Cantors Paradise The infinity on earth
cantor’s paradise
zermalofrankel set theory a left numbers
the actual infinity you might be
wondering what
madness drove the 17 year old kid to
spend his entire christmas holiday
studying and researching the
mathematical nature of the actual
infinity
well the answer is actually quite simple
when i was preparing for this ted talk
the first thing i did was go onto the
tedx website and see
well what topics could i talk about well
as a christian and
someone who’s quite interested in
philosophy i thought well wouldn’t this
be a great opportunity to present some
philosophical arguments for these ex
the existence of god or perhaps i could
use this time
to prepare a powerful case for the
historical resurrection
yet the moment i looked at the rules the
first thing i saw
was well no religious or political
agenda so
while that had to go out of the window
then i thought okay i know a bit of
science how about talking about
evolution
and young earth creationism and how
they’re kind of there’s some problems
here
but then i then saw another thing
no bad science so well that had to go
out to the window as well
so i was left with nothing but the
actual infinity
and it’s not that i’m complaining about
it because it turns out the actual
infinity
is actually a very interesting topic no
pun intended there
and you might think well i’m just paid
by a math teacher to tell you
oh maths is very fun now let’s all apply
for maths
for a levels but no the actual infinity
is actually a very
interesting and profound topic that has
significant implications on
our physical world and also the
conceptual world of platonic realism
and conceptual realism so what i would
like to do here
since this topic the actual infinity is
a massive topic
i like to discuss its implications on
the physical world
and discuss whether it can actually be
applied to the physical world
whether an actually infinite amount of
things or
quantities can be realized or actualized
in our world around us both in this
temporal space
and also a spatial sense can the past be
actually infinite
and can space as a whole be actually
infinite
to discuss this i would like to turn to
the use of paradoxes
the use of paradoxes has been used
throughout the history of philosophy
to discuss or show that something cannot
happen or
that some result or some argument leads
to contradictory results
a good example of this would be zeno’s
paradoxes which is one of the arguments
i’ll be raising today
and also grim reaper’s paradox which
i’ll be also discussing in this video
so without further ado let’s get started
what is the xenos paradox
well zeno actually raises multiple
paradoxes throughout his life
which are recorded by aristotle in his
book physics
while we don’t have any direct works
from zeno we do
see a lot of his paradoxes in
aristotle’s books including
the dichotomy paradox which we’ll be
discussing in this video arcolas and the
tortoise
the stadium paradox and the arrow
paradox and others
so what is the dichotomy paradox while
there’s more responses to this paradox
i think that i love this paradox i think
it’s very interesting because of its
numerous variations not only do you have
to accept
the first paradox that zeno raises you
can also
discuss and develop it further to suit
your needs and that’s essentially what i
do
so before we get started or before we
delve deeper into these second or third
variations
i would like to discuss what the xenos
paradox is
and what idea it is trying to get at so
zeno’s paradox is best
represented by an analogy while zeno
uses a runner
i like to use a painter since i
coincidentally have a painting right
behind me
and i’m not in a museum of modern arts
i’m just in my dining room
so imagine there’s this painting a
painter is trying to draw a paint
painting well that’s what painters do so
in order to draw half the painting
you first have to draw a quarter of the
painting and before you could draw a
quarter of it
that’s to draw an eighth of its so on
add infinitum so as you can see
there there’s this potentially or this
divisions you could potentially divide
this so on
add infinitum towards the side you’re
starting to draw from
so you have half a quarter an eighth
and so on and for this infinite series i
would be saying
i’ll be referring to it as the zed
series because that would just be easier
named after a zeno so as you can see
this goes for
running for any action as well
before you move the entire thing you
have to move half of it and the z series
so what xeno tries to say that it’s
impossible for anyone to move
because in order to move they first have
to cross an actually infinite number of
series which is impossible so what are
some
preliminary responses to this argument
well the most normal one or the most
common one
is raised by aristotle in his book the
physics he basically writes
that this series is only a potential
infinite and not an actually infinite
series
however this is wrong because if you
look at the nature of the potential
infinite and the nature of the actual
infinite
what we do see is that while the
potential infinite is basically a number
or a series of numbers
going up one two three four five six or
any series like that
add infinitum the actually infinite
series
is an actually definite set of numbers
while
some of these actually infinite series
are denumerable and others are
non-denumerable
what we see isn’t a defined or an
actually infinite set of numbers which
already exist as a whole
so essentially how do we know something
is a potentially infinite or is
it an actually infinite this is what we
sometimes call a one-to-one
correspondence you see
whether one set could be put into
one-to-one correspondence with another
set if they can
then they’re equal in size for example
if you have five lobes of red and five
fish
you put one loaf of bread with another
fish and then the second
bread with a second fish third fourth
third and so on like that
so you can see that five loads of bread
have the same number of the fish because
they could put
be put into one to one correspondence so
as you can see if we
put the number of divisions in a
one-to-one correspondence with the
natural numbers which as
we’ve known or cantor discusses isn’t
actually infinite series you basically
have quite an interesting result
one of the natural numbers goes to one
in the
z series two to one half
three to one quarter and so on and into
an item there’s a one to one
correspondence so what we do see
is that with these pairs the zed series
is actually
an actually infinite series and that’s
what ben ardex writes in his
book or essay infinity so we can see
that this response that this uses a
potential infinite
it doesn’t really work so what else can
someone say in response to this
well ben ardet says well let’s imagine
time itself could also be divisible in
this set series
one minute could be divisible into one
half and a half
to a quarter and so on like the z series
now with that to mind it seems that at
face value there’s enough
points on the finite time to correspond
with the points on a finite line
so time the restriction of time on this
crossing a fine infinite or a finance
line
does not actually happen because you
actually have enough time
by one to one correspondence to actually
paint the entire painting or run the
entire distance
so what can we say to this i’ll raise
variation two of the zenos paradox
imagine you have a log and a
metaphysical knife which you cut your
exact precision to whatever you want to
cut
you have log you cut half a quarter like
the z series
now assuming that bernardette’s paradox
works it does seem that you will cut
towards the entire log and in
fact finish the entire log because you
actually have enough time to cut
from one side to another but that’s
clearly absurd if we took 10
they’ll always converge towards one side
but they’ll never actually cut or cross
the line
so i don’t think ben ardex response
works here
so how would i respond to the xenos
paradoxes because
we all know that we can move i can move
finite distances
and definitely i disagree with the
conclusion of this primary
xenos paradox that movement cannot
happen so
while it’ll be fallacious to argue that
just because i don’t like the conclusion
i have
that means that i could throw away the
arguments we have to respond to the
arguments
so what’s the best response to the
arguments in my opinion the best
response to these arguments
is the idea that these these
deconstructions
or these divisibilities do not actually
go for infinity
while you might like to say you can
conceptually devise them by infinity
it doesn’t mean that there’s physically
and actually infinite number of
divisions
why well it’s actually quite simple
because
imagine we have a painting here for
those chemists out there and i’m not
saying i’m a really good chemist but
we could see that this painting is
broken down into
paint molecules and those molecules into
elements and those elements into atoms
and they could be deconstructed into
these final quantum
particles like the quartz which cannot
be further deconstructed
into further physical elements so at the
grounding of everything there’s actually
these
indivisible things which although are
conceptually divisible
are not chemical or physically divisible
so we have
each of these indivisible things so in
fact there is no
infinite thing we have to transverse
traverse but there’s actually a finite
things we have to traverse
so would that defeat zeno’s paradox if
it does why did i raise it in the first
place
well that’s because i would like to
raise the final iteration of the final
variation of
zeno’s paradox imagine you live in a
universe well we all do live in a
universe and that universe is
according to the eighth year of the time
the dynamic theory of time the idea that
past
and future are real phenomena so
when we live in an infinite theory of
time we soon realize
that if the past was actually infinite
what while we age for example
on our first birthday or the 100th
birthday or thousandth birthday if we
lived that long
the actually infinite past would stay
exactly the same the universe will
always stay the same age
despite us aging within that universe
and that is i think
well quite absurd because how could us
age in the universe
in which the universe does not
mathematically age and you might say
well where am i getting this doesn’t age
from
well if we look at the kantour’s
infinity or his arithmetic what we do
see
is that a left null and actually
infinite number plus
n or any finite number is a left null so
any finite number when added to a in
transfinite number does not actually
change so we can see that while we are
aging in a
infinite universe the infinite universe
doesn’t actually age so that does seem
to be absurd
and furthermore imagine we have an
infinite space around us an actual
infinite side
to one side and actually infinite all
around us whether
we move right or left our coordinate
will always be the same because when we
look at coordinates it’s relative to a
different point for example
uh point five seven on a
graph would just be 5 from the x and
7 from the y so what we can see
is that each points or our coordinates
are developed or based on the idea of
finite spaces
so how does bernard respond to this or
this development of the xenos paradox
well basically what we do see is that
then our debt suggests that the entire
world
although actually infinite can be
divisible into finite places there are
finite
spaces in in every part of a transfinite
plane
but then that raises a problem since we
cannot add up to infinity
it’s absurd to suggest that the entirety
of trans-finite
spaces are built out of finite parts
because if all parts of
the finite or the universe are finite
it follows logically that in totality of
the universe
is finite because if you have finite
numbers and you add to them
you will always get a finite number so
it seems that zeno’s paradox or this
final variation of xenos paradox
does seem to defeat the idea of an
actually infinite past
or an actually infinite surroundings but
what is the grim reaper’s paradox
diaries
well the grim reaper’s paradox is
essentially the idea that
imagine there’s a guy at fred i’m sorry
her name’s fred because fred well kind
of gets killed a lot in this paradox
imagine a sentence if fred’s dead
or if fred’s alive at 12 then a grim
reaper will kill him
if fred’s alive at 11 30 a grim reaper
would spawn and kill him
so as you can see what’s happening here
is that from 11 to 12
there has to be a time where the grim
reaper kills him as the z
series decreases either a half an hour
past
12 a quarter of an hour past twelve an
eighth of an hour past 12
a gram will kill him if he’s not already
dead so we have this
actually infinite series this said
series of
grim reapers spawning in to kill fred
ever since
the bell struck at 11. so
soon if we follow the series we have to
ask ourselves a few questions
well is fred alive or dead at the end of
the series well the answer is actually
quite weird
he has to be dead because if he wasn’t
dead something would have killed him
but at the same time nothing actually
had killed him because before each grim
reaper could kill him
something before him must kill fred
already so there’s actually no definite
grim reaper which kills fred
so what do we conclude from this i think
we could conclude that
it’s impossible for us to have an
actually infinite series of causes
because if there was an actually
infinite series of causes there will not
actually be any
definite explanation for anything in the
world around us or
anything at all so the series of causes
in the past
has to be finite so now that we’ve
concluded that
causation has to be finite space has to
be finer and time has to be finite
what implications do they have on the
world around us
well i wouldn’t talk too much about this
unless i violate the
rules of tedx and i will start fearing
into the religious agenda thing
but if we accept this arguments that
i’ve raised
we soon realized that we were faced with
uh croatia
or uh existence x nilo
the universe came out of nothing because
time or space cannot be
infinite and hence there must be a
beginning to space so
as the irreligious talk this is i’ll
just leave you like that
you can find your own conclusions but
we’re all faced with
a creation out of nothing i hope that
you’ve liked everything
and i hope you’ve enjoyed this ted talk
hope you found it informative
if you want to learn more about the
actual infinity feel free to do
any more research for yourself
i hope you have a good week stay safe
and thank you