more random space brains than animals

on rocky planets? After all, there are at the very least billions of ordinary observers alive today here on Earth, and there have been billions more in the past, and there hopefully will be billions more in the future. And that's just Earth. I mean, we could think about millions, billions of other planets out there that are full of

ordinary observers. To all those aliens, and so we have evidence of at the very very least billions of ordinary observers, and we don't have evidence of a single random brain floating in space assembled out of random particles ever existing in the history of the universe. So how could these random space brains out number normal evolved organisms That, on the face of it doesn't seem to make any sense, right, But this is where our old friend time comes in, and time, as we know kind of us, is up

our ideas of probability with all kinds of things. Like if somebody tells you, well, I was drawing poker hands and I just drew five royal flushes in a row, you wouldn't believe them because that's impossible, that that will never happen in the history of Earth. But if you also know that they had been attempting to draw hands for I don't know, tend to the ten to the ten to the ten to the ten years, then okay, sure, yeah,

it becomes a little more believable. Now. Right now, our best looking scientific model of the universe, the one that the most evidence today seems to be pointing toward, is

the lambda cold dark matter model. And this model includes everything we know about the past and present, so it's got the idea that our local universe is about thirteen point eight billion years old, that it began in an incredibly hot, dense state, and then for about the past thirteen point eight billion years, it's been expanding and cooling. And if we look at what this universe is doing and what it's made of right now, we can actually make pretty decent predictions about what it's going to do

in the future. And it appears that what the universe will do in the future is that it will continue to do what it's doing. It will expand and cool, and entropy will steadily increase, meaning that order is going to tend to disorder, and specialness and uniqueness will tend

to unspecialness and equilibrium. An energy that can be used to do work will turn into useless ambient heat that can't do anything, and stars are going to use up their fuel and burn out and they'll collapse into black holes, and black holes eventually themselves will dissipate due to hawking radiation, and everything will just run down and cool and even out, until the universe eventually becomes a vast, undifferentiated, cold grave of thermal energy, no stars, no planets, no animals, and

no ordinary observers going on into the future for more time than you can imagine, and it sounds like a grim outcome for things. But again, we're talking about very vast measurements of time. Yeah, I mean, you don't personally really need to worry about this. This is so many billions of years away that it's not going to be a problem for you. Yeah, it's more of a blow to one's sort of a loose worldview than it is

an actual existential threat to you specifically. Yes, but I would guess more worldviews are actually compatible with this model of things, then you would guess at first blush. Well, I mean, I guess it certainly lines up with Ragnarok, but sure. But also, I mean, people it can feel kind of depressing when people say, like, oh, you mean, like human civilization can't exist forever. At some point all the usable energy would run out and we'd have to

go extinct somewhere out there in the dark. Well, yeah, technically that does appear to be physically true, but on the time scales we're talking about, human civilization wouldn't be human civilization anymore. It would be some kind of thing evolved so far beyond what human civilization is now that you wouldn't recognize it at all. We would be will be so pretentious by that point that we're able to to glimpse these future post humans would say, oh, screw

those guys often too, the twilight uh entropy with them. Well, I mean, if existence is endless variation on change, eventually one of the changes worth exploring might be the change of not existing. Yeah yeah, or or kind of to to invoke Ian and banks of sublime ng and taking this kind of alternate mode of existence that that that comes after you've you've you've had your full shot at the imperial conquest game. I like that sublime NG. That's nice.

It's like a chemical process almost. Yeah. That in in his um uh constructed universe of the of the of the culture. That's what you see these super advanced elder civilizations doing. Uh. They they don't destroy themselves. They reach a point where they just sublime and they they just

kind of leave everything that they've built behind. Well, as more and more of our existence tends to be uh trending toward being less physical and being more encoded as digital information, one wonder is if the ultimate transcendence is just to sort of like become a a radiation imprint on the background of the universe. Yeah, I mean that's kind of a beautiful afterlife in its own right. But anyway,

back to Boltzmon brains. Um, So what what? What? What happens when the universe just goes on and on and on like that for periods of time you can't even begin to comprehend. Well, here's where the weird statistical argument comes in, that you're more likely to be a space brain than one of the kinds of creatures we're pretty sure we actually are. During all that vast time in the future, random fluctuations in this empty universe will occasionally

depart from this dead equilibrium before returning to it again. Now, how will that happen? Well, Sean Carroll, the physicist who has written a good bit about Boltzmon brains, who we quoted in the last episode, will continue to quote today. He writes about this in his paper why Boltzmann Brains

are Bad. Just to give one example of what's going on here, so he says, quote, there can be collisions between rare high energy photons or gravitons, which could pair produce electrons and positrons or protons and anti protons and so forth, and the general tendency of such pairs would be to re annihilate rather quickly. They would annihilate each other, he writes. But occasionally the new particles will have enough

momentum to travel far apart from each other. Sometimes rarely, as should henceforth be understood, many such collisions will happen nearby, producing enough nearby matter to assemble itself into a macroscopic object such as a brain. Now again that sounds that's so improbable. So you're just saying random particles that are getting pair produced out there in space by random events are going to collect into enough atomic matter that they would form an object like a brain, like a specified

object like that. Again, that is super improbable. If you're thinking that would never happen, you're right, Except you're never is not including enough time, and including enough time, it actually would happen. As I think we mentioned in the last episode, one is instantly reminded, of course, of the monkeys pounding on the typewriters to produce the complete works of William Shakespeare. Right, and in given normal time, that

will never happen. But given enough time, if you just stretch the t variable out to arbitrary lengths that actually will happen. It's almost guaranteed to happen. But in fact, Carol writes that it it doesn't really matter exactly what the process of producing brains is as long as you've

got two assumptions. And these two assumptions that give us the Boltzman Brand universe are that the universe either last forever or last for an extraordinary long time, much longer than ten to the ten to the sixty six power years, and that is a long long time. And then the second criterion is that it undergoes random fluctionations that could

potentially create conscious observers. Now, as long as those conditions are met, randomly assembled brains or conscious computers or whatever other type of conscious agent you want to imagine, will eventually become more numerous in the history of the universe than normal conscious animals like us. And so Carol points out in his paper that the lambda cold dark matter model, the best current scientific model of the universe, looks like

it fulfills exactly those criteria. It's laws of physics are going to allow random fluctuations that can randomly assemble particles into macroscopic objects. Like dust, like rocks, stars, planets, but also lobsters, blue rays of Jean Claude Van Damn, movies, uh, Christoph Lambert, even human brains. And it appears that the universe will go on existing for so long that there will be an unbelievably huge number of opportunities for random

fluctuation objects like this to come and do existence. And so this gives us the argument there's going to be more time out there for random brains to as symbol out of particles in the void than there is going to be opportunities for ordinary observers to exist in the time space of the universe where there's usable energy to

power things like stars and planets and evolution. Our evolution window is actually extremely small compared to the whole life of the universe, a lifespan that is so vast that from our finite perspective it might as well be infinite. And for all we know, we can't prove that it's not infinite. But of course, the question of what it would mean for time to be infinite is I guess kind of Uh, that's a vexing question itself, right, what

would that mean in reality? All? Right, Well, let's take a quick break, and when we come back we will discuss the identity of infinity. Than alright, we're back, all right, So today in the podcast, we're gonna be discussing the concept of infinity and infinite sets, since this is where I made an erroneous statement in the first time we covered Boltzmann brains. The last time, we were in the middle of trying to illustrate the idea that it can

be proved that some infinities are larger than others. Because if you've got a universe going on and on in an equilibrium, how can you compare how numerous two different sets of objects within them, like Boltzmann brains and ordinary observers are. If it just keeps going on, you'd want to have some way of saying, well, even though it keeps going on, one group within the universe is bigger than the other group within the universe. And so the

idea here is that you look at relative frequencies. Right, So here's what I said last time. I said, some infinities are larger than others. And the example I gave was comparing two sets of integers. Those sets were even numbers everything divisible by two and numbers divisible by seven. So even though you could keep naming items in each of these sets forever, I claimed that there were more

even numbers the numbers divisible by seven. And that seems quite intuitive, right, I mean, it made sense to me when I said it, and I thought I had actually read that in the past. But that turns out to be not true. Now on a sort of face value level, it's kind of hard to see how that's not true. Right, Even numbers happen more often if you just keep counting up the number line, you will hit even numbers more

often than you hit numbers divisible by seven. Yeah. I think I kind of played off of this by saying with that Connor McCloud from Highlander, if he lived forever, he's going to cut off heads and he's going to use the bathroom, and he is going to use the bathroom more than he cuts off heads, right, right, And so that was that I had said, Yeah, when I when I when I take your example and I line it up with my Highlander heavy example, it seems to

be true in both cases. We were right until you invoke the concept of infinity, and if you actually say this process extends infinitely, then everything gets ruined. Everything gets jammed up because, believe it or not, if the process mathematically were to go on for infinite time, and each set of events happens infinitely, then it's provable by something called set theory that you can show that those sets are of equal size. And that might be hard to understand,

but we will explain that. It comes down to the work of a German mathematician named gay Org Canter that there are ways in which some infinite sets that seem like they should be of different sizes are actually the same size. And yet at the same time there actually are different sizes of infinite sets those so some infinities really are bigger than others, just not in the example

that I gave of of integers. And I guess it's more arguable when you're talking about events like what Connor McCloud does, because then you're talking about things happening in the physical world, where the concept of infinity gets even more complicated. What would it mean for events to be recurring infinitely? Well, I think the problem is that by by dragging infinity into it, we're basically dragging God into the scenario, right, and you know how God reacts to

mortals trying to to boss uh it around. Well, you know, infinity has historically been a concept of that's very, very wrapped up in ideas about theology and religion and the universe and the afterlife. Uh I just mentioned Georg Cantor. I mean, Georg Cantor had a lot of strange religious beliefs that kind of read to us as eccentric today, deeply wrapped up in his mathematical discoveries about the nature of infinity. Yeah. Yeah, the more you look at it,

the more intertwined they are. Well, I guess we should come back to Cantor in a bit and talk specifically about the types of infinity that he dealt with, but more generally, maybe let's look at the concept of infinity and in cultures throughout history. All right, Yeah, so this is it's really fascinating when you think about it, because we obviously as humans live limited lives, finite lives of

finite observation within a finite world. Yet we advanced to think beyond these limitations and in doing so envision to some degree infinity, or at least two toy with the concept of something being infinite. In fact, through the use of language. In particular, we used a to quote our old friend Julian James, finite set of terms that by metaphor, is able to stretch out over an infinite set of circumstances. Oh,

I mean that's essentially the library of Babel, right. I mean that you have a finite set of encoding features. But those bits of code can make every statement in the world. Yeah, and we alone, of all the animals, are able to stand in wonder and horror of infinity. You look over your cat, your dog, you check out the smartest dolphin in the sea. They don't. They don't

have any concept of infinity, and maybe they're happier for it. Well, I mean you might wonder if they have some kind of intuitive concept, not of in infinity, but of the idea of recurrence. There are different ways of picturing something is infinite. One of the things that I often think about is that if you go back into ancient times, you'll see these different ways of imagining, say time, even if you had the idea that there's not necessarily a

beginning and end of time. Of course, many religions have that. They are totally different ways of emphasizing how that lack of beginning and end works. Is there a boundless infinity where time stretches all the way back forever and goes all the way into the future forever, or is there a bounded infinity where time endlessly repeats like along a

loop that never stops going. You know, one of our probably the you know, the earliest ideas that gets tied up and eventually becomes adfinity is that, of course, of the ocean. And we still invoke this, uh, the old saying. You know, somebody breaks up with their high school breaks up with his girlfriend. He's he's he or she they're they're distraught about it. What's the nuggeta whizd them that is leveled at them? They're always more fish in the sea.

But there's a finite number of fish in the sea, right, But there are so many fish that it's almost a kind of infinity, like there's a there's a boundlessness to it um And likewise, just the idea of the ocean itself is certainly a finite realm, but it is so vast, it is so hard, especially for for ancient people to comprehend that it was considered almost a kind of boundless realm. It was and it was tied up in these notions

of primordial chaos. Well, it's one of those pseudo effective concepts that's useful to us in mathematics in the same way that our lives are full of useful pseudo random numbers. Like people cannot actually generate truly random sequences of numbers. We talked about this in the Eaching episode, But you can generate what feel like pseudo random numbers. You know, it's like at least seems kind of random, even if

it's not truly mathematically random. And we use that kind of inference to you know, guess things and come up with a random solutions to things all the time in our life, in the same way we use pseudo infinities as a metaphor for understanding all kinds of things. Uh. You know, the stars in the sky or the fish in the sea were constantly picturing infinities that are not truly infinite, their finite numbers of things, but they're so big they stand in for infinity, which we can't conceive.

But we've tried to conceive infinity. Uh, and we've been doing so for hundreds and hundreds of years from millennia. Uh. So I wanted to run through some sort of I guess you could say great moments in UH infinite naval gazing right here, just some of the UH, just a few of the important individuals and movements. One of the the earliest that's worth talking about, UH, it has to

do with the religion of Jainism. Okay, So after the decline of the Vedic religion on in in on the Indian subcontinent around a four hundred b c. To other religions rose to prominence in India. You had you have Buddhism, which we've talked about on the show before, as well as Jainism. And it's a jain is um is really its own distinct religion. It does involve some uh properties and concepts that pop up in Hinduism or Buddhism, but uh, you shouldn't think of it as just a meteor offshoot

of Hinduism or Buddhism. Its name derives from the Sanskrit verb to conquer. That's interesting because I tend to think of Jainism as embodying non violence to an incredible extent. Well, yes, but the thing you're conquering. Think of it in terms of conquering one's passion. It's the internal struggle. So James don't hold up a traditional founder. Uh. They have a number of key teachers that are called turth and Carras

or ford Makers. Uh. So, for instance, UH, there was an important one named part of an author who may have lived in the seventh century BC, and he would have been the twenty three of these ford makers. But then the twenty four and last, Uh, turthen Cara was a man by the name of a vard Anna and known by the title Maja Vira or great Hero, and he would have been a contemporary of the historic Buddha. It was during his lifetime and the years immediately to

follow some of the greatest contributions to Jane mathematics were made. Uh. And Jane philosophy is big into pondering the enumeration of large numbers. They classified numbers into three categories, innumerable, innumerable, and infinite. In fact, Jainis m recognizes five different forms of infinity. There's infinite in one and in two directions, there's there's infinite in area, there's infinite everywhere, and there's infinite perpetually. Infinity is also tied up in their metaphysics

of the metaphysics metaphysics of the Jains soul. There are an infinite number of souls in the universe and liberated souls or siddhas, which are free from the cycle of samsara uh. They have in finite knowledge, infinite vision, infinite power, and infinite bliss. Now I wonder what it would mean to have infinite knowledge. This is something that's always been kind of interesting to me about the idea of omniscience.

It seems to me that the idea of omniscience inherently invokes or involves things that appear to be contradictions in the same way that omnipotence does. Right, So people classically say, oh, if you've got an omnipotent soul or an omnipotent being, could it create a rock that it itself could not lift? You know that that's the classic one. The same thing would come through with omniscience, meaning could an omniscient being with all knowledge know what it is like to not

know things well? Or would you know things if to be complete knowledge? Is that maybe just a complete absence of knowledge in a way, like you if you try and imagine a person that is defined, but you know what we have finite knowledge, finite vision, finite power, it's certainly finite bliss. If you ramp all those things up to the infinite, then do we just bleed into the fabric of reality? We vanished completely? We I mean maybe

that that is that's the ultimate liberation right there. Well, all of these concepts that are being taken to the transcendent level of infinity here, like vision, power, bliss, knowledge, they're all in a way kind of defined by the bounds of limitation. What what would it mean to see everything? Then? Are you even really seeing? I mean seeing is an act of like focusing and an act of perceiving, And so if it's everything, I don't know what what it

would necessarily mean. What is bliss of? Is it if it is experienced outside of the contrast with with suffering, right, or even something that's slightly less than bliss. Now I want to be clear, I'm not trying to like rag On Jane metaphysics or whatever. I mean. These are kind of beliefs that appear in all kinds of religions, but I think they highlight some of the inherent qualities of contem lating the infinite that makes it seem wholly and

kind of mind boggling to begin with. It's exactly these types of contradictions that make it so attractive as an idea. Yeah, certainly. And of course the Greek scrappled with infinity as well. In fact, if we look back to the to the the ideas of an Aximander of Militiasts who lived six ten through five b c. Uh, he's considered the the

earliest Greek philosopher to commit his ideas to writing. The only fragments of his work remain, and he introduced the principle of the apron or boundless, and this is uh what he would discussed as the original state of the universe. And he's certainly describing infinity that you can also see clear the clear influence of the Greek cosmological concept of primordial chaos here as well. Oh yeah, and this would be this would be an important concept that that subsequent

Greek philosophers would toy with as well. One of the big names is of or says Zeno of Elia, who lived four ninety through b c. And he played with this concept of the apron uh, and he found that infinities breed paradoxes such as the fabulous paradox of Achilles

and the tortoise. Uh and uh, this this is a this is a fun little thought experiment that I think will that lines up rather nicely with the Boltzmann brains thought experiment, because the idea here is that you have a tortoise and then you have the the the god blessed um hero Achilles. And the tortoise says, hey, I would like to challenge you to a foot race, and Achilles is like, what are you doing. I'm gonna I'm

gonna of course, I'm gonna beat you. I am Achilles. Uh, you know, nothing stands in my way, and I'm certainly going to outrun a tortoise. And the tortoise says, well, uh, I would need a head start, of course, because I am a tortoise and you are Achilles. But I'll still beat you if you just give me a reasonable head start, and they agree on a head start something to the

effect of like ten feet or so. There's nothing crazy, and Achilles is like, well, I'm still gonna be you, don't I don't understand, and the tortoise explains, well, think of it this way, You're gonna have to catch up with me. Before you pass me, and Achilles says, sure, yeah, I'm gonna have to I'm gonna have to run to where you start. And then the tortoise says, but by the time you reach my starting point, I'll be ahead of you, and uh, and then you're gonna have to

catch up with where I've gotten to. And and he keeps, you know, carrying this out one step further, one step further, until he convince his Achilles that he can never catch up with the tortoise, and a Kelly's just concedes the whole race, and the tortoise wins. I remember reading about this paradox when I was younger, and I love stuff like this. It always like, I mean, it seemed right.

It's one of those things that seems very true in the same way that it certainly seems like there must be more even numbers than there are than there are numbers divisible by seven, right, Yeah, yeah, I mean, like another version of this would be imagine how long it would take you to walk across a basketball court. Yeah, well, first you have to walk halfway across it. Before you reach that you gotta walk halfway across that distance, and

then halfway across that distance. So you can take something finite and if you divide it up into infinite portions, then something that is very possible seems impossible. Uh. One of the take homes is that, according to the argument

made by the tortoise, movement is impossible. Yeah, exactly. Now to move the analogy over to the Boltzmon brain argument, I would say, actually, if you wanna say, like the tortoise represents Boltzman brains and Achilles represents ordinary observers, all you don't really have to do is say, okay, tortoise does not get ahead, start, they start at the same time. Achilles runs, say a thousand times faster than the tortoise, and they both start the race and they just go.

Except the difference is the tortoise is immortal, all right, So Achilles is gonna win the early part of the race by a long shot. Like tortoise isn't gonna come close until Achilles gets really exhausted and dies. I know, if if only, if only he was full God, he might have a shot. But then you've got eternity for the tortoise to just keep walking out ahead. Yeah, slow and steady right right, wins the race. Uh. The Great thinkers, of course, tackled infinity as well. Aristotle three twenty two

BC he considered the the apperon as well. Um. Yeah, here he is in physics talking about the boundless quote. Everything has an origin or is an origin. The boundless has no origin, for then it would have a limit. Moreover, it is both unborn and immortal, being a kind of origin. For that which has become has also necessarily an end, and there is a termination to every process of destruction. Another thinker of note, h Thomas Aquinas, came along twelve

seventy four. He focused on the quality of existence rather than the quantity. Of course he didn't uh. So he considered, you know, God is infinite in quality more so than in quantity. So he saw infinity as a mode of existence and identified a separation between mathematical infinity and religious infinity. So we see a curious principle emerge. Even an infinite

God cannot create an infinite object. He talks about this at length in his work Assuma Theologica, uh, sort of taking both sides, like saying, all right, well if it if God is infinite, then X. If God is fine eye, then why But I think he says that nothing except God can be infinite, right right, Well, that kind of point of view is going to hold some sway for a while. Um. Another individual worth noting Nicholas of Cusa fourteen o one through fourteen sixty four. He argued that

everything is within the infinite. The world itself must be within God. So he used mathematical examples to describe the relationship ship between God and the world. There's no circumference to God, and the center is everywhere. This is the concept of a maximum God, unlimited, transcendent, and also unreachable and unfathomable to a species that is defined by its finite limitations. So this say, is starting to sound kind of like the like God is the universe type belief.

Then you have Spinosa comes around two through seven and he says, if God is infinite, then God is the one substance. The substance must have infinite attributes, and we must be modes of this one entity. So God is not the not a personal God. God and Nature are one and the highest ethic, according to Spinoza, is to live in accordance with the laws of nature, to be part of the infinite. So the ideas that were beings of mind and body, both of which are composed of

this universal substance. That's an interesting idea that if if one thinks of God as infinite, then how could there be things that weren't God? Right? Because wouldn't God necessarily encompass those things? If God had no limit and went on forever, what what could be outside of him that would seem to suggest that there was a limit to him? Yeah? Yeah, what could what could be outside of the absolute, the

maximum God? Yet again, I feel like infinity is one of those things where you start playing with language in a way, it's kind of a game that you you start using certain words thinking you know what they mean. But when you use words like infinity or everything or forever thinking you know what they mean, they end up kicking up conclusions that you couldn't expect because you you

contemplate them more and more deeply all the time. Uh, and they tend to transcend the ways you originally invoked them. Does that make any sense? Yeah? Well, I mean I think that the big one is. Of course, it's one thing to talk about infinity in terms of just pure philosophy, but then when you start lining it up with mathematics, right, and when you start bringing the raw numbers in and crunching those numbers. Uh, that's where you get into some

of these these real conundrums. That's where you you you hear these arguments of someone saying, well, you're talking about infinity, but you're not talking about mathematical and infinity. Because here's what happens when you throw the numbers in, right, And this brings us back to Gayard Cantor. So Gayard Canter was a I think I mentioned this earlier, but he

was a German mathematician. He was born in Russia in eighteen forty five and he lived until nineteen eighteen, and he is the main person responsible for modern set theory and the theory of what are now called transfinite numbers, which was revolutionary and very controversial in its time, but in many ways is widely accepted now and has proven extremely useful. So in writing about infinity, Cantor made use

of the idea of sets. Sets are both simple in the core principle and extra ordinarily complicated and powerful as a tool for mathematical reasoning. And so set theory is basically just it's a framework for grouping items into sets. So you've got some items, you could group them together, and then you've got a set of items, and you

can treat that set as a thing. And one of the things set theory does is that it gives you a way of comparing the size of sets of things by matching the items in those sets in a one to one pairing off process. And the size of a set, meaning how many items it contains, is known as its cardinality. No, so I know that's a lot of terminology. We want

to try to avoid getting too abstract here. But basically, if you think, okay, I've got a set with uh five objects in it, my five fingers on my right hand, and then I've got another set that has all my fingers on my left hand, are those sets equal in size? Well? I can check. I mean, obviously I can tell because I can count to five. But even if I couldn't count to five, I can check by pairing off the items in each set and seeing if they pair up in the same extent, or do I have leftover items

in one set that can't go with the other. Now I have five fingers on each hand, so yes, I can pair up the sets. They match up all right, So you would say that the cardinality or the size of both of these two sets of fingers on each hand is five, and the sets are equal in size. But now let's go to the example that I that I got wrong in the last boltzmon Brain podcast. So that would be looking at the sets of all even numbers in one set and all numbers divisible by seven

and the other set. Even though if we just count up the natural number line, we hit way more even numbers the number is divisible by seven. Cantor could show that the cardinality of these two sets is exactly the same because each item in each set can be paired one to one with an item from another set. So imagine you've got set A that's even numbers, and then you've got set B that's numbers divisible by seven. Well,

let's pay off the first items in each set. You've got Set A is to set B is seven, and then the second item in each set that's Set A is four, set B is fourteen. Here's the question, what would stop you from counting forever this way? Would you ever hit a number in set A that did not have a corresponding number in set B. Clearly you wouldn't you can match them off until the end of time. Uh though, actually, if the end of time comes and stops you from counting more, then suddenly the even numbers

set is much bigger. But because you know there's more frequency, you hit even numbers more often. But if you grant that they're infinite and you don't have to stop and tallly them up, but you treat them as ongoing sets, then they are in fact demonstrably the same size. Okay, I'm with you. I'm surprised you are, because that's it's messing with my brain. I mean, that seems wrong, right. There can't be the set of even numbers and the set of numbers visible by seven cannot be the same size.

There's clearly more of one than the other. But cantor can prove that they're the same size. It's because we've invoked we've invoked infinity, and that changes things. It messes everything up. Suddenly all your intuitions go out the window and nothing makes sense anymore. Our listener Jim in New Jersey, who is a great, great email writer. He's always been hearing from from Jim for years. Uh, and Jim especially on like mathematical logical computer science type type topics. Jim

has really great emails. He sent us an excellent email, uh gently correcting are my mistake in the first episode, and he had a really good analogy. He said, quote, think of this as a marathon race without a finish line. The hair will always be ahead of the tortoise, but they will always pass the same mile markers, just at different times. They both cover the same distance. It's just

that one racer reaches each milestone quicker. They all reach the same milestones since there are an infinite number of them an infinite time. I think that's a nice way of picturing it. Slus. We got to work the tortoise in for like a third time here. I like it. How many tortoises have we done so far? Three? I guess because we talked about the tortoise and Achilles, and then you talked about the tortoise as boltzman brain, and

now we have proper tortoise and hair example. So now it's tortoise as numbers divisible by seven, whereas the hair is even numbers. Yeah. So even though the hair goes faster, they eventually go the same distance if they go forever. But this is so weird. Right, because with this type of reasoning, I don't know, you might begin to sense some deeply troubling implications. One thing is, any infinite set of things that can be counted in order is equal

in size to any other. Because if you can count them in order and it's clear what the order is, you can pare them off like this. And if you can pare them off like this, you can pare them off forever equally. I mean, I just keep coming back to the idea. If you're willing to accept the albeit loose idea of something going on forever that is just going to go on into infinity, then you can you can buy the fact that these two sets are equal.

I guess so, I mean it's cutting me, man, It's well be because it comes back again to the idea that we are finite beings in a finite world. That is all we've ever evolved to be, and yet we imagine things that are beyond that, and so of course it breaks our our our our normal ability to to comprehend things. Yeah, let's keep imagining. So the implications of

set theory and Cantor's work are obviously extremely profound. For example, here's one thing that feels pretty obvious which set is bigger All the natural numbers, and that means all the positive integers starting with zero, so all the counting numbers, you know, zero, what's bigger that set? Or all the rational numbers, which is another way of saying anything that can be expressed as a fraction with a quotation of U with positive eneger, So like one half or two

over one or one third. Which of those sets would you think would be bigger? Well, I mean obviously it would seem to invoke the Akelles and Tortoise situation, right, you would think, well, if you just start dividing everything we've been talking about up, you can just you can you can just have infinite divisions that just just you know,

blows that number out of proportion. Yeah, you think, obviously the set of rational numbers is bigger, right, that includes all the fractions, because the set of fractions includes every natural number. Every everything in the first set is also in the second set. One over one, two over one, three over one, So all the natural numbers are in the set of rational numbers, but it also includes every fraction in between every natural number. So you've got one half,

one third, three, seventh, and nine tenths. Those are all rational numbers. So there must be more rational numbers or fractions than there are natural numbers. Right. Wrong, Yet again, Cantor used set theory to prove that these sets are of equal size. Now you might wonder again, how could you prove that? Like, remember from a minute ago that the key to being able to show that the items in accountable infinite set are equal is that you can

arrange them in accountable order. You've got to be able to order it so that you can pare them off in an orderly way, making sure you're not missing anything. Right, So how could you count fractions? Right? You can't start with the lowest fraction and count up from there. That doesn't make any sense. You know. You can't say I'll start with one tenth and then go to two tenths because you have lower numbers than that. So what can you do? Well, here was a real stroke of genius.

Cantor rearranged every possible fraction into an ordered table. Robert f a picture of this table in in our notes for us here, So what are we looking at? Well?

I'll have to include a link to this image on the landing page for this episode stuffable you mind dot com, But for me, the first thing I think of is that if you ever did those charts in like P class that show the volleyball rotation plan when you're playing like team volleyball, that's kind of It's like, imagine, uh, a bunch of fractions got together to play some sort of strange team sport and they needed a guide to

show how they're moving around. Well, it is like that in terms of how you navigate, but it's incredibly orderly. So here's how the table works. Is very simple. You've got a top line in a sideline. The top line is numbers at the bottom of the fraction, right, so, and that goes up with every digit one to and there's a column that goes down with fractions that have

that number on the bottom. And then you've got a sideline that goes down with rows that are all the integers going on for a and those integers have rose beside them, where the top number in the fraction is that number. So you can actually move through this list in a diagonal back and forth pattern that allows you to make sure you're counting every possible fraction there could be.

So it starts with one over one, then two over one, then one over two, then one over three, then two two three, one four one three two, and so forth. And you actually, through this method, could make a table where it was possible to count every fraction that you never actually get to the end, but you can organize them in a way that you know you're not leaving anything out. So it's basically volleyball, because in team volleyball everyone moves so that everyone has to serve during the

course of the game. Um, you know, unless the gamings are like that, let's ignore that. Uh, everything must serve, everything must be counted. And that is exactly what kills its supremacy in terms of competing with the eyes of the set of natural numbers, because remember from before, if you can count them all in an order, then you

can pare them off with natural numbers like one five. So, believe it or not, it can be mathematically shown that the size of the set of natural numbers and the size of the set of rational numbers is the same, even though that makes absolutely no sense to us. Cantor was actually writing about this to a mathematician friend of his named Richard did kind uh, And he said about some of his own discoveries that you know this this

type of stuff. He was coming across j evoir me jana quapa, which means I see it, but I do not believe. Al Right, hold that thought, Joe. We're gonna take a quick break and when we come back back to infinity. And we're back to infinity. Now you can use the same logic to violate your intuitions lots of other ways. There are plenty ways to stab your brain with this. So here's an obvious one, which is bigger

infinity or infinity plus one. Well, on the surface, it would sound like infinity plus one because it's all that plus one extra dude, exactly. I mean you remember this from playing games as a kid, right, yeah, Well, well I am you know, I am infinity strong. Well, I'm infinity plus one. Well, I mean it's it gets into not to summon the specter of the infinity hotel. But that's kind of the concept there. You have a hotel with infinite rooms and then infinite guests are staying in it.

What do you do when one extra guest shows up? Is there room in the hotel? Of course there is. You just have everybody move one room over and it opens up one room for the new guests. And apparently that's how infinite sets work, because as crazy as it is, an infinite set plus one is still the same size as a regular infinite set. So you've got set A that's an infinite number of objects. Set B contains an infinite number of objects plus one extra object at the beginning.

Can you still pair them off forever? Yep? Will you ever run out of items to pair off from each group? Nope? They are somehow still equal. And this is because, I mean, one of the things that's hard for us to remember is we often try to treat infinity as a number, and infinity is not a number. And infinity is a concept. You know, it's a mathematical tool, but it's not a number. Like if you add plus one to any number, it

is more than that number. But plus one to infinity is not more than infinity because infinity wasn't a single finite number to begin with, right, and and and explaining this to my son, I don't tell him infinity is the largest number. I tell him numbers go on forever. That is infinity. That's good parenting. Well, I want him to have a proper you know, understanding of the boundless. But he's not, is he He's not? Kinna? Uh well,

a proper understanding of the boundless. That's the kind of thing you spend your whole life trying to to wrap your head around. Yeah, well you ought to be careful. You might have a child that grows up to be a philosopher, mathematician. We'll see, we'll say, playing with fire there. All right, So where are we with with cantor though?

What what is cant are saying at this point? Well, I mean, so we're at a weird place already, just with what we've talked about before, because especially when you take this and try to extrapolate it back to the more weird less mathematical, spiritual, theological, philosophical types of ideas of infinity, Like what does it mean to all these people who want to invoke infinity in their religion for us to discover that for some types of infinite quantities

in set theory, a part of the whole is actually exactly the same size as the whole itself. Yeah, it doesn't leave much room for God. I mean, I guess it leads infinite room for God. That's the That's the confounding thing about all there you go. Uh So the other thing, though, is that Canter also showed it's absolutely true that some infinite sets are bigger than other infinite sets.

Now we've just been messing everything up by violating our intuitions to show that sets that seem like they should be of different sizes if they're infinite, are actually the same size. How could it be possible that some infinite sets are bigger than other Cantor guides us again. Let's think about real numbers versus rationals. So last time we looked at rationals. That's fractions, right, anything that can be expressed as a normal fraction. Uh, then you've got real numbers.

So real numbers include all of those numbers, but also include irrational and transcendental numbers, which are numbers that cannot be expressed as fractions, things like pie in the square root of two. You know you've seen people trying to calculate these out, like you can write out many many digits of pie three point one, four or so on forever, but you'll never get to the last decimal place of pie because it does not have a last decimal place.

And if it did have a last decimal place, you would actually somehow be able to express it as a fraction. But despite the fact that it never terminates, pie is a real number, you can only write it with decimals. And there are presumably lots of numbers like this, But how many are there you might be guessing, give and what We've just been learning that there would be accountable

infinity of these types of numbers. So maybe there's the same number of numbers that are rational versus numbers that are real, but no. Cantor showed that real numbers, including irrationals like maybe the square woradi of two, cannot be arranged into accountable list including all of the infinite possibilities. If you tried to make such a list. Cantor showed that you can always point out real numbers that don't

appear on the list. So how would you show that? Well, for a simplified version, and this is a kind of dumb down version, Cantor was trying to do this with like binary numbers, but uh, for for a simplified version, imagine trying to make a list of all real numbers lining up their decimal values. So you've got like one point three, four, five, six, seven, eight, nine, ten, and

then one point seven to five. What you know like that where you line up their digits in columns, and Cantor said, whatever is in that list, I can find a number that it's not already in it. And you can try to make a list like that that goes on forever. But no matter how you do it, I'll find numbers that are real numbers that aren't on the list. And the way he did this is is this genius

thing where he went down the list diagonally. Now we already had a kind of diagonal squirmy line thing through the rational numbers, but this is a straight diagonal line through all these numbers that are listed. So what you do is you take the first decimal digit of the first number, and then the second decimal digit of the second number, and then the third decimal digit of the third number, and then take each of those digits and change them into something else, and then put them in

sequence to create another decimal number. Now, by definition, this number is not on your list. It can't be, because you guaranteed at least one decimal place is off from every possible number on the list. So no matter what, an attempt to count real numbers fails, you can't possibly create an ordered way of listing them all. All this means they're not countable infinities. There's an uncountable infinity of them. And thus you can prove that some infinities actually are

bigger than up than others. There are more real numbers than there are rational numbers or natural numbers. Now, let's try to bring this back to a concrete example, because I know we've we've been in the math for a while. I apologize for that, but I did want to try to set that straight here. So do you want to talk about Connor McCloud. Sure, yeah, How does Connor McLoud fit into office? Then let me get because at this

point I'm no longer sure anymore. Certainly, I would think during the course of a normal immortals life, ah, he is pooping more than he is beheading. But how do we compare those two infinities now if he's living forever, well, that is a really difficult question. I am also in a in a strange place with you here now, because it's clear that the hair in this race, the one that accumulates faster, is that the highlander goes to the

bathroom more often than he beheads. People. And if you just keep watching this process forever, any time you stop to check how many times each of these things has happened, the bathroom visits will be larger, and it will just keep getting relatively larger. It'll go on like that unless

you say it goes on forever. Now, it's hard to say exactly what it would mean to say a sequence like that goes on forever, because you're talking about organisms that live in a universe that you know, even if you call him immortal, you'd think at some point he'd run out of usable energy in the universe. It's Highlander. With all things Highlander, it's best just not to ask

too many questions about it. But we're in this weird conundrum, right because we've discovered that if it were actually possible for this to go on forever, whatever that means for events in real time, if it were to go on forever, then he would do those these different activities equally. But I don't know that it makes any sense to say events in the physical space go on forever in the same way that say a list of integers in an infinite set can go on forever. Yeah, I mean it,

it comes back once again. To the idea that our lives, our experience, our world is finite numbers. This thing that we have either created to match up with the with the the universe, or that we have discovered in the fabric of the universe. These go on forever, These have true infinity. Yeah, I think that's a really great point. But then again, what if the laws of physics don't

tell you that you can ever stop counting? You know what if you look at the laws of physics and you look at the universe and you say, I don't find anything that says time stops. So what do you do? Then you keep counting? I guess you keep pooping, you

keep beheading, and you just and things balance out. Right, Then that seems like you've got to wonder, like, does that actually undercut the Boltzman brain argument then, because if you actually have a universe going on forever, obviously you have early universes where there is low entropy, and you've got all these stars and planets and we're we're all we're making aliens, you know, making all kinds of crazy organisms evolving through biological evolution, biochemistry, and you've got all

these ordinary observers, and then you've got this long dead period where those things die out and you're generating boltzman brains out in space. But if it really all goes on forever, then it kind of doesn't matter how many of those things happen at either point, right, because it

would just keep recurring. You'd eventually get an entropy fluctuation that would take you back to something like a big bang, and you just like started all over again, and then this would go on forever and you cannot compare them. That's true. So I actually wrote to Sean Carroll, did you ask him about Highlander infinity to poop? And it didn't mention a highlander. I don't know if he would have replied to me if I mentioned highlander, but he he was incredibly generous with his time to to respond

to me. We really appreciate that. But yeah, he he basically acknowledged this. So Dr Carroll responded to exactly this kind of question and said that it's true that two accountable infinities are equal to each other. He said, however, it's also true that one is five times bigger than the other, or tend to a hundred times bigger than

the other. That's how infinity works. And I think he's recognizing the problem there that like, you'll have these things where frequencies are obviously very different, but if you truly extend them to infinite sets of things, then they're accountably the same. So he says, quote, so comparing infinities is clearly not what you want to do. If what you care about is the relative frequencies of two kinds of events,

you have two options. One throw up your hands and say there is no way to compare or to regularize, i e. Consider only a five night region of space time, so that all numbers are finite. Calculate the ratio of Boltzman brains to ordinary observers, and then take the limit as that region gets infinitely big. If you do the ladder, you will generally find Boltzman brains vastly outnumber ordinary observers.

And and this is what they do in their papers. Now, to be clear, Carol is not saying he thinks we are Boltzman brains. He has physical arguments and to some degree, you might say, philosophical arguments for why we are not Boltzman brains, and we discuss those in the previous episode, and one of them was that he says, you know, assuming you are a Boltzman brain is cognitively unstable. It doesn't make any sense because what's your reason for thinking

you're a Boltzman brain. It's like all this science and math and stuff that we only know because we think we have an accurate picture of the universe. If you were a Boltzman brain, you'd have no reason to think you had an accurate picture of the universe, and thus the method that you use for arriving at the conclusion that you're Boltzman brain would be totally worthless. So you're saying, there's a chance, do you everything about how subversive the

slogan to infinity and beyond is? Yes? Unfortunately, um, it kept popping up during preparation for this. I'm like reading about these like that the Jane concepts of infinity, and it's just that that that stupid catchphrase from toy story sounding off in the back of my head, and and it gives me nothing. It provides no insight into what I'm and what into what I'm trying to wrap my head around. But it keeps going off like a like a malfunctioning and sprinkler or something that Yeah, I I

can understand that. So I'm not one of these people who had like a deep emotional experience as an adult watching a Toy Story movie, because I don't think I ever saw anything after the first movie when I was a kid. But good, well, yeah I've heard that. But I remember that first movie and I didn't think about it back then. But now I'm thinking, wait, to infinity and beyond that's infinity plus one. Yeah, maybe he's he was. He was really contemplating infinity on a level that we

just weren't prepared for. To infinity and incoherence to infinity and stay there. Well, anyway, I guess that's gonna have to wrap us up for today. I think we are we are out of time. Unfortunately, I wanted to get to some of the emails we've gotten about boltzmon Brains, but we do not have time to address the finite time in this podcast. We sadly do not have infinite time. And I'm sure also you have finite patients for for

for infinity. You know, there's a fine line between things that are the most fascinating on Earth and things that really start to grade on your brain. It's it's like pulling on the tail of a of an of an infinite serpent, right You're never going to reach that point where you reach the you you find the creature's head and have a complete understanding of its anatomy. You're just gonna keep tugging and you will get pooped on. Yeah, you probably get pooped on because you really chose the

wrong end to to tug. But I still think we we managed to have a good time here. We got to discuss uh infinity and a little more depth. We got to uh iron out some of the lingering questions remaining Boltzman brains, not all of them, because the mere concept of the Boltzman brain is kind of meant to uh to stir prolonged contemplation. Yeah, uh, And we I do not want you to walk away with the impression today that we're advocating the belief that you are a

Boltzman brain. If you haven't heard that first episode, you should go back and listen to that where we actually talk about reasons you're not a Boltzman brain. Come on, yeah, don't uh don't lose any sleep over it. All right, Well, on that note, it's time for us to close out here. As always, I would like to direct you to stuff to blow your mind dot com. That's the mother ship that's we will find all the podcast episodes and links

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