October 9, 2014 • 31 mins
Imagine a hotel with infinite rooms and infinite guests. Do you fancy they have space for one more late arrival? Join Robert and Julie as they check into the Infinity Hotel to discover how the concept of infinity affects mathematics, physics and the possibility of a boundless multiverse.
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Speaker 1 (00:03):
Welcome to Stuff to Blow Your Mind from how Stuff
Works dot com. Hey, you're welcome to Stuff to Blow
your Mind. My name is Robert Lamb and um Julie Douglas,
and this is our second episode on infinity. If you
missed the first one, if you recommend you go back
and listen to that one. But we are going to
do a little recap here in case any of the
(00:24):
information has fallen out of your head in the time being. Yeah,
so let's talk about what infinity is and what it
is not, because that's our nice little baseline there. It
is not a real number, because we discussed before. You
can't say that infinity is X number. You can't write
it on a check. You can use a symbol, I guess,
but good luck cashing it. Uh, why have I never
(00:46):
done that? Uh? It cannot be measured. Of course, that
is tied back to it not being a real number.
It's not something that's growing, it's not doing anything. It's
the ultimate Zen concept. It just is yes. And of
course this has been an important topic to philosophers, uh, theologians,
(01:07):
great thinkers throughout time, and that's one of the primary
things we discussed in our last episode because Ultimately you
get into discussions about the boundless, about the limits of things.
I mean, you're dealing with everything from the basic human experience.
You know, how do I deal with the fact that
my life only stretches on until so so long? How
do I deal with the fact that I can only
(01:27):
perceive the universe with a certain bubble of perception. And
then you start imagining a god and you're trying to
figure out how that works. Is God is God? Infinite?
Is God physically infinite? Is God infinite in quality as
opposed to quantity. It's to say the biggest idea is
it is the the the infinite idea that the humanity
(01:47):
has tackled with over the ages. Yeah, and we talked
about George Tvorski writing for Ion nine, and he used
the chess board to illustrate this idea, saying that all
the other pieces having number assigned to it except for
the king, whose number value is infinite. Yeah, that chess
(02:07):
is It is interesting because we mentioned before, you have
a finite number of playable games for chess, and then
you then you have this king that has an infinite
importance within the game, So you have two different types
of infinity going on there. Yeah, you lose, you lose
the king, You lose the game because zer or some
game if you get rid of the king, and so
(02:28):
again it ties back to this whole idea about eternity
even consciousness uh embedded in there with infinity. So of
course it becomes with this great uh media area for
philosophers to really tussle with, because it's got the afterlife there,
it's got the idea and the question of God, and
by its very nature, it's an idea that we cannot
(02:51):
fully contain and we cannot really completely grasp it, and
yet we try to and so did philosophers as we
discussed the last episode, and they used to try to
prove this out or try to make their arguments even
more persuasive. So today we're gonna look at some examples
in math and physics to see how well we can
contain the idea of infinity and reach some sort of meaning,
(03:15):
which we already talked about as sort of impossible, like
meaning is infinite in itself for various reasons. To just
to recap mathematics and infinity, you voluencountered numbers like twelve
point one to nine, eight, seven, six, five, three, two, one, zero, one,
just on on and on forever. You just have to
stop counting those numbers. After a while, you have to
(03:37):
stop recording them because it becomes a pointless exercise. You
just round up, right. Is that because humans are finite
and we kind of continue to count infinitely. Yeah, that's
that's Uh, that's kind of the big topic of discussion here.
And so you kind of have two schools of thought
in mathematics, right when when it comes to to struggling
(04:01):
with the infinite, and it kind of comes down to
some of the core arguments with mathematics itself. So is
is mathematical infinity? Is this a mere human creation, an
extrapolation into the world where there's no truth? Or does
mathematical infinity actually exist? Well, this comes down to the
question of God to rite. There are some observable things
(04:21):
of how infinity works in mathematics, um, and I say observable,
but you can it's just as we discussed in your example.
There numbers can go on and on forever, and we
have finite lives. Uh, we can't see God in Therefore,
some people would say he or she or it does
not exist. So you're grappling with some of the same
(04:44):
concepts in math and in infinity. So if you were
to ask the constructivists or the institutionists, they would say
that classical mathematics, well, they deal with the sort of
math that God would do. An infinite God he can
tackle with infinite numbers. Uh, and and and the idea
of a mathematical infinity. But we are humans, we are finite,
(05:05):
and we would rather focus with human mathematics with finite
numbers because that ultimately is what relates to what we do.
You know what I hear with that, my brain hurts.
I'll just I'll just leave it to God. He or
she or it can deal with it. Yeah, I mean,
it's kind of the idea that infinity is an interesting
topic to tackle with as an abstraction, but does it
(05:27):
really relate to the work that I'm doing here, the
work that mathematics needs to do in the world. And
then they're the formalist Now. Formalism is a theory that
holds that statements of mathematics and logic can be thought
of as statements about the consequences of certain certain string
manipulation rules. Um systematic formulation of the concept of mathematical
formulism arose directly as a reaction to the paradox discovered
(05:50):
within set theory, which studies the concept of infinity. According
to twenty century mathematician Abraham Robinson, infinite totalities do not exist,
either really or ideally. Any mentioned of infinite totalities is meaningless. Still,
the business of mathematics must continue as if infinite totalities
actually do exist. Right, we said, there are a couple
of things that we're talking about. A system here in
(06:12):
this idea that you can again capture infinity. You can
define it um and then set theory, which is something
that a man named Georg Cantor came up with. And
he's sort of the rabble rouser here. According to Natalie
while cover writing for Quantum magazine, infinity was boxed and
(06:32):
sold to the mathematical community in the late night concentury
by the German mathematician Georg Cantor, who invented a branch
of mathematics dealing with sets collections of elements that ranged
from empty the equivalent of the number zero to infinite.
So Cantra stepped in there really and began to define
infinity in a much more specific way. Yeah, and uh
(06:57):
he was he was a one of these individuals who
was hated as well as love, depending on who you're asking,
because he was bringing up some really mind blowing, disturbing
takes on infinity. Yeah. Natalie Walquiver says that other mathematicians
initially despise what they called this mess of infinities. And
(07:18):
we'll talk more about this because it's largely due to
his ideas of set theory, and one of his colleagues
called them a grave disease. Another called him a corrupter
of you. Uh. Sadly, he was so vilified that he
actually um fill into a lifelong depression after that. Um.
But I think it tells you how radical some of
(07:39):
these ideas were at the time and still are really,
because what it does is it just sets minds ablaze
as to what what reality is and isn't. Essentially, Yeah,
I mean, one of the core concepts to come out
of it is that there are different kinds of infinity
and some are bigger than others. Well, that that's thinking
for a second. Yeah, let me let's get into the
(08:01):
math sweats here. I'm having the math sweats right now. Uh.
Set theory is essentially a useful language for describing mathematical objects.
And what we're talking about is a nine item list
of rules that Cantor came up with called Zermelo Frankel
set theory with the axiom of choice or the z
(08:22):
f C, established and widely adopted in math by the
nineteen twenties. Okay Um. One of the axioms says that
two sets are equal if they contain the same elements.
Another simply asserts that infinite sets exist. So you can
have these infinite sets that are one to one equal
to each other, and you have them going on forever
(08:43):
and ever and ever and ever. And Cantor showed that
for any infinite set, forming a new set made of
all the subsets of the original sets represents a bigger
infinity than that original set. So once you have one infinity,
you can always make a bigger one by creating a
subset the original set, and then an even bigger one
by making a set of all the subsets, and so
(09:04):
on and so forth. And there are an infinite amount
of infinities in different sizes. And he also proved doubt
that an infinite set of even numbers like two, four,
or six could be put in that one to one
correspondence with all counting numbers like one to three. So
that came to this idea that there are as many
evens as there are odds and evens in an infinite set.
(09:28):
There's one problem with this, though, real numbers, yes, like
the one that you talked about earlier. You know that
for example point zero zero zero one or pie right
three point one four or so on and so forth,
blah blah blah blah. They go on for an ever
and they are uncountable, and they don't correspond in a
one to one fashion with counting numbers. But hold on
(09:48):
because we will get back to that. If that didn't
thinking completely. Will return to the unccountability of real numbers
in a minute, once we have a structure in which
to house them. All right, I am going to dab
at the infinite number of sweat molecules dripping off of
me right now. Uh No, it's not too bad actually,
Uh And let's take a break. When we get back,
(10:09):
you'll talk more about the infinite hotel. If you're out
there and you're thinking, I don't know what they're talking
about there, they're losing me. I'm sinking beneath the waters
here in the It is the in the deep end
of the mathematical, philosophical, theological, physical pool. I need help,
(10:32):
while we need help to for the love of infinity,
give me an example. Yes, we do have, We do
have an example. We have a flotation device. And it
is a thought experiment, because the thought of experiments, as
always they can take some very difficult to grasp ideas
and concepts and put them into into a metaphorical form
that we can latch onto and then and then actually
(10:55):
explore the concept a little more easily and sometimes a
little a little deeper. Right, And what could be more
of a form inaccessible than a building, which is essentially
what we're talking about. Um, this is an infinite building,
an infinite hotel. And this was created by German mathematician
David Hilbert who was obsessed with cantors work and and
(11:17):
came up with us little ditty to try to explain
a hotel with an infinite number of rooms. All right,
it's so it's a hotel, infinite number of rooms? How
many how many rooms? Does it have to like an
infinite amount? Oh but accountable infinite amount, which is going
to turn out to be important. All right, So let's
say just to just to roll out the basic entry
(11:38):
level portion of the thought experiment. I show up to
the Infinity Hotel, all right, and and I need a
room for the night. I know there's a convention in town. Uh,
the local set theory convention maybe, and uh, and I
I know that all the hotels are booked, but I
really need a room. So the Infinity Hotel sounds like
a good place to go, all right, So let me
(11:59):
a my jaunty little manager's bellhop cap here. What I
am going to do is, I'm going to ask the
guests in room do you have vacancy? Always? We have
an infinite amount of rooms. Okay, I see a lot
of cars in the parking lot. Hold up, okay, just
look at a magazine or something. We'll figure this out.
I'm going to ask the guests in room one to
(12:21):
move to room two, and then the guests in room
two to move to room three, and so on and
so forth. Every guest moves to room N plus one.
And since there are an infinite amount of rooms here,
then there's room for every guest to move into a
different room. So you're saying that even though they're already
(12:42):
infinite guests at the infinity hotel. Occupying the infinite rooms,
you can still make everyone move over one room and
thus open up a single room for me to stay
in this evening. You, a whole person, a whole number,
a natural person, a natural number, may and are into
this hotel in fact, and if forty other people want
(13:03):
to join you, they can, I mean not in your room. Obviously,
you don't want to sleep with forty other people in
your room. But forty other people would like to get
forty rooms. Well, hey guess what they can. And all
I have to do is have everybody gather their luggage
and move to room N plus forty. So if you
are in room two, now you move to room two.
(13:25):
That sounds reasonable. I'm gonna pick up my bags, I'm
gonna move on in and hopeful I'm gonna be a
sleep in an hour or two. Oh. By the way,
I see a bus coming around the corner. Looks like
some more guests are arriving. Thus containing a countardly infinite
number of people arriving. No problem. All I have to
do is ask each guest to move to room N
(13:47):
to room number two N. So room one moves room
to room two to four Room three to six and
four to eight, and this fills up all the infinite
rooms and empties all the infinite odd rooms. So you're
telling me that the Infinity Hotel, which is currently filled
up with infinite number of guests, has room to take
(14:09):
care of the guests the infinite number of guests arriving
in an infinity bus. That's right, because all it is
is shifted to everybody to the infinite even number rooms
to accommodate all the infinite amount of people coming off
the infinite buses. I'm gonna put them all in odd
number rooms. And you remember Cantor in his infinite sets.
This is an example of those one to one matches,
(14:30):
not to mention, underscoring the idea that Cantor had that
there are an infinite odd and even amount of numbers.
So so, so far, the Infinity Hotel thought experiment is
holding up. But of course, one of the the fun
things about thought experiments is that you can continue to
experience with them in an attempt to break them, to
(14:52):
push them to the absolute limits. Break come on, Okay, Well,
so far the Infinity Hotel has been able to deal
with one new occupant. It's been able to deal with
a bus full of infinite new occupants. But what does
it do of coming around the corner? Are infinite buses
and each bus has infinite yes, I mean just an
(15:16):
infinite line of infinitely filled buses. All right, well, maybe
I remember that youklid said that they're an infinite number
of prime numbers. Yep, I just remembered that. And so
current occupants of rooms are assigned the prime number two.
So for instance, the current occupant of room number seven
goes to two to the seventh power or room. Then
(15:41):
the next group they're all assigned to the prime number three,
and they take their bus seat numbers to figure out
their room numbers. So seat number seven goes to three
to the seventh power or room two thousand and seven.
And I want to throw in it. This is important
because if you just did them bust by bus, you
would never finish onload owning the first bus. Well, yeah,
(16:03):
first you gotta take uh, you gotta take care of
your career that's already in in the hotel, right that
you've given that prime number two, And then yes, you
have to start referring to bus seat numbers to begin
applying all the infinite number of prime numbers, so you
can do this. Each each new group gets a new
prime number, so you know, you go to the you know,
(16:25):
prime number five, then prime number seven, eleven, and thirteen,
and everybody just has to reference that number on their
bus seat. So again, this is just really it's it's messy,
but there's an elegant system to take place or to
make sure that everybody gets their place, and that ensures
that there are no overlapping room numbers for the infinite
(16:46):
amount of people filing out of the infinite number of buses.
Here's here's the thing, though, here's the thing. Okay, here's
what I don't want to see coming out of those buses.
I don't want to see any irrational numbers. I'm talking
about you, pie. I don't want to see you tumbling
off the bus because this is not going to work.
I don't need to see a negative seven a representation
(17:08):
human representation of a negative seven. First, you're you're negative, right,
you're killing the vibe here in the hotel. Second, I
don't have negative numbers. I don't have basement number rooms,
you know, extending infinitely down into my my building here,
and I don't have half fractional rooms available, because that
(17:29):
ultimately breaks the hotel, that brings everything crashing down, because
there's one infinity that's too big, and that's the infinity
contained in a continuous flying, a continuous line of real
numbers that goes on forever. If it showed up at
the hotel, they wouldn't be room for everyone. You wouldn't
be able to, uh to take that real number and
count it. You wouldn't be able to put it on
the guest list. Yeah, and that's yeah exactly, that's the problem.
(17:51):
I mean, because we can deal with this lowest level
of infinity, the countable infinity of the natural numbers, but
that's uh, that real number, that continuum that Cantor had
described before, that is uncountable, that doesn't behave in that
one to one set way. And that's what I love
(18:12):
about the Infinity Hotel is that it describes both these
different levels of infinity, these larger amounts of infinity, but
how certain numbers don't work within it or are uncountable.
Plus you've seen being John Makovitch, right, you know what
happens when you have a half demension, right, you had
(18:35):
the half floor of the buildings right where where everything's
cut in half and you have to stoop. And what happens,
people will will use that as a vessel to somehow
infiltrate your mind and hijack your body if you're placed
in a half room or a half floor. Yeah, and
that nobody wants that. It's gonna hard hard to retain
(18:55):
tenants when you have that going on. Exactly, so again
you have Cantor's continum hype offices. Fit says that there's
there's no set whose cardinality is strictly between that of
the integers and the real numbers. Can't prove it's true,
can't prove it's false, can't measure it, can't hunt for
a particle or measure anything about it. Yes, those real
numbers are uncountable in this scenario of infinity. And then
(19:16):
along uh of of and about nine gun named Goodill
has a pair of proofs that are just spectacular to
the math community because essentially he shows that you can
never prove that the continuum hypothesis is false. And this
feels like, uh, you're like you're moving the needle here
(19:37):
on infinity just a bit right, Like we think we've
grabbed onto something here. The problem is that in nineteen
sixty or they're about Paul J. Cohen shows that you
can never prove that the continuum hypothesis is true. And
why does it all of this matter is because it
shows that there are unanswerable questions in mathematics, particularly dealing
(19:59):
with infinity, which just is really a microcosm of the
macrocosm problem of the unknowability of life in general. So
if your brain is exploding right now, don't worry, because
now we're going to take it back to physics. We're
gonna take it to discussions of the physical world as
complicated and mind blowing is that whole realm can get
(20:22):
at least we're dealing with observable physical reality, right, Sure,
you say that, And now I have the physics sweats,
which is an entirely different stench here. And that's understandable. Yes,
but because certainly physics gets very complicated as well. But
when but here's the thing. When we see infinities in physics, generally,
it means that we have a problem and means that
(20:43):
something's really catastrophic wrong unless you get into theoretical physics,
as we will in a second. Yes, but you know
basic physics, you're trying to use somebody that's designing. You know,
you know a new building and you look at the plans.
If you see infinity on there, you know that something
is wrong. You do not want to try and stand
in that building or or or or certainly have an
(21:05):
office in it. Um it's never an actual measurement. It
doesn't correspond to to to reality as as we we
deal with it in our lives right today, for instance,
quantum mechanics okay um and and and in such an
important area of studies. So many of our greatest technological
achievements in recent memory have arisen out of quantum mechanics.
(21:27):
But even here we see this crisis. The crisis being
that even though we can predict the the light heat
power emission of a lamp, it's a finite amount of energy,
but the various wigglings of waves and atoms creates this
answer infinite energy, which which we simply can't deal with.
Right we know that there's not infinite energy coming out
(21:50):
of off of a lamp, so we afterwards we have
to begin applying quantum physics to the forces of our universe.
Quantum field theory, which is of course initially played infinities
as well, and it takes ends up taking decades to
flog through it all. You end up, you know, having
all these infinities, all of these problems in the in
the theory that have to be eradicated, like like enemies
(22:12):
floating around in a video game, or which becomes incredibly
important if you're looking at particle physics, right, because if
you're trying to isolate the Higgs boson, you're taking these
finite number of particles that we know and have named
in seeing how they behave right, you do not need
infinity in the mix if you're trying to figure out
(22:33):
the composition of the universe, both from the time in
which it became to now I mean, take the Higgs
for example, the search for the Higgs boson, this was
such the so called god particle, right. Uh. This uh, this,
this of course been one of the predominant science stories
in recent years, because the whole idea was that we
(22:54):
had infinities in the theory, we had to remove them,
and the thing that could remove it would be this
Higgs particle. Right, So the whole quest for the Higgs
particle comes out of this, this this necessity, this need
to eradicate the infinities from the theory, because again, infinite's
(23:14):
just don't work in the set and the infinities here
in the theory said said, all right, we have an infinity.
That means there's something finite in the world that we
haven't discovered yet, and if we discover it, then we
find our way around that answer. Now, if you sort
of back up and then begin to take the large
view of the universe and not just try to contain
it in this this one model, right, you know that
(23:36):
the universe is expanding how much infinitely? These are questions
that come up, and then you begin to say, well,
how how far does space go? Anyway? And this is
where theoretical physicists get into infinity and they really have
some fun um And it's fascinating because here you see
some examples of infinity played out in ways that you say,
(24:00):
this is a possibility, because infinity otherwise we just see
is this sort of you know, the line on the
piece of paper and mathematics that goes on forever. But
then you've got these ideas like multiverse existing. Now, when
I think of the multiverse, I always think of Jorey
Lewis Borges the Library of Battle, the library that first
(24:20):
of all contains all books, all written books, but then
beyond that also contains all possible books, and and and
and and so being is is ultimately kind of a
model for a multiverse. Uh, an existence that contains everything
that is and everything that could be. Again, you go
back to Cantor, and you're talking about larger infinities, right,
(24:40):
And essentially that's what we're talking about with multiverses. And
I'm going to read this from Brian Green's The Elegant Universe. Um,
we could talk about multiverse in its very own episodes.
We won't go too deep here, he says, Imagine that
what we call the universe is actually only one tiny
part of a vastly larger cosmological expanse, one of the
nor almost number of island universes scattered across a grand
(25:03):
cosmological archipelago. Although this might sound rather far fetched, and
in the end it may well be, Andrei Lynd has
suggested a concrete mechanism that might lead to such a
Gargangan universe. Lynn has found that the brief but crucial
burst of inflationary expanse may not have been a unique
(25:24):
one time event. Instead, he argues the conditions for inflationary
expansion may happen repeatedly in isolated regions and the peppered
throughout the cosmos, which then undergo their own inflationary ballooning
and size, evolving into new separate universes. And each of
the universes, the process continues, with new universes sprouting from
(25:46):
far flung regions in the old, generating a never ending
web of ballooning cosmic expanses. And this is what he
calls the multiverse. And that's pretty mind blow I mean,
even if you you shrink back down to are single universe. Um.
There was an interesting point that was brought up by
physicist Raphael Bosso in that World Science Festival talk I
(26:09):
was mentioning earlier. We will include a link to in
the landing page for this podcast episode. But he pointed
out that that when you start start looking at the
way light travels across the universe, when you start looking
at this expansion of the universe, you end up with
a universe that's arbitrarily large. Light can never reach you
from rapidly expanding regions regions, and so any given observer
(26:31):
is trapped within a finite sphere of observable universe within
an expanding infinite. So you kind of get this again.
You get these ideas of here's this, here's the sphere
of the finite within the infinite. You can think about
the infinite, but you're ultimately trapped within that sphere of
the finite. Yeah, and this is a similar idea that
(26:52):
Lee Smollen has in terms of the perception of the
infinite at least. Smallen, by the way, is a cosmologist
at and State, and his idea is that the conditions
at the Big Bang and at the centers of black holes,
each being characterized by a colossal density of crushed matter.
I suggested that every black hole is the seed for
(27:14):
a new universe that erupts into existence through a Big
Bang like explosion, but is forever hidden from our view
by the black holes event horizon. And of course, any
of these discussions of the physical universe are are even
more twisted when you when you have coorse draw in
the fact that time and space are one, and if
you play with space, you're playing with time, and if
(27:36):
it were to be flat, then it could stretch out
for an infinity, right, And then you get all these
different ideas of well maybe in this case that supports
this idea of some sort of repeating going on because
you have a finite number of particles that you have
infinite space and time, and maybe that repeating pattern creates
more universes, right, And then you have ever many worlds
(28:01):
interpretation that says that the universe branches off into distinct
worlds to accommodate every single possible outcome. And so maybe
we live in an infinite web of alternate timelines. I mean,
it gets crazy and crazy as you go along. So yes,
I mean I don't even have to everyone else's mind
is extrapolating the possibilities on that. I don't have to
(28:23):
push you in that direction. And if you guys would
like some articles to accompany this, uh, we have ever
many worlds interpretation on how stuff works, and we also
have some good amount of string theory and multiverse um,
so we definitely have some stuff for you guys to
dive into. Yes, on the Internet, which is of course
(28:43):
a finite world. Even though it is it is a
finite world. But yet we we can't even grasp what
We can't even put a number on it. We can't
say how many pages there actually are on the Internet.
We can't actually even comprehend it's a finite thing we've created,
and it is already, at least as far as human
perception goes, bordering the infinite. Let's supposed to just fine,
(29:04):
since we can see its growth and we can put
a number on it. But it feels infinite at times. Alright,
So we're gonna walk you guys out of this um
and we're gonna take a little walk into something called
the infinite monkey theorem to end this section. Yeah good.
This is an you know, a thought experiment that I
often forget even really entails infinity, maybe because I just
(29:28):
get too caught up on the idea of monkeys banging
on typewriters and now awful that writing room, let's be,
and I've I've written in some awful rooms before. Uh.
This states that a monkey hidden keys at random on
a typewriter keyboard for an infinite amount of time will
almost surely type a given text, like say, the complete
(29:50):
works of William Shakespeare. Okay, okay, And so in this context,
almost surely is a mathematical term, alright, and it has
a precise meaning, And that monkey is not an actual
monkey but a metaphor for an abstract device that produces
a random sequence of letters for an infinity. In the
theorem illustrates the perils of reasoning about infinity by imagining
(30:13):
this vast that finite number, and vice versa. So the
probability is really tiny, right, The probability exists that a
monkey could eventually write a work as cohesive as say
Shakespeare's Hamlet, just by pure dumb accident of banging on
the keys forever and ever and ever endever, which gives
(30:35):
my fiction writing some hope. Oh you're you're better than
a monkey. I don't know. I don't know about that,
and I'm fine with that actually, all right, So if
you're still with us, then hopefully you have you maybe
you have a better idea of what infinity is all about.
Maybe you have more nuanced idea, more expansive idea. Maybe
(30:58):
this is uh forced you to sort rearrange your your
contemplation of the infinite and of the boundless, in terms
of of our human experience, in terms of our cosmos,
in terms of our ideas of God. So we'd love
to hear from you. We'd love to hear your thoughts
about infinity, your personal takes on infinity, your favorite uses
of infinity and fiction. All of that is fair game. Yeah,
(31:20):
and before you do that, make sure you stopped at
Stuff to Blow your Mind dot com. That's right, That's
where you'll find all the podcast episodes, all the videos,
all of the blog posts, a finite amount of all
of those, but certainly plenty of stuff to keep you occupied. Yeah,
So share those thoughts with us, why don't you? And
you can do that at below the mind house to
works dot com. For more on this and thousands of
(31:45):
other topics, visit how stuff works dot com
Welcome to Stuff to Blow Your Mind from how Stuff
Works dot com. Hey, you're welcome to Stuff to Blow
your Mind. My name is Robert Lamb and um Julie Douglas,
and this is our second episode on infinity. If you
missed the first one, if you recommend you go back
and listen to that one. But we are going to
do a little recap here in case any of the
(00:24):
information has fallen out of your head in the time being. Yeah,
so let's talk about what infinity is and what it
is not, because that's our nice little baseline there. It
is not a real number, because we discussed before. You
can't say that infinity is X number. You can't write
it on a check. You can use a symbol, I guess,
but good luck cashing it. Uh, why have I never
(00:46):
done that? Uh? It cannot be measured. Of course, that
is tied back to it not being a real number.
It's not something that's growing, it's not doing anything. It's
the ultimate Zen concept. It just is yes. And of
course this has been an important topic to philosophers, uh, theologians,
(01:07):
great thinkers throughout time, and that's one of the primary
things we discussed in our last episode because Ultimately you
get into discussions about the boundless, about the limits of things.
I mean, you're dealing with everything from the basic human experience.
You know, how do I deal with the fact that
my life only stretches on until so so long? How
do I deal with the fact that I can only
(01:27):
perceive the universe with a certain bubble of perception. And
then you start imagining a god and you're trying to
figure out how that works. Is God is God? Infinite?
Is God physically infinite? Is God infinite in quality as
opposed to quantity. It's to say the biggest idea is
it is the the the infinite idea that the humanity
(01:47):
has tackled with over the ages. Yeah, and we talked
about George Tvorski writing for Ion nine, and he used
the chess board to illustrate this idea, saying that all
the other pieces having number assigned to it except for
the king, whose number value is infinite. Yeah, that chess
(02:07):
is It is interesting because we mentioned before, you have
a finite number of playable games for chess, and then
you then you have this king that has an infinite
importance within the game, So you have two different types
of infinity going on there. Yeah, you lose, you lose
the king, You lose the game because zer or some
game if you get rid of the king, and so
(02:28):
again it ties back to this whole idea about eternity
even consciousness uh embedded in there with infinity. So of
course it becomes with this great uh media area for
philosophers to really tussle with, because it's got the afterlife there,
it's got the idea and the question of God, and
by its very nature, it's an idea that we cannot
(02:51):
fully contain and we cannot really completely grasp it, and
yet we try to and so did philosophers as we
discussed the last episode, and they used to try to
prove this out or try to make their arguments even
more persuasive. So today we're gonna look at some examples
in math and physics to see how well we can
contain the idea of infinity and reach some sort of meaning,
(03:15):
which we already talked about as sort of impossible, like
meaning is infinite in itself for various reasons. To just
to recap mathematics and infinity, you voluencountered numbers like twelve
point one to nine, eight, seven, six, five, three, two, one, zero, one,
just on on and on forever. You just have to
stop counting those numbers. After a while, you have to
(03:37):
stop recording them because it becomes a pointless exercise. You
just round up, right. Is that because humans are finite
and we kind of continue to count infinitely. Yeah, that's
that's Uh, that's kind of the big topic of discussion here.
And so you kind of have two schools of thought
in mathematics, right when when it comes to to struggling
(04:01):
with the infinite, and it kind of comes down to
some of the core arguments with mathematics itself. So is
is mathematical infinity? Is this a mere human creation, an
extrapolation into the world where there's no truth? Or does
mathematical infinity actually exist? Well, this comes down to the
question of God to rite. There are some observable things
(04:21):
of how infinity works in mathematics, um, and I say observable,
but you can it's just as we discussed in your example.
There numbers can go on and on forever, and we
have finite lives. Uh, we can't see God in Therefore,
some people would say he or she or it does
not exist. So you're grappling with some of the same
(04:44):
concepts in math and in infinity. So if you were
to ask the constructivists or the institutionists, they would say
that classical mathematics, well, they deal with the sort of
math that God would do. An infinite God he can
tackle with infinite numbers. Uh, and and and the idea
of a mathematical infinity. But we are humans, we are finite,
(05:05):
and we would rather focus with human mathematics with finite
numbers because that ultimately is what relates to what we do.
You know what I hear with that, my brain hurts.
I'll just I'll just leave it to God. He or
she or it can deal with it. Yeah, I mean,
it's kind of the idea that infinity is an interesting
topic to tackle with as an abstraction, but does it
(05:27):
really relate to the work that I'm doing here, the
work that mathematics needs to do in the world. And
then they're the formalist Now. Formalism is a theory that
holds that statements of mathematics and logic can be thought
of as statements about the consequences of certain certain string
manipulation rules. Um systematic formulation of the concept of mathematical
formulism arose directly as a reaction to the paradox discovered
(05:50):
within set theory, which studies the concept of infinity. According
to twenty century mathematician Abraham Robinson, infinite totalities do not exist,
either really or ideally. Any mentioned of infinite totalities is meaningless. Still,
the business of mathematics must continue as if infinite totalities
actually do exist. Right, we said, there are a couple
of things that we're talking about. A system here in
(06:12):
this idea that you can again capture infinity. You can
define it um and then set theory, which is something
that a man named Georg Cantor came up with. And
he's sort of the rabble rouser here. According to Natalie
while cover writing for Quantum magazine, infinity was boxed and
(06:32):
sold to the mathematical community in the late night concentury
by the German mathematician Georg Cantor, who invented a branch
of mathematics dealing with sets collections of elements that ranged
from empty the equivalent of the number zero to infinite.
So Cantra stepped in there really and began to define
infinity in a much more specific way. Yeah, and uh
(06:57):
he was he was a one of these individuals who
was hated as well as love, depending on who you're asking,
because he was bringing up some really mind blowing, disturbing
takes on infinity. Yeah. Natalie Walquiver says that other mathematicians
initially despise what they called this mess of infinities. And
(07:18):
we'll talk more about this because it's largely due to
his ideas of set theory, and one of his colleagues
called them a grave disease. Another called him a corrupter
of you. Uh. Sadly, he was so vilified that he
actually um fill into a lifelong depression after that. Um.
But I think it tells you how radical some of
(07:39):
these ideas were at the time and still are really,
because what it does is it just sets minds ablaze
as to what what reality is and isn't. Essentially, Yeah,
I mean, one of the core concepts to come out
of it is that there are different kinds of infinity
and some are bigger than others. Well, that that's thinking
for a second. Yeah, let me let's get into the
(08:01):
math sweats here. I'm having the math sweats right now. Uh.
Set theory is essentially a useful language for describing mathematical objects.
And what we're talking about is a nine item list
of rules that Cantor came up with called Zermelo Frankel
set theory with the axiom of choice or the z
(08:22):
f C, established and widely adopted in math by the
nineteen twenties. Okay Um. One of the axioms says that
two sets are equal if they contain the same elements.
Another simply asserts that infinite sets exist. So you can
have these infinite sets that are one to one equal
to each other, and you have them going on forever
(08:43):
and ever and ever and ever. And Cantor showed that
for any infinite set, forming a new set made of
all the subsets of the original sets represents a bigger
infinity than that original set. So once you have one infinity,
you can always make a bigger one by creating a
subset the original set, and then an even bigger one
by making a set of all the subsets, and so
(09:04):
on and so forth. And there are an infinite amount
of infinities in different sizes. And he also proved doubt
that an infinite set of even numbers like two, four,
or six could be put in that one to one
correspondence with all counting numbers like one to three. So
that came to this idea that there are as many
evens as there are odds and evens in an infinite set.
(09:28):
There's one problem with this, though, real numbers, yes, like
the one that you talked about earlier. You know that
for example point zero zero zero one or pie right
three point one four or so on and so forth,
blah blah blah blah. They go on for an ever
and they are uncountable, and they don't correspond in a
one to one fashion with counting numbers. But hold on
(09:48):
because we will get back to that. If that didn't
thinking completely. Will return to the unccountability of real numbers
in a minute, once we have a structure in which
to house them. All right, I am going to dab
at the infinite number of sweat molecules dripping off of
me right now. Uh No, it's not too bad actually,
Uh And let's take a break. When we get back,
(10:09):
you'll talk more about the infinite hotel. If you're out
there and you're thinking, I don't know what they're talking
about there, they're losing me. I'm sinking beneath the waters
here in the It is the in the deep end
of the mathematical, philosophical, theological, physical pool. I need help,
(10:32):
while we need help to for the love of infinity,
give me an example. Yes, we do have, We do
have an example. We have a flotation device. And it
is a thought experiment, because the thought of experiments, as
always they can take some very difficult to grasp ideas
and concepts and put them into into a metaphorical form
that we can latch onto and then and then actually
(10:55):
explore the concept a little more easily and sometimes a
little a little deeper. Right, And what could be more
of a form inaccessible than a building, which is essentially
what we're talking about. Um, this is an infinite building,
an infinite hotel. And this was created by German mathematician
David Hilbert who was obsessed with cantors work and and
(11:17):
came up with us little ditty to try to explain
a hotel with an infinite number of rooms. All right,
it's so it's a hotel, infinite number of rooms? How
many how many rooms? Does it have to like an
infinite amount? Oh but accountable infinite amount, which is going
to turn out to be important. All right, So let's
say just to just to roll out the basic entry
(11:38):
level portion of the thought experiment. I show up to
the Infinity Hotel, all right, and and I need a
room for the night. I know there's a convention in town. Uh,
the local set theory convention maybe, and uh, and I
I know that all the hotels are booked, but I
really need a room. So the Infinity Hotel sounds like
a good place to go, all right, So let me
(11:59):
a my jaunty little manager's bellhop cap here. What I
am going to do is, I'm going to ask the
guests in room do you have vacancy? Always? We have
an infinite amount of rooms. Okay, I see a lot
of cars in the parking lot. Hold up, okay, just
look at a magazine or something. We'll figure this out.
I'm going to ask the guests in room one to
(12:21):
move to room two, and then the guests in room
two to move to room three, and so on and
so forth. Every guest moves to room N plus one.
And since there are an infinite amount of rooms here,
then there's room for every guest to move into a
different room. So you're saying that even though they're already
(12:42):
infinite guests at the infinity hotel. Occupying the infinite rooms,
you can still make everyone move over one room and
thus open up a single room for me to stay
in this evening. You, a whole person, a whole number,
a natural person, a natural number, may and are into
this hotel in fact, and if forty other people want
(13:03):
to join you, they can, I mean not in your room. Obviously,
you don't want to sleep with forty other people in
your room. But forty other people would like to get
forty rooms. Well, hey guess what they can. And all
I have to do is have everybody gather their luggage
and move to room N plus forty. So if you
are in room two, now you move to room two.
(13:25):
That sounds reasonable. I'm gonna pick up my bags, I'm
gonna move on in and hopeful I'm gonna be a
sleep in an hour or two. Oh. By the way,
I see a bus coming around the corner. Looks like
some more guests are arriving. Thus containing a countardly infinite
number of people arriving. No problem. All I have to
do is ask each guest to move to room N
(13:47):
to room number two N. So room one moves room
to room two to four Room three to six and
four to eight, and this fills up all the infinite
rooms and empties all the infinite odd rooms. So you're
telling me that the Infinity Hotel, which is currently filled
up with infinite number of guests, has room to take
(14:09):
care of the guests the infinite number of guests arriving
in an infinity bus. That's right, because all it is
is shifted to everybody to the infinite even number rooms
to accommodate all the infinite amount of people coming off
the infinite buses. I'm gonna put them all in odd
number rooms. And you remember Cantor in his infinite sets.
This is an example of those one to one matches,
(14:30):
not to mention, underscoring the idea that Cantor had that
there are an infinite odd and even amount of numbers.
So so, so far, the Infinity Hotel thought experiment is
holding up. But of course, one of the the fun
things about thought experiments is that you can continue to
experience with them in an attempt to break them, to
(14:52):
push them to the absolute limits. Break come on, Okay, Well,
so far the Infinity Hotel has been able to deal
with one new occupant. It's been able to deal with
a bus full of infinite new occupants. But what does
it do of coming around the corner? Are infinite buses
and each bus has infinite yes, I mean just an
(15:16):
infinite line of infinitely filled buses. All right, well, maybe
I remember that youklid said that they're an infinite number
of prime numbers. Yep, I just remembered that. And so
current occupants of rooms are assigned the prime number two.
So for instance, the current occupant of room number seven
goes to two to the seventh power or room. Then
(15:41):
the next group they're all assigned to the prime number three,
and they take their bus seat numbers to figure out
their room numbers. So seat number seven goes to three
to the seventh power or room two thousand and seven.
And I want to throw in it. This is important
because if you just did them bust by bus, you
would never finish onload owning the first bus. Well, yeah,
(16:03):
first you gotta take uh, you gotta take care of
your career that's already in in the hotel, right that
you've given that prime number two, And then yes, you
have to start referring to bus seat numbers to begin
applying all the infinite number of prime numbers, so you
can do this. Each each new group gets a new
prime number, so you know, you go to the you know,
(16:25):
prime number five, then prime number seven, eleven, and thirteen,
and everybody just has to reference that number on their
bus seat. So again, this is just really it's it's messy,
but there's an elegant system to take place or to
make sure that everybody gets their place, and that ensures
that there are no overlapping room numbers for the infinite
(16:46):
amount of people filing out of the infinite number of buses.
Here's here's the thing, though, here's the thing. Okay, here's
what I don't want to see coming out of those buses.
I don't want to see any irrational numbers. I'm talking
about you, pie. I don't want to see you tumbling
off the bus because this is not going to work.
I don't need to see a negative seven a representation
(17:08):
human representation of a negative seven. First, you're you're negative, right,
you're killing the vibe here in the hotel. Second, I
don't have negative numbers. I don't have basement number rooms,
you know, extending infinitely down into my my building here,
and I don't have half fractional rooms available, because that
(17:29):
ultimately breaks the hotel, that brings everything crashing down, because
there's one infinity that's too big, and that's the infinity
contained in a continuous flying, a continuous line of real
numbers that goes on forever. If it showed up at
the hotel, they wouldn't be room for everyone. You wouldn't
be able to, uh to take that real number and
count it. You wouldn't be able to put it on
the guest list. Yeah, and that's yeah exactly, that's the problem.
(17:51):
I mean, because we can deal with this lowest level
of infinity, the countable infinity of the natural numbers, but
that's uh, that real number, that continuum that Cantor had
described before, that is uncountable, that doesn't behave in that
one to one set way. And that's what I love
(18:12):
about the Infinity Hotel is that it describes both these
different levels of infinity, these larger amounts of infinity, but
how certain numbers don't work within it or are uncountable.
Plus you've seen being John Makovitch, right, you know what
happens when you have a half demension, right, you had
(18:35):
the half floor of the buildings right where where everything's
cut in half and you have to stoop. And what happens,
people will will use that as a vessel to somehow
infiltrate your mind and hijack your body if you're placed
in a half room or a half floor. Yeah, and
that nobody wants that. It's gonna hard hard to retain
(18:55):
tenants when you have that going on. Exactly, so again
you have Cantor's continum hype offices. Fit says that there's
there's no set whose cardinality is strictly between that of
the integers and the real numbers. Can't prove it's true,
can't prove it's false, can't measure it, can't hunt for
a particle or measure anything about it. Yes, those real
numbers are uncountable in this scenario of infinity. And then
(19:16):
along uh of of and about nine gun named Goodill
has a pair of proofs that are just spectacular to
the math community because essentially he shows that you can
never prove that the continuum hypothesis is false. And this
feels like, uh, you're like you're moving the needle here
(19:37):
on infinity just a bit right, Like we think we've
grabbed onto something here. The problem is that in nineteen
sixty or they're about Paul J. Cohen shows that you
can never prove that the continuum hypothesis is true. And
why does it all of this matter is because it
shows that there are unanswerable questions in mathematics, particularly dealing
(19:59):
with infinity, which just is really a microcosm of the
macrocosm problem of the unknowability of life in general. So
if your brain is exploding right now, don't worry, because
now we're going to take it back to physics. We're
gonna take it to discussions of the physical world as
complicated and mind blowing is that whole realm can get
(20:22):
at least we're dealing with observable physical reality, right, Sure,
you say that, And now I have the physics sweats,
which is an entirely different stench here. And that's understandable. Yes,
but because certainly physics gets very complicated as well. But
when but here's the thing. When we see infinities in physics, generally,
it means that we have a problem and means that
(20:43):
something's really catastrophic wrong unless you get into theoretical physics,
as we will in a second. Yes, but you know
basic physics, you're trying to use somebody that's designing. You know,
you know a new building and you look at the plans.
If you see infinity on there, you know that something
is wrong. You do not want to try and stand
in that building or or or or certainly have an
(21:05):
office in it. Um it's never an actual measurement. It
doesn't correspond to to to reality as as we we
deal with it in our lives right today, for instance,
quantum mechanics okay um and and and in such an
important area of studies. So many of our greatest technological
achievements in recent memory have arisen out of quantum mechanics.
(21:27):
But even here we see this crisis. The crisis being
that even though we can predict the the light heat
power emission of a lamp, it's a finite amount of energy,
but the various wigglings of waves and atoms creates this
answer infinite energy, which which we simply can't deal with.
Right we know that there's not infinite energy coming out
(21:50):
of off of a lamp, so we afterwards we have
to begin applying quantum physics to the forces of our universe.
Quantum field theory, which is of course initially played infinities
as well, and it takes ends up taking decades to
flog through it all. You end up, you know, having
all these infinities, all of these problems in the in
the theory that have to be eradicated, like like enemies
(22:12):
floating around in a video game, or which becomes incredibly
important if you're looking at particle physics, right, because if
you're trying to isolate the Higgs boson, you're taking these
finite number of particles that we know and have named
in seeing how they behave right, you do not need
infinity in the mix if you're trying to figure out
(22:33):
the composition of the universe, both from the time in
which it became to now I mean, take the Higgs
for example, the search for the Higgs boson, this was
such the so called god particle, right. Uh. This uh, this,
this of course been one of the predominant science stories
in recent years, because the whole idea was that we
(22:54):
had infinities in the theory, we had to remove them,
and the thing that could remove it would be this
Higgs particle. Right, So the whole quest for the Higgs
particle comes out of this, this this necessity, this need
to eradicate the infinities from the theory, because again, infinite's
(23:14):
just don't work in the set and the infinities here
in the theory said said, all right, we have an infinity.
That means there's something finite in the world that we
haven't discovered yet, and if we discover it, then we
find our way around that answer. Now, if you sort
of back up and then begin to take the large
view of the universe and not just try to contain
it in this this one model, right, you know that
(23:36):
the universe is expanding how much infinitely? These are questions
that come up, and then you begin to say, well,
how how far does space go? Anyway? And this is
where theoretical physicists get into infinity and they really have
some fun um And it's fascinating because here you see
some examples of infinity played out in ways that you say,
(24:00):
this is a possibility, because infinity otherwise we just see
is this sort of you know, the line on the
piece of paper and mathematics that goes on forever. But
then you've got these ideas like multiverse existing. Now, when
I think of the multiverse, I always think of Jorey
Lewis Borges the Library of Battle, the library that first
(24:20):
of all contains all books, all written books, but then
beyond that also contains all possible books, and and and
and and so being is is ultimately kind of a
model for a multiverse. Uh, an existence that contains everything
that is and everything that could be. Again, you go
back to Cantor, and you're talking about larger infinities, right,
(24:40):
And essentially that's what we're talking about with multiverses. And
I'm going to read this from Brian Green's The Elegant Universe. Um,
we could talk about multiverse in its very own episodes.
We won't go too deep here, he says, Imagine that
what we call the universe is actually only one tiny
part of a vastly larger cosmological expanse, one of the
nor almost number of island universes scattered across a grand
(25:03):
cosmological archipelago. Although this might sound rather far fetched, and
in the end it may well be, Andrei Lynd has
suggested a concrete mechanism that might lead to such a
Gargangan universe. Lynn has found that the brief but crucial
burst of inflationary expanse may not have been a unique
(25:24):
one time event. Instead, he argues the conditions for inflationary
expansion may happen repeatedly in isolated regions and the peppered
throughout the cosmos, which then undergo their own inflationary ballooning
and size, evolving into new separate universes. And each of
the universes, the process continues, with new universes sprouting from
(25:46):
far flung regions in the old, generating a never ending
web of ballooning cosmic expanses. And this is what he
calls the multiverse. And that's pretty mind blow I mean,
even if you you shrink back down to are single universe. Um.
There was an interesting point that was brought up by
physicist Raphael Bosso in that World Science Festival talk I
(26:09):
was mentioning earlier. We will include a link to in
the landing page for this podcast episode. But he pointed
out that that when you start start looking at the
way light travels across the universe, when you start looking
at this expansion of the universe, you end up with
a universe that's arbitrarily large. Light can never reach you
from rapidly expanding regions regions, and so any given observer
(26:31):
is trapped within a finite sphere of observable universe within
an expanding infinite. So you kind of get this again.
You get these ideas of here's this, here's the sphere
of the finite within the infinite. You can think about
the infinite, but you're ultimately trapped within that sphere of
the finite. Yeah, and this is a similar idea that
(26:52):
Lee Smollen has in terms of the perception of the
infinite at least. Smallen, by the way, is a cosmologist
at and State, and his idea is that the conditions
at the Big Bang and at the centers of black holes,
each being characterized by a colossal density of crushed matter.
I suggested that every black hole is the seed for
(27:14):
a new universe that erupts into existence through a Big
Bang like explosion, but is forever hidden from our view
by the black holes event horizon. And of course, any
of these discussions of the physical universe are are even
more twisted when you when you have coorse draw in
the fact that time and space are one, and if
you play with space, you're playing with time, and if
(27:36):
it were to be flat, then it could stretch out
for an infinity, right, And then you get all these
different ideas of well maybe in this case that supports
this idea of some sort of repeating going on because
you have a finite number of particles that you have
infinite space and time, and maybe that repeating pattern creates
more universes, right, And then you have ever many worlds
(28:01):
interpretation that says that the universe branches off into distinct
worlds to accommodate every single possible outcome. And so maybe
we live in an infinite web of alternate timelines. I mean,
it gets crazy and crazy as you go along. So yes,
I mean I don't even have to everyone else's mind
is extrapolating the possibilities on that. I don't have to
(28:23):
push you in that direction. And if you guys would
like some articles to accompany this, uh, we have ever
many worlds interpretation on how stuff works, and we also
have some good amount of string theory and multiverse um,
so we definitely have some stuff for you guys to
dive into. Yes, on the Internet, which is of course
(28:43):
a finite world. Even though it is it is a
finite world. But yet we we can't even grasp what
We can't even put a number on it. We can't
say how many pages there actually are on the Internet.
We can't actually even comprehend it's a finite thing we've created,
and it is already, at least as far as human
perception goes, bordering the infinite. Let's supposed to just fine,
(29:04):
since we can see its growth and we can put
a number on it. But it feels infinite at times. Alright,
So we're gonna walk you guys out of this um
and we're gonna take a little walk into something called
the infinite monkey theorem to end this section. Yeah good.
This is an you know, a thought experiment that I
often forget even really entails infinity, maybe because I just
(29:28):
get too caught up on the idea of monkeys banging
on typewriters and now awful that writing room, let's be,
and I've I've written in some awful rooms before. Uh.
This states that a monkey hidden keys at random on
a typewriter keyboard for an infinite amount of time will
almost surely type a given text, like say, the complete
(29:50):
works of William Shakespeare. Okay, okay, And so in this context,
almost surely is a mathematical term, alright, and it has
a precise meaning, And that monkey is not an actual
monkey but a metaphor for an abstract device that produces
a random sequence of letters for an infinity. In the
theorem illustrates the perils of reasoning about infinity by imagining
(30:13):
this vast that finite number, and vice versa. So the
probability is really tiny, right, The probability exists that a
monkey could eventually write a work as cohesive as say
Shakespeare's Hamlet, just by pure dumb accident of banging on
the keys forever and ever and ever endever, which gives
(30:35):
my fiction writing some hope. Oh you're you're better than
a monkey. I don't know. I don't know about that,
and I'm fine with that actually, all right, So if
you're still with us, then hopefully you have you maybe
you have a better idea of what infinity is all about.
Maybe you have more nuanced idea, more expansive idea. Maybe
(30:58):
this is uh forced you to sort rearrange your your
contemplation of the infinite and of the boundless, in terms
of of our human experience, in terms of our cosmos,
in terms of our ideas of God. So we'd love
to hear from you. We'd love to hear your thoughts
about infinity, your personal takes on infinity, your favorite uses
of infinity and fiction. All of that is fair game. Yeah,
(31:20):
and before you do that, make sure you stopped at
Stuff to Blow your Mind dot com. That's right, That's
where you'll find all the podcast episodes, all the videos,
all of the blog posts, a finite amount of all
of those, but certainly plenty of stuff to keep you occupied. Yeah,
So share those thoughts with us, why don't you? And
you can do that at below the mind house to
works dot com. For more on this and thousands of
(31:45):
other topics, visit how stuff works dot com
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