Recent content by Gjmdp

I have seen that most PhD Math students struggle a lot with these examinations. It is fair to assume that any PhD student is already very talented and commited to the field of maths, so I am wondering whether professors (including those who write the exam) could actually ace the exam (get a...

Why the scepticism? There are already several automated theorem provers that have proved hundreds of theorems. AI is getting smarter at an ever faster rate (see GPT-3) and at Microsoft many researchers in AI and Mathematics believe they might be able to create algorithms that solve IMO problems...

It does terrify me the idea of waking up with the news that some AI is able to prove theorems better than any mathematician. That would mean humans would be useless in the field of Mathematics. I just saw that Microsoft is currently working on creating an AI that can beat humans at the...

You are right. I guess you could just assume S is complex enough to discuss about provability of statements (like in ZFC).

I will try again. I have read carefully all your posts and I am still trying to learn about this topic, I am sorry if I still make basic mistakes. Since I am not familiar with decidability (I am familiar with it only in the context of computability), I will refer to provability of statements...

I came up with a proof for that there the undecidability of statements is decidable: Let A, B be questions answered with "yes" or "no": B: is P undecidable? A: is B undecidable? If P is decidable, then B is "no". Hence B is decidable, thus A is "no". This means that P decidable -> (A = "no")...

You are right. A better description of what I meant is the undecidability of undecidable statements.

I would like to ask whether is there some research/papers on the unprovability of the unprovability of some true statements. It seems like an important concept in Mathematics, but haven't found much research/interest on it.

Isn't it pessimistic that some problems will remain open forever and we won't have a clue about whether they can be proved or not? Maybe we won't ever know whether there is a proof or disproof for problems such as P vs NP...

Yes, but after we prove that they are unprovable there is nothing more to do and we can leave the problem. However, what would happen if there is no such proof of its unprovability? This is, its unprovability is unprovable. We would never come up with a proof that tells us we can't proof the...

This is, are there theorems for which we can't ever proof that no proof exists for them? That would mean that there might be theorems that, no matter how much effort we put into proving/disproving/proving their unprovability, will always remain open. I would like to know whether there is some...

Thank you for your answer. How can you prove that?

Let u, v be column vectors n x 1 and M a m x n matrix over a field K. If M*u= M*v, then (M^-1)*M*u=(M^-1)*M*v, thus, I*u=I*v. Hence u=v. But that shouldn't be the case. What is wrong in my reasoning? Thank you.

So... can you tell me the answer? 🤔 I did the same and all the results were british universities. I have searched Stanford, MIT and Harvard university, and it wasn't stated what the requirements were for international (british students), only the requirements for US students... So, no, they...

In many british universities, like the University of Cambridge, the UK bachelor (1st class), with a duration of 3 years, is the minimum accepted requirement to apply for a PhD. This is, some PhD students only hold a UK bachelor, without a Master's degree. I was wondering if a british bachelor...