Відео

What! You can't tease an O(n log(h)) algorithm!

not sure why in Query, if you skip the right node in a To, parent will always catch it.. what it loop terminates before that? I'm so stupid

Thank you for clear explanation.

awesome explanation to give the intuition, now I just have to think about finding the right pile in logn for each card, and nice shirt btw!

God bless you zen programmer this information is super useful

Clear like crystal.

orz

Awesome explanation! Thanks

Why we are defining such function which can not give all possiable Gödel numbers ..I'm talking about negation function you defined 10:20 ..F j(n) => ~P(F n(n))..this function checks provability of statement of diagonal Gödel number only ...Or I'm still not getting please explain .?????🤯🤯

The best possible explanation of both algorithms; The clear and exact visual representation of the processes, made it easier to understand. The explanation was quite precise and to the point.

Que gran explicación muy fácil de entender

amazing video!

What about if result is negative after subtract value of i from target?

Best explanation of fenvik tree I have seen

Excellent. This is by far and away the best exposition of Godel's work I have come across. I appreciate the programming language approach at the end. I believe this was how Alan Turing came at the subject.

Your content was great; it's too bad you stopped, but thanks for the stuff that you did share with the rest of us.

Other than the mistake with Pr(s) being a predicate instead of a set of Godel numbers, the first proof is incorrect in defining F_i(n) as functions that generate natural numbers. It needs to generate statements since the input A of P(A) is a statement not a natural number, otherwise P(F_n(n)) wouldn't make sense.

🫡

At 13:07 : we just care about "valid" functions, requiring that their character string is of finite length. Defining Q(n) as F_n(n), its character string would need to include the character string for the formula of each F_i(n) of which there are infinitely many. Doesn't this mean that Q(n) is of *_infinite length_* and hence not a "valid" function? Q(n) would thus not be a contradiction to the original assumption that it is possible to list all "valid" functions, i.e., list all possible formulas with one free variable?

This is what I have been looking for

lovely video, thank you so much!

nice vid bro, but switch the shirt

4:14 so cute

This content deserves millions of subs, I hope you make more videos soon!

wow i learned something from this video

Your 2nd proof is quite interesting (my background is also CS though I love math). I think, though, that it's a bit off, or that your conclusion is. The problem I'd assumption/criteria #4 for the functions/programs. You can't know for all programs which do or don't halt (I'm pretty sure you know that.) So you absolutely CAN make a sequential list of programs that compile (setting aside my spidy sense that certain programs probably are valid but don't compile because they would require the compiler to solve the halting problem...). What you can't do is write a generic program that diagonalizes across those programs or arbitrarily simulates any of them and then ads 1 AND always gives an output, because you still haven't solved the halting problem. I think you basically recreated a proof of the halting problem, rather then proving that the programs aren't innumerable. The set of all valid programs is clearly a proper subset of the set of all strings (interpreted as programs), and the set of all strings is clearly of the same cardinality as the rational, which Cantor showed to be numerable (countable) by the other kind of diagonalization. Let me know if you think I misunderstood you. Thanks.

3:25 I thought we also knew that no system could prove its own consistency?

2:05-2:25 I don't like how vague these words are and how there seems to be no consensus on what these words actually mean.

Every second of this 9 mins video is worth watching. You sir really are a talented teacher, showing the source code makes the concept so much easier to comprehend.

10:01 Isn't the set of sequences here uncountable, i.e. there exists no such enumeration? If so, that would be a big issue in this proof idea...

You are the goat

Amazing explanation, you are a god

how come the point (13,3) is closer to (9,4) than (8,7)? and because of that we never traverse to the point (10,2), why does this happen?

Really nice explanation, thank you!

My question is: When you call A = for example, F1(n) = n+1, g(A) is the Gödel number of this, and there's a proof of F1(n) that's for example, Fn(n) = B, and g(B) = p, that's the proof of statement A. When you call P(A), this is the proof of statement A, that's p (g(B) or is ANOTHER proposition that's "A is provable"? Because both can be associated with a Gödel number and, if there's a number for both of them, there's a formula Fj(n)...

FINALLY,!!!! i understand this :D, this video helpme a lot, thanks : )

I really do appreciate that you described the theorem rigorously and didn’t “dumb it down” , while maintaining attraction.

With luck and more power to you.۔ hoping for more videos.

My solution involved bitmasks. I think that can bring it down to constant time, but I may be overlooking something

Not that kind of Hash

the actual Goat!!!!

I believe all proofs of Godel's FIrst Incompleteness Theorem are invalid, but the theorem itself is true. Self-contradictory statements shouldn't be allowed (and aren't even a normal feature of theorems we write in math). Think about all the hand-waving these arguments have. And in Godel numbering, attempting to encode "This statement is not provable" wouldn't be so easy. There is no "this statement" symbol, so to encode this concept in the Nth statement, we'd have Statement(N): "Statement(N) is not provable". Encoding this would create a large number much greater than N, which would result in this NOT being the Nth valid statement. So the Godel numbering itself would prevent against self-referentiality, unless the arithmetic specifically had a symbol for encoding the concept "this statement". So for any system, you could add a "this statement" symbol to its Godel numbering, but by doing so you'd be making your system 'weird', in that most mathematical systems don't bother with self-referentiality. Basically, I'm conjecturing that Godel's First Incompleteness Theorem is "incomplete" (unproven) for non-self-referential systems. (Meaning, it is possibly true, but hasn't been proven.) Put another way, Godel's First Incompleteness Theorem is the Tyler Durden of mathematics. This is still the best video I've seen on the topic so far.

Eagerly awaiting your Second Incompleteness Theorem video!!

You are a fantastic educator. If you've ever watched competitive programming streams where they explain the answers you'll know being good at something and being able to explain something are two very different things. You have done the community quite a service in bringing such clear explanations!

Thank you, I loved this video. It was short and simple.

isso é sensacional, fico muito triste que não pude ver isso na minha turma de lógica matemática!

Hello sir, please can you explain how does query work for array with size being non power of 2?

Gödel's first incompleteness theorem is fundamentally logically flawed. Even the math could be complete. - A fact of scientific history. He is afraid, but rather very close to the whole, a profession in the field of mathematics, he laughed and laughed when Gödel already formalized the mathematical formalization. He just wanted to use numbers. It didn't even go through, it already failed. In mathematics. This, in turn, helped Neumann in computing. Because, paradoxically, Gödel's only usable theory was Gödel numbering / coding. Which also had practical benefits. Of course, only at first in computer technology. When every character and bit had to be saved. Well, what was useful was exactly what failed in mathematics. What is useless and wrong has been raised on the altar.

Great explanation!!! Thank you

7:35 The Gödel's encodings clever but it does NOT result in finite numbers, even with a finitely long statement (in general). Is that a problem? I don't know. I.e. the empty statement is always 0, but even a single-letter statement seems problematic. While e.g. the single-letter "f" is 2^3 = 8 *given* the table of letters there ("f" = 3 is ok, it doesn't have to correspond to ASCII). Unlike ASCII and Unicode which are finite encodings, for any possible statement the set of letters needs to be infinite, and then "f" could be an arbitrarily high number depending on where you put it into the encoding table, and having it in the first few elements of the table is an arbitrary ordering of letters. I'm not saying it needs to be "last", infinitely high number, but it could be. There ARE infinitely many numbers, integers, AND rationals, so it's unclear to me if it's a problem, i.e. you can encode each pair of numbers for a statement and a proof into a rational (only not divide by 0, doesn't seem like a real limitation, though I'm not sure).