at twenty four Godel proved
that math has a wall you can never climb over
September 1930
at a math conference in Konigsberg Germany
the big names on stage were fiercely debating one grand question
are the foundations of mathematics actually solid
is it possible with one perfect axiom system
to derive every mathematical truth
during the roundtable on the third day
a slight young man stood up
and spoke for less than a minute
almost nobody in the room understood him
but those few sentences
later turned out to be the most important remark
in the intellectual history of the twentieth century
the young man was Kurt Godel
he was twenty four that year
in that same city
the next day
David Hilbert the pope of mathematics
gave his retirement speech at another conference
he ended by declaring we must know
we will know
that line was later carved on his gravestone
what he did not know
was that just one day earlier
the young man a few streets away
had already proved that declaration could never come true
hello
hi everyone
I am Leo Wang
today I want to spend a whole episode
taking you inside the Godel incompleteness theorems
we have mentioned these theorems more than once before
and people keep asking in the comments where they actually apply
some even bring up self reference and consciousness
but every time we only brushed past it
never really going inside to see what the proof does
today
I am paying that debt
one warning first
this episode is denser than usual
because I do not want to stop at the textbook line
that a system cannot prove its own consistency
I want you to see clearly
what Godel actually did
and why it still shakes
mathematics philosophy and artificial intelligence today
alright
let us start with the background
from the late nineteenth into the early twentieth century
mathematics went through an unprecedented crisis of trust
it started with set theory
set theory was meant to give all of math one unified foundation
then in 1901
the British mathematician Russell found a paradox
you may have heard it a barber announces
that he has exactly one rule
he shaves everyone who does not shave himself
here is the problem
does he shave himself
if he does
then he is one of the people who shave themselves
so by the rule he must not shave himself
if he does not
then he is one of the people who do not shave themselves
so by the rule he must shave himself
this is not a riddle
Russell proved
that the loosely defined set theory of the time
contained exactly this self contradicting structure
in other words
there was a crack in the foundation of mathematics
the whole field erupted
then Hilbert stepped forward
with a grand repair plan
later called the Hilbert program
three goals complete consistent decidable
what do they mean
complete
every statement can be proved true or false inside the axioms
with no blind spots
consistent
the system never derives a contradiction
it cannot say one plus one is two
and also say one plus one is not two
decidable
there is a mechanical procedure
that decides whether any statement is true or false
Hilbert believed all three would be achieved
in his Konigsberg speech he insisted that for us
there is no such thing as the unknowable
and he closed with that ringing line we must know
we will know
but the words had barely left his mouth
when Godel arrived
in 1931
Godel published a paper
the title translates as
on formally undecidable propositions
of Principia Mathematica and related systems
it was only about twenty pages
and it completely ended the dream of Hilbert
Godel proved two things
the first incompleteness theorem any consistent formal system strong enough
to describe arithmetic with axioms you can list one by one
contains statements that cannot be proved true
and cannot be proved false inside it
in plain words
if your math system is consistent
then it must be incomplete
there will always be true things
you can never prove
the second incompleteness theorem
such a system cannot prove its own consistency from inside
in plain words
math cannot vouch for itself
you want a health check on a math system
sorry
the report has to come from outside the system
these two results shattered the first two goals of the Hilbert program
Turing and Church later shattered the third from another angle
but that is another story
now here is the key part
how exactly did Godel prove it
this is what most popular explanations skip
and it is the most brilliant and elegant part of the whole thing
the core weapon of Godel was self reference
the proof has three steps
step one is Godel numbering
Godel invented a method
to translate mathematical language itself into natural numbers
every symbol every formula every proof sequence
gets a unique number of its own
like assigning a code to every Chinese character
think about how clever that is
before this
mathematics and talk about mathematics
lived on two completely different levels
math studies numbers
metamathematics studies math
the numbering merged those two levels
numbers became both the object of study
and the language describing it
suddenly math could talk about itself
step two is building the Godel sentence
using that numbering
Godel built a special arithmetic statement
translated into plain language
it says this statement is not provable in this system
notice
this is not vague philosophy
it is a rigorous arithmetic statement
you can write it in exact mathematical symbols
the logic resembles this sentence is false
but it is fundamentally different
this sentence is false gives you a paradox a dead loop
the Godel sentence does not give a paradox
it gives incompleteness
why
follow me through it
if the statement is provable
then what it says is not provable is false
so the system proved something false
the system is inconsistent
a contradiction
if the statement is not provable
then what it says is not provable is true
a true statement that cannot be proved
so the system is incomplete
two roads
which one do you take
if you want to keep the system consistent
you have to accept that the statement is true
but unprovable
truth is always one size bigger than your system
step three is the diagonal argument
the name comes from how Cantor proved the reals are uncountable
Godel moved that trick to the level of logic
making the number of a statement point back at itself
think of a snake biting its own tail
but the bite does not produce a contradiction
it produces a gap
a gap the system can never close from inside
every time I think about this proof
I feel a kind of aesthetic shock
a twenty four year old
used the language of mathematics itself
to make mathematics confess its own limits
this is not an attack from outside
it is mathematics exposing itself from within
like a mirror suddenly seeing the mirror
and finding a corner it can never reflect
alright
that is the core math
now why does this theorem still shake us
the first shock is philosophical
the Godel theorems reveal something deep
in any sufficiently complex formal system
true and provable are not the same thing
the territory of truth
is always larger than what proof can cover
this reminds me of Wittgenstein
we made a video about him before
about how the limits of language shape the world you can think
Wittgenstein said
the limits of language are the limits of the world
what cannot be spoken
must be passed over in silence
Godel said something similar from another angle
but he went further
Wittgenstein said what cannot be spoken
stay silent about it
Godel pointed elsewhere the unspeakable not only exists
it is also true
not being able to say it
does not mean it is not there
the silent part is not a void
it is real
the second shock comes from the British physicist Roger Penrose
in 1989 he published The Emperors New Mind
and in 1994 Shadows of the Mind
the books make a bold argument
human consciousness cannot be pure computation
the chain of reasoning goes like this first
the Godel theorems prove
every consistent formal system has true statements it cannot prove
second
a computer is essentially a formal system
every step of a Turing machine encodes as a formal rule
third
but human mathematicians can see that the Godel sentence is true
we understand what it says
and we know it is true
even though the formal system cannot derive it
fourth
so human mathematical intuition exceeds any formal system
fifth
conclusion the human mind is not a Turing machine
consciousness is not computation
this argument caused huge waves in academia
supporters think Penrose hit the nerve
critics think he got step three wrong
the strongest objection says
human mathematicians do not see the Godel sentence as true
from inside that particular formal system
we actually jump into a bigger system to make the call
in other words
your seeing may itself depend on
some higher level of formal reasoning
if that is so
that higher system is bound by the Godel theorems too
it has true statements it cannot prove
you can jump up one more level
but with every jump
a new blind spot appears
so the critics say
humans do not transcend all formal systems
we just keep hopping between systems at different levels
and every level has its own ceiling
does this structure sound familiar
the moment I heard it
a Buddhist idea jumped into my head
in the Diamond Sutra
Subhuti tells the Buddha that the dharma the Tathagata teaches
cannot be grasped and cannot be spoken
the Buddha used language to teach the dharma
and at the same time told you
that what language says is not the dharma itself
you can treat language as a raft
once you reach the far shore you put it down
but standing there after dropping the raft
you still use some subtler framework to understand the world
and that framework itself
is it not just another raft
is that not exactly the shape of the Godel theorems
every system has a blind spot
you jump out and stand in a higher system
and you can see the old blind spot
but where you stand now has a new one
the finger pointing at the moon
is not the moon
you drop the finger and look at the moon
but you look with your eyes
the eye is also a finger
close your eyes and feel with the heart
the heart is also a finger
so when do you ever reach the real moon
maybe when every finger has been put down
but the very idea of putting down every finger
is that not one more finger
you see
this is the deep structure of the Godel theorems projected onto practice
each layer of awareness lights up the attachment of the layer below
and each layer of awareness can become a new attachment
unless there is an ultimate vantage point
one that sits inside no system at all
Buddhists call it awareness
Laozi calls it the Tao
physicists are still looking for it
back to AI
large language models today
are statistical inference systems in very high dimensional parameter space
they are stunningly powerful
but in the end they run on Turing machines
they are formal systems
if Penrose is right
then AI can never have
that power to step outside a system and see truth
but what if the critics of Penrose are right
what if human seeing is just a higher form of computation
and we simply do not understand the mechanism yet
then one day AI might make the same jump
just in a different way from us
honestly
I do not have a settled answer to this
but I do have one observation
we discussed a hypothesis before
consciousness may not be a byproduct of the brain
but a fundamental property of the universe
the brain is a receiver not a generator
if that holds
everything changes
consciousness is not produced by computation
computation happens inside consciousness
the ocean is not produced by the waves
the ocean is what the waves rest on
if that is true
then Godel tells us more than formal systems have limits
any system trying to fully understand itself from within
is doomed to have blind spots
because the one who truly understands
always has to stand outside the system
which reminds me of a line practitioners use you are not your thoughts
you are the one watching the thoughts
thoughts are computation inside the system
the observer is outside it
but if you cling to being the observer
then the observer becomes a new system
with its own Godel sentence
and its own blind spot it cannot see
so Zen keeps one more layer of caution even the word awareness
can become a new prison
it is one more raft you have to put down
a viewer asked a wonderful question in the comments the Sartre line
existence precedes essence the Buddhist idea of no self
and the Godel incompleteness theorems
how are these three related
I think from three completely different directions
they point at one truth every fixed definition is incomplete
Sartre says
you are not predefined by some essence
you exist first
then you define yourself through your choices
but you can never finish that definition
because you keep choosing
definition never catches up with existence
Buddhism says
the self is not a fixed entity
it is a flowing process of conditions coming together
clinging to a frozen definition of self
that clinging is the source of suffering
Godel says
any formal system strong enough to describe itself
cannot complete a full definition of itself from inside
there are always truths about itself
the system can never speak out loud
three people
three roads
ending up on the edge of the same cliff
what is below the cliff
the thing no language no logic no formal system can reach
you can call it the unspeakable
call it emptiness
or call it the Tao
the name does not matter
what matters is that it is there
and that it is true
back to the story we started with
at that 1930 conference
the only person who chased him down for details
was von Neumann
he pulled Godel aside and talked with him for a long time
a little over two months later
von Neumann independently derived the second incompleteness theorem
and excitedly wrote to Godel about it
only to find Godel had proved it long before
and Hilbert himself
was probably the last to grasp what it meant
the tower he spent a lifetime building
had its foundation blown up by a paper from a twenty four year old
Godel later went to the Institute for Advanced Study at Princeton
where he became the closest friend of Einstein
they say the two walked to work together every day
one studying the limits of time and space
the other the limits of logic and proof
maybe the two people on earth who best understood the loneliness of the other
but the ending for Godel was bleak
in his later years he suffered severe paranoia
convinced someone was trying to poison him
he ate only food cooked by his wife Adele
in late 1977 Adele was hospitalized with a stroke
and Godel almost stopped eating altogether
on January 14 1978
Kurt Godel died in Princeton
weighing only twenty nine kilograms
the cause was malnutrition and starvation
a man who used reason to light up the limits of reason
was in the end imprisoned inside a system by his own fear
you can almost read his fate as a metaphor for his own theorem
he could see the gap in formal systems
but not the gap in his own mind
a system cannot save itself from inside
but the proof he left behind
is still shining today
because it does not say reason is useless
it says reason has a boundary
and beyond that boundary there is still truth
being able to see the boundary
is already a kind of transcendence
that light
and the words of the Buddha even the dharma must be let go
how much more what is not the dharma
and the words of Laozi the Tao that can be spoken
is not the eternal Tao
across two thousand five hundred years
all shine on the same wall
what do you think
is on the other side of that wall
I am Leo Wang
see you next episode