Veritasium recently released a very good video that explains Gödels incompletenes theorem such that even I could understand it: https://www.youtube.com/watch?v=HeQX2HjkcNo
https://www.youtube.com/watch?v=nsZsd5qtbo4
[1] https://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/081...
Edit : removed a line
One thing I'd like to see in a companion video would be an explanation of why Turing machines represent computation. That video (like many others) skims over the why, and only talks about what they can/can't do after we've already decided to use them.
Turing's 1936 "On Computable Numbers" paper gives a nice philosophical justification for his model, which (from my reading) boils down to the following:
- Mathematicians can do their work within a finite volume of space (their brain, or a room, or the whole Earth if we want to be conservative). Also, they could (in principle) do their work using only written communication, on standard pieces of paper, each of which also has a finite volume.
- Finite volumes can only have finitely-many distinguishable states (having infinitely many states would require them to be infinitesimally similar, and hence indistinguishable from each other within a finite amount of time)
- Hence we can label every distinguishable state of a brain with a number; and likewise for every distinguishable piece of paper (at least in principle)
- Since these are both finite, any mathematician's behaviour could (in principle) be completely described by a table detailing how one state (brain + paper) leads to another
- Given such a table, the actual content of the states (e.g. the wavefunctions of particular protons, the electrical potential of particular synapses, the placement of ink molecules, etc.) is irrelevant for the behaviour; only the transitions from one numbered state to another matter
- Hence we could (in principle) build a machine with the same number of states as one of these tables, and the same transitions between the states, and it would exactly reproduce the behaviour of the mathematician
This is the philosophical justification for why a (Turing) machine can calculate anything a human can (in fact, the same argument shows that a Turing machine can reproduce the behaviour of any physical system).
However, this is still a rather hand-wavey "in principle" thought experiment about unimaginably huge numbers. Turing managed to take it further.
For simplicity we assume all the papers are arranged in a sequential "tape", we'll call the distinguishable states of the papers "symbols" and those of the mathematician/machine "states":
- One thing a mathematician can do is read a tape with one of these tables written on it, followed by a sequence of numbers representing the symbols of another tape, and emulate what the described machine would do when given the described tape (i.e. they could keep track of the current state and tape position, and look up the transitions in the table, to see what would happen)
- Since a mathematician can emulate any given table (in principle), so can a machine. This would be a "universal machine", able to emulate any other. (The video talks about such a machine, in the proof that the halting problem is undecidable)
So the question becomes: how unfathomably complicated would such a universal machine have to be?
- These transition tables and tapes can be very big, and may contain very large numbers, but we can write them down using only a small alphabet of symbols, e.g. "start table", "new row", "the digit 7", etc.
- Reading a sequence of such symbols, and emulating the described machine, can get tricky. Turing described a universal machine "U", but he did so in a rather indirect way, which also turned out to have some mistakes and glaring inefficiencies. Davies later worked through these and ended up with an explicit machine using only 147 states and 295 symbols.
Hence we can use a machine with only a few hundred states to exactly reproduce the behaviour of any mathematician (or any physical system), as long as it's given an appropriate description (i.e. "software"). Later work has found universal Turing machines with only 4 states and 6 symbols.
One reason Turing's justification for his model is important, rather than just proposing the model and seeing what happens (like in the video), is that Alonzo Church had already proposed a model of computation (called Lambda Calculus), but didn't have such a justification.
Gödel himself dismissed Lambda Calculus, proposing his own system (General Recursive Functions) as a better alternative. When Church proved they were equivalent, Gödel took that as reason to dismiss his own system too! Yet Turing's argument did convince Gödel that a fundamental limit on computation had been found. Turing proved his machines are also equivalent to Lambda Calculus, and hence General Recursive Functions; so all of these proposed models turned out to be 'correct', but it was only Turing who could explain why.
Personally I consider this reduction of physical behaviour to transition tables and then to machines, to be the main reason to care about Turing machines. Proving the undecidability of the halting problem was also a great achievement of Turing's 1936 paper (as shown in that video), but that can also be explained (I would argue more easily) using other systems like Lambda Calculus.
Without Turing's justification of his model, undecidability comes across as simply a flaw. Sure Turing machines may be (distantly) related to our laptops and phones, but if we found a better model without this 'undecidability bug' we could make better laptops and phones! Turing's argument shows that there is no better model (just equivalents, like Lambda Calculus).
I don't think this assertion follows. I don't think an argument like this can work without delving further into reasonable description of a "physical system".
If you just just use the mathematics we employ to describe physical systems unreservedly it is possible to construct "physical systems" that exhibit non-computable behavior. For instance you can have computable and continuous initial conditions to the wave equation that produces a solution that is non-computable. See : https://en.wikipedia.org/wiki/Computability_in_Analysis_and_...
I think it's important to emphasize that Turing stated his arguments in regards to "effective procedure" (which I see you mention in a different post). I don't think the substitution of "effective procedure" with "physical system" is justified.
Thanks, that's a really nice example which I hadn't come across before (or at least not spent too much time studying). I may have to refine the language I use in future; although a cursory look seems to be compatible with my own understanding (my mental model is roughly: "if we found a halting oracle, we would have no way to tell for sure")
> I think it's important to emphasize that Turing stated his arguments in regards to "effective procedure" (which I see you mention in a different post). I don't think the substitution of "effective procedure" with "physical system" is justified.
Yes, Turing did not say as much (at least in his 1936 paper). He was essentially abstracting over 'whatever it is that a person might be doing', in an incredibly general way. Others have since taken this idea and applied it more broadly.
Another useful caveat is that Turing machines are framed as (partial) functions over the Natural numbers. It's quite a leap from there to a "physical system". An obvious example is that no matter how cleverly we program a Turing machine, it cannot wash the dishes; although can simulate the washing of dishes to arbitrary precision, and it could wash dishes by controlling actuators if we decided to attach some (but even that would run into problems of time constraints; e.g. if calculating the first instruction took so long that the water had evaporated).
Whether or not a finite region can have an infinite number of states (countable or otherwise) is irrelevant; we can only distinguish finitely many in finite time. Two states being indistinguishable means they'll give rise to the same output behaviour (e.g. from a mathematician in a room, carrying out some procedure).
I'm not sure we can say this part even though we might believe it to be true. I mean maybe now we can since we have no deeper theories than quantum field theories, and we can posit all of physical reality reduces to QFTs and GR and simulate all of QFT computationally either with quantum computers (in real time) or classical (not in real time + need a random number generator). But the we'd still have computational limits by which observables we can measure and how big of computations we can run (think size of computer limited by black hole density for one crude example - although I guess a blackhole could be used as a quantum computer. So then I'd say the speed of light is the other main limit, we've already lost the information to run certain calculations to the depths of space).
And to focus back in on the random number generator requirement for classical to simulate quantum...nothing fundamentally random can be simulated classically in principle. Yet randomness is needed to describe the most basic layer of reality. I can't get true randomness from classical no matter what.
Turing's scenario essentially applies to a person in a room, with an unlimited amount of paper (this could be a movable tape extending through the walls, to prevent the room filling up with paper!).
However, the same argument works if we replace the person with any other physical system (e.g. some mechanical device, or some physics experiment), and replace the room with any finite region of space (e.g. the Milky Way galaxy).
I don't remember Turing himself making this argument (let alone in his 1936 paper), but it's known as the "physical Church-Turing thesis" https://en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis#V... (the Church-Turing thesis is the argument that all 'effective methods of calculation' can be carried out by a Turing machine)
As for your points about quantum physics, etc. that's certainly true, and it's why the Church-Turing thesis isn't a theorem (and can't be, since we can never know if any particular model of reality is correct). However, it's certainly a valid physical law, akin to nothing moving faster than light, etc. i.e. it's empirically true, theoretically sound, and we have no better explanation for the moment.
Perhaps its origin in mathematics, where it is a second-class citizen compared to provable theorems, prevented the Church-Turing thesis getting the recognition it deserves, e.g from physicists.
I believe that quantum computation is a richer more powerful theory. In essence BQP is a superset of BPP in your link.
I think this happens with all sorts of mathematical objects. Non commutative algebra is much richer than commutative algebra. There are also more interesting non commutative things than just deformations of classical objects. Its the classical commutative case that is the weird limit.
It's interesting to consider P = NP as a physical law, although that's certainly more tenuous than the Church-Turing thesis, etc.
Can you please flesh out the following? I don’t quite follow it..
> having infinitely many states would require them to be infinitesimally similar, and hence indistinguishable from each other within a finite amount of time
Let's consider the states which just contain a single electron, somewhere in the room (I'm not bothering with quantum effects, but we would do the same thing just in Hilbert space instead of 3D space). The region is finite, so the coordinates of this electron are bounded: if the boundaries are at, say, 0 and 100 in each axis, then putting the electron at position (101, 0, 0) would be the same state as the empty one we started with (since the electron is outside the region we care about, and there's nothing else).
Now let's say we do some measurement of this electron's position, which is only accurate to whole number coordinates. That gives us 100^3 = 1,000,000 distinguishable states with a single electron. We might imagine all sorts of different states 'in between', but they can't affect that measurement due to its limited resolution.
If we increase the resolution of our measurement by 10, we get 10^3 times as many distinguishable states; another factor of 10 gives 10^3 times as many states again; and so on. However, regardless of how finely we can resolve the electron's position, we can only distinguish between finitely-many states. Any differences finer than that limit are irrelevant to the measurement, and hence to any behaviour that depends on that measurement.
If we could resolve between infinitely-many positions for the electron, that would require distinguishing between states which are infinitesimally-close together, i.e. on an infinitely-fine grid; this would require distinguishing between two numbers whose decimals only differ after infinitely many digits. This doesn't seem like a reasonable capability.
The same principle applies when we have two electrons in the room, or trillions, or any combination of electrons, protons, photons, etc. in any arrangement we like.
A similar thing happens when we reach really large numbers too, e.g. trying to put as many particles in the region as we can: at some point the relative difference between, say, a googolplex and five particles versus a googolplex and six particles becomes too small to distinguish. There will eventually be a limit. (This is like the resolution problem, but instead of adding decimal places to the right of each number, we're adding factors of 10 to their left)
One way to get around this problem of limited resolution is to avoid an explicit 'measurement', and instead have the region itself behave differently for different states. A good example is chaotic behaviour, where tiny changes in an initial state will grow exponentially until those differences eventually become distinguishable. However, the infinitesimally-similar states described above will remain infinitesimally close for all finite lengths of time; in order to distinguish between them, we would need to wait an infinite amount of time.
So putting aside Plank and all the experimental stuff, even on a philosophical level there are problems with thinking of space as continuous. For, that with which we measure space is necessarily discrete and the time-frames in which we measure are necessarily finite. So, even if space was continuous, two ‘states’ that are infinitesimally similar could for ALL practical purposes be considered identical.
Is that it?
As a far less formal analogy, it's similar to how people complain that digital audio in general has lower quality than analogue, just from a discrete vs continuous standpoint. When in fact there's no limit to the quality of a digital representation (if we liked, we could store a gigabyte per sample, or a petabyte, or whatever). Yet we can always draw the line somewhere, beyond which there are no meaningful distinctions.
In fact, we only ever need 2 symbols, e.g. 0 and 1. This is because any set of symbols can be replaced by a set of numbers, and those numbers can be written in binary (with a fixed width, to avoid needing 'spacers'). This can require a lot more machine states, since those states allow the machine to 'remember' the bits it's read so far (e.g. if we're using 8 bits at a time, then we need enough states to "remember" what the first 7 bits were as we're reading the 8th).
For a calculation like pi, we can use the tape to (a) write down the digits (or bits) of pi, and (b) keep track of intermediate variables needed for the calculation. One simple approach to keeping track of intermediate variables is to store them in order on the tape, where each variable is a sequence of 1 symbols, separated by 0 symbols. For example a tape like: 111010010 represents four variables with values 3, 1, 0 and 1 (the number of 1s we see before we hit a 0). This actually corresponds to Peano arithmetic, if we use the symbols S and Z instead like SSSZSZZSZ.
An algorithm which spits out digits of pi forever (e.g. http://www.cs.ox.ac.uk/people/jeremy.gibbons/publications/sp... ) could work by moving its 'variables' section further and further to the right, allowing the digits of pi to accumulate, e.g.
Start
^
Initialise variables and output/variables separator
|VARIABLES
Move variables to make space for next digit
_|VARIABLES
Calculate and write next digit
3|VARIABLES
Move variables to make space for next digit
3_|VARIABLES
Calculate and write next digit
31|VARIABLES
and so onWhether the abstract infinite pi ‘exists’ is up for metaphysical debate :)
def pi_decimal_digits():
q, r, t, j = 1, 180, 60, 2
while True:
u, y = 3*(3*j+1)*(3*j+2), (q*(27*j-12)+5*r)//(5*t)
yield y
q, r, t, j = 10*q*j*(2*j-1), 10*u*(q*(5*j-2)+r-y*t), t*u, j+1
That said, one can perform all of arithmetic using that system and represent any property of numbers that could be represented otherwise.
Really? I measure it as exactly 200 bytes (not counting the 4-space indentation that HackerNews requires, and not including a trailing newline)
> and, more importantly, an infinite amount of time to produce pi to infinite precision
I can see it right now, with infinite precision. If it's taking an infinite amount of time to show up I'd contact your ISP ;)
There it is, in all it's infinite splendor.
in geometry notation :)
Maybe we will make progress in our understanding this millennia :)
Note that quantum computers are no more powerful than Turing machines; although it's thought they might be faster at certain tasks. For example, we know how to factorise integers quickly using Shor's algorithm on a quantum computer, but we don't yet know a quick algorithm for factorising integers on a classical computers. We can still factorise integers on a classical computers: we can do it with a non-quantum algorithm, or we can even do it by emulating a quantum computer and using that emulator to run Shor's algorithm (in this case the emulation is so slow that it exactly cancels out the speedup from Shor's algorithm).
Backpropagation through time is also a pretty simple algorithm, which is already implemented on classical computers.
1) Where does the randomness of QM come in though? You need a random number generator alongside your Turing machine.
We coud debate that one can never prove true randomness as it may always be due to some deeper determinism. But if we take the wavefunction of QM as complete, which Many Worlds does - the most compsci-like interpretation - then randomness is fundamental according to theory.
I can have all the TM's I want, but I still need this extra component. Just like the human Turing machine could have all the pen and paper and math in the entire universe and beyond, a human can't produce a truly random number (or can we, then we get into a debate about true free will and whether free will can produce a "random" number).
2) Let's say we instead use quantum computation instead of classical. There are still physical systems we can't "compute", like the state of the entire observable universe, the state of the entire universe, or anything requiring information that has forever left our light cone. I know TM's and quantum versions of TM's are infinite, but they can't bypass the speed of light. So we need some softer version of saying all of physical reality is computable imo.
MWI is an entirely deterministic interpretation of quantum mechanics. Absolutely no randomness is involved which is one of its most appealing properties.
That's why I said it's up to your own preference.
It's interesting that you mention a many-worlds approach, since that doesn't actually involve any randomness. Rather than choosing a single result probabilistically, many-worlds keeps them all. This can be represented quite nicely with what I referred to above as 'nondeterministic functions', i.e. functions which may return multiple values (in some sense, those values are literally the "worlds"). This would look like a logic language a la Prolog, but wouldn't be restricted to depth-first traversal: it could run breadth-first (very slow), concurrently (nice in principle, but would swamp any finite machine with overheads), or iterative deepening (probably the nicest implementation IMHO). Haskell programmers would call this a "breadth-first list monad" (AKA LogicT, AKA Omega). Instead of "returning multiple values", we could also do a continuation-passing transform and invoke the continuation multiple times (concurrently).
This would run exponentially slowly, but that's a whole separate issue ;)
> We coud debate that one can never prove true randomness as it may always be due to some deeper determinism.
We could get a Copenhagen-like approach by doing the same as above, but only returning one of the results, chosen "at random" according to the square of the wavefunction (plus some hand-wavey Copenhagen-craziness, like defining whether or not Wigner's Friend is an 'observer')
When it comes to "randomness" my own approach is to mentally substitute the word "unpredictable", since it tends to be more enlightening. Turing machines can certainly produce results which are "unpredictable", in the sense that the only way to know what will happen is to run the program; in which case that's more like a (repeatable) observation rather than a prediction. (BTW I also think that's a nice resolution of the "free will" paradox: my behaviour may be predicted by observing what an exact copy of me does; but it was still "decided by me", since that copy essentially is "me")
In any case, the outcome of a Turing machine using some pseudorandom algorithm is indistinguishable from "quantum randomness". There is certainly a limit to the unpredictability of any pseudorandom events (given by its Kolmogorov Complexity, which is upper-bounded by the Turing machine's program), but on the other hand I think it's quite presumptuous to think "quantum randomness" has infinite Kolmogorov Complexity.
> There are still physical systems we can't "compute", like the state of the entire observable universe, the state of the entire universe, or anything requiring information that has forever left our light cone.
That's a very different question, since it involves how we choose what to put on the tape, rather than what a Turing machine is capable of computing. It's like criticising a genie for being unable to grant a wish, when in fact it's we who can't think of what to ask for.
Besides this, there is a way for a Turing machine to compute the whole universe, including what's outside our visible lightcone: we simply run every program. That's actually a pretty straightforward task (and remarkably efficient), e.g. see the FAST algorithm in section 9 of https://people.idsia.ch/~juergen/fastestuniverse.pdf
Note that if we run such a program, there is no way to know which of the outputs corresponds to our universe. However, our ignorance of where to look does not imply that such a Turing machine has not computed it!
This doesn’t seem true since all predictions are ‘running the program’
When a system is Turing-complete, equations like this will inevitably involve some expression with t-1; and solving that involves some expression with t-2; and so on back to the initial condition. Hence there's no 'shortcut' (this is a consequence of Rice's theorem BTW, which itself is a consequence of the halting problem).
This seems to still involves ‘a program’ in that there are foundational mathematical principals that build your confidence/logic that the periodicity of sin(x) is so well defined that any even integer multiple of pi will be zero. e.g. you could write out the logical ‘programatic’ steps that would also teach a math newbie how to make the conclusion you just made about the sin() function.
What's interesting about systems like Turing machines is that the "program" is a specific, isolated thing like a tape of symbols. In contrast, those "foundational mathematical principles" you mention are more of a diffuse web, with lots of interchangable pieces, all mixed up with unrelated facts. This gives lots of different ways to arrive at the answer. With Turing-complete systems, we always have to come back to the code.
But computers with self-consistent time travel are more powerful. It's perhaps physically possible to construct one of those, depending on whether we ever end up with a more-correct timeless theory of physics that makes different predictions to existing physics.
> Backpropagation through time is also a pretty simple algorithm, which is already implemented on classical computers.
I read “backprop through time” as meaning that you have the result of the backprop at the beginning of the process – that's more powerful, if we're using continuous computation. (With discrete computation, you're right that it's no more powerful.)
performed by a Nondeterministic Turing Machine.
See the following for an example:
Recent advances are compatible with a long tradition with
contributions by Euclid, Dedekind, Frege, Russell, Church, Turing, von Neumann, etc.
The following article is in furtherance of the long tradition:
Can you suggest any improvements?
nondeterminism. That is, for a given input a Nondeterministic
Turing Machine can only explore the number of alternatives
bounded by its input. Actors do not have the limitation.
expressions. See proof in the following:
determined by initial input for the number of possibilities
that a system can explore given that it must always come
back with an answer. A Nondeterministic Turing Machine has
the property of bounded nondeterminism because it starts in
a global state and from each state there are finitely many
possible successor states. Because the machine must always
produce an answer, each path must be finite and consequently
the total number of states that can be explored is finite.
In the development of a theory of concurrent computation,
Dijkstra remained committed to a global state model of
computation. This commitment gave rise to the “unbounded
nondeterminism” controversy. Digital systems can be
physically indeterminate by making use of arbiters in order
to resolve reception order in communication between systems.
Because of physical indeterminacy, a digital system can have
the property of unbounded nondeterminism in that no bound
exists when it is started on the number of possibilities
that an always-responding digital system can explore.
Consequently, there are digital computations that cannot be
performed by a nondeterministic Turing Machine. Being
restricted to bounded nondeterminism is one the reasons that
the lambda calculus and Turing Machines models are
inadequate for digital computation.
Nondeterministic Turing Machine has only bounded
nondeterminism. That is, for a given input a Nondeterministic
Turing Machine can only explore the number of alternatives
bounded by its input. Actors do not have the limitation.
He built up the whole damn video for that one moment, when he says "And godel went through all this trouble to come up with this card..." and flips over the card. But how godel did it is at least as cool as the proof itself.
I'm mostly being tongue in cheek with the Veritasium nose-thumbing. But if you saw it and want to understand it more deeply (and in my opinion more satisfyingly), I recommend "What Godel Discovered": https://news.ycombinator.com/item?id=25115746
It explains Godel's proof using Clojure. Quite accurately, too.
The thread is also hilarious. Shoutout to the MIT professor who took offense at someone linking to his wikipedia page: "Wikipedia is contentious and often potentially libelous." ... and then you check his wikipedia page, and it's a super positive article about all of his discoveries and life's work.
Carl Hewitt is no doubt a famed and accomplished professor, but why does nobody point out that he is claiming that Gödel’s proof is incorrect[0]?
I already heard of his conviction that “The Church/Turing Thesis is that a nondeterministic Turing Machine can perform any computation. […] [T]he Thesis is false because there are digital computations that cannot be performed by a nondeterministic Turing machine.”[1]. (Which, to be fair, is a strongly related argument to the one that Gödel was wrong.)
I don’t personally know whether he is right, and his digression about millions of people risking death as a result does not inspire confidence, but in my understanding, neither statement is widely accepted.
[0] https://news.ycombinator.com/item?id=25160970
[1] https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3603021
It would be helpful if you added some links or some explanation, or just anything beyond what you've written there.
orders on propositions play in maintaining the consistency
of foundations.
Neglecting to include the order of a proposition in in its
Gödel number makes it possible to mistakenly conclude that
the proposition I'mUnprovable exists as a fixed point of a
recursive definition.
See the following for details:
Each proposition is assigned to an increasing prime power, and the increasing list of primes has total order, such that swapping propositions yields a distinct Gödel number.
His claim is that the proposition "This proposition is not provable", formalized as "P equivalent to P is not provable" is not well formed in a typed logic, as "P"'s type is first-order, while "P is not provable"'s type is second-order. Therefore, his claim is that the proposition is simply not well-typed and therefore not interesting. Godel's proofs were discussing an untyped logic, but according to Hewitt that is not an accurate representation of mathematics.
I don't think anyone in the space agrees with him, though, as far as I could tell from some cursory reading.
mathematics with orders on propositions.
The [Gödel 1931] proposition I'mUnprovable is certainly of
historical interest and may perhaps be of philosophical
interest.
However, including the proposition I'mUnprovable in
foundations makes the foundations inconsistent.
Gödel number but did not include the order of the proposition.
then they have the same Gödel numbers.
Unfortunately, Gödel number of a proposition leaves out the order of the proposition :-(
same characters in the same order.
(Of course there can theoretically be bugs in a computer verified proof. If there are bugs in the verifier. But if 4 independent proof assistants verify a proof of a theorem, this is extremely unlikely.)
All of the computer-verified proofs of existence of [Gödel
1931] proposition I'mUnprovable left out the order of
proposition in its Gödel number :-(
Of course, on actual computers, Turing's result applies, Rice's Theorem manifests, and such a program will likely run indefinitely, until we get bored and give up.
The point that Hewitt has made in another thread is typical misdirection. See, it doesn't matter which order of logic we're talking about; in general, we have results for higher order logic, all the way up to Lawvere's fixed-point theorem for any Cartesian closed category. To fight this, Hewitt has to disclaim the entirety of 20th-century mathematics. (At least he's internally consistent -- he does deny modern maths!)
Sad fact is that, ever since 1976 (http://www.laputan.org/pub/papers/aim-353.pdf) folks have recognized that Hewitt's claims about the foundations of computation are not just wrong, but flamebait. But he never had the humility to learn from his mistakes.
More interesting facts:
His Twitter account has been relatively recently deleted
https://mobile.twitter.com/drcarlhewitt/status/3972794314824...
And his PhD advisor was Seymour Papert.
https://professorhewitt.blogspot.com/
See following for a simple ActorScript program that cannot be
implemented using a Nondeterministic Turing Machine:
I think I understand why, but could you provide some more details?
Maybe there can be physical machines which run ActorScript, but I would expect them to come with new laws of physics, and also I think that there are probably more comfortable programming languages. We already have https://en.wikipedia.org/wiki/Constraint_Handling_Rules for example.
more here: https://www.erlang-solutions.com/blog/lets-talkconcurrency-p... and here: https://codesync.global/media/almost-actors-comparing-pony-l...
See discussion in Related Work section of the following article:
contribute to the the discussion?
proper accountability for material that it publishes.
Furthermore, Wikipedia is heavily censored by an anonymous group.
See [von Plato 2018] on how Gödel deceived von Neumann in
developing one of his fundamental proofs in [Gödel 1931].
I should post the wiki link so that we can have the whole discussion again :)
I'll admit that threw me for a loop for a second.