Eh… unless you are subbing to every streaming service out there it’s not. A new BluRay release is circa £20-30, DVDs are cheaper and also get actual discounted.
When it gets really expensive is when you have kids at which point Disney+ which is £7.99 pays for itself in about a day.
Oh I meant cinema movies before they end up on streaming sites. The Dune 2 BluRay had cost 1 less than to borrow and stream it. Might be different from country to country but that was really an eye opener for me. Okay the stream was 4k the BluRay not, thats maybe the only point for the stream, but really that price, incredible.
That said, if you have a system X, it can't prove it's own consistency, but a stronger system Y can prove its consistency (and perhaps some other stronger system can prove Y's consistency. This gives us a chain of systems, each proving the consistency of some weaker system).
That doesn't absolutely prove that the system is consistent--if Y was inconsistent, it could prove X is consistent (and also could prove that X is inconsistent). Nonetheless, it is still valuable. After all, part of our use of Y is the fact that we know of no inconsistency in it. And since formal systems often are subtly inconsistent, "consistent assuming some other system is consistent" is a lot better than "we have no proof whatsoever of consistency".
What’s interesting to note is that even if a strong system could prove its own consistency then it wouldn’t tell you anything. An inconsistent system can prove its own consistency. So if a system has a proof that it is itself consistent then you still wouldn’t know if it is consistent.
It would not. If a system can prove (I know sufficiently rich systems can’t do this but suppose it could) its own consistency you still can't conclude it is consistent.
EDIT: I'm working under the hypothetical situation in which PA could prove its consistency. I know it can't but assuming that it could prove it's own consistency you still couldn't conclude that it was consistent since an inconsistent system can prove it's consistency.
GP is (I think correctly) stating that a system that can "prove" its own consistency is definitely inconsistent. Inconsistent systems can "prove" anything; if a system can appear to prove its own consistency, it isn't.
Yes. I know this. What I'm saying is that even if a consistent system that was strong enough could prove it's consistency then it still wouldn't tell you anything. There are system that can prove their own consistency for which it is known that they are consistent.
These are weaker than the systems Godel's theorems are referring to, as discussed in the opening paragraph. Do these systems are not "strong enough" in the sense described in this thread.
Obviously I’m aware of this. As stated several times, my original comment refers to the hypothetical situation in which PA could prove its consistency without Godel’s theorems being true/known. One would not be able to conclude anything.
The point being, having PA prove its own consistency couldn’t tell you anything of value even in the case that Godel’s theorems were not true. This is an interesting phenomenon. The only way to know a system is consistent is to know all of its theorems.
You should take this proof as saying: if Lean 4 is consistent then New Foundations is consistent. There is no contradiction of Godel's incompleteness theorem.
I don't think the system here is trying to prove any of its underlying assumptions, just building on some set of existing ones. I doubt the theorem you are thinking of is applicable.
You can't expect a software distributor to be aware of all of the Easter eggs developers added to its software.
Even after communicating the fact to them several times, and one country or another asking the prison of some high executive. Those things are just too hard to keep track of.
Would have been interesting if they would have explained a bit better what they think will actually be achievable with such an infrastructure. If physicists build something like the LHC I understand even as a layperson what they try to prove. With this whole exascale computing I sometimes think: we do it because we can and applications are more motivated by the solution in search for a problem (disclaimer: I was part of an luckily unsuccessful H2020 proposal on exascale computing in a totally unrelated area) . But maybe some neuroscientist here can enlighten me about the impact this can have on practical research.
Computers are infinitely flexible. The LHC has no chance of providing any insight on, say, large population behaviour, but that’s one of the many things computers can do.
By making the point that Microsoft is a follower in this regard, yet the community complains about them as if Apple and Google eco-systems were some kind of angels.
