As someone who regularly looks up things I read "a while back," her experience is very common and insanely frustrating.
There always do exist magic combinations of words that you can put into google that will find the thing you're looking for. But the search space doesn't feel differentiable in a mathematical sense: you can't iteratively improve your terms because you either hit on a combo that works, or you get the same wrong results as you saw for your past 10 searches.
Can one do QFT in an ultrafinitistic foundations? My guess is no.
Also, I don’t think ZF sans the axiom of infinity works as an ultrafinitistic theory? It still has every natural number, just not the set of all of them.
"In constructive mathematics, proof by contradiction, while not universally rejected, is treated with caution and often replaced with direct or constructive proofs."
(gemini llm answer to google query: constructive math contradiction)
"Wiles proved the modularity theorem for semistable elliptic curves, from which Fermat’s last theorem follows using proof by contradiction."
So, will the Lean formalization of FLT involve translation to a direct or constructive proof? It seems not, I gather the proof will rely on classical not constructive logic.
"3. Proof by Contradiction:
The core of the formal proof involves assuming ¬Fermat_Last_Theorem and deriving a contradiction. This contradiction usually arises from building a mathematical structure (like an elliptic curve) based on the assumed solution and then demonstrating that this structure must possess contradictory properties, violating established theorems.
4. Formalizing Contradiction:
The contradiction is formalized in Lean by deriving two conflicting statements, often denoted as Q and ¬Q, within the context of the assumed ¬Fermat_Last_Theorem. Since Lean adheres to classical logic, the existence of these conflicting statements implies that the initial assumption (¬Fermat_Last_Theorem) must be false."
(gemini llm answer to google query: Lean formalization of fermat's last theorem "proof by contradiction")
FLT is a negative statement ("there are no nonzero integers x, y, z such that..."), and proofs by contradiction are constructively valid for proving negative statements.
"It’s fine to use a proof by contradiction to show something doesn’t exist. When the assumption that it does exist leads to a contradiction, then that shows it can’t exist.
It’s not so fine to use a proof by contradiction to show something does exist. Here’s the situation. The assumption that it does not exist leads to a contradiction. What can you conclude from that? You would like say “therefore it exists”. But you haven’t got any idea what it is. You may know it’s out there somewhere, but you have no idea how to find it. It would be better to have a proof that tells you what it is.
That’s a difference between what’s called “classical logic” and “intuitionistic logic”. In classical logic, proof by contradiction is perfectly accepted as a method of deductive logic. In intuitionistic logic, proof by contradiction is accepted to show something doesn’t exist, but is not accepted to show something does exist."
As far as I understand, The lead of this project Kevin Buzzard is a mathematician first. And the majority of mathematicians are untroubled by non-constructive proofs. I would imagine that proof directions that result in the most interesting additions to Mathlib would be chosen.
I have no idea why Gemini is saying that. Proof my contradiction is totally fine. Sure, many people prefer a more direct proof as they are nicer to read, but proof by contradiction is totally fine and sometimes the only way to prove important results.
Neither will bans and prohibitions, unless you are willing to go full north korea with cameras everywhere and computers locked down. And you'll probably fail with that.
Of course you can. 20 years in prision for using Bitcoin. Not a lot of people would wanna risk that. And when 99,99% chooses to not touch it with a pole the value will collapse. Then a few people can sit there with Bitcoins that no one wants to buy from them.
Is it a good idea? Mayne not. But obviously one can crush Bitcoin if one wants.
> Without bitcoin, you collect the ransom in gold with a dead drop.
Where the perpetrator needs to:
1. Be in the general area.
2. Only ask for enough currency that's easy to physically move (an actual real limitation in many countries)
3. Be sure the bills aren't marked (practically impossible). Because of this:
3(1). Be sure to not deposit the currency, ever.
3(2). Be sure not to use the currency with anyone who knows you who will every deposit the currency ever.
3(3). Allow the victim only enough time to procure a large amount of currency (likely days), but not enough time to procure a large amount of marked currency (this is an inherent conflict).
Obviously kidnapping is possibly via use of physical currency, but the practical limitations of cash over anonymous digital currency with regards to kidnapping are massive.
Your concerns with hyperinflation are alleviated by investing in any commodity, and trading on a black market. The fact that the commodity is a blockchain asset is effectively moot. The days of states not forcing individuals to be up front with capital gains on blockchain assets are over.
Cash is just a paper IOU for some other thing. With 100 dollar bill, that thing is 100 USD.
You could also issue paper for gold backed IOU, or gold itself, or bitcoin.
Counterfeiting paper is a problem, so you need force against this, and this is why the government is generally involved. But what kind of paper counterfeiting is enforced against, is just a legality. Currently the wost thing to counterfeit is 100 dollar bills. Law could be changed to make it equally bad to counterfeit gold bills, or bitcoin bills.
Saying bitcoin is less private than cash is a category error.
It would be extremely hard to issue cash as non government assuming the government doesn't want cash to exist, it would basically require a centralized entity that could be trivially shut down by the government.
I'm more of an Ethereum/PoS guy but what is for sure a fact, now that we're talking of decentralized issuance, is that you need PoW, at least for bootstrapping a cryptocurrency that has an even remotely fair distribution of tokens (instead of almost all being in the hands of the people that were there from the beginning).
Even then, almost all PoW are tuned in a way to pump the value and enrich the early people, not fairly distribute tokens or make the crypto functional as a currency.
The victim is described, sure. But their memory? Not so much - they get a line maybe. Then a dozen stanzas about the murderer. These ballads are dated, tone-deaf and disturbing.
It's rather inappropriate that we don't change names of victims when they are no longer relevant to active investigation. I would rather die in a slip and fall accident than become famous and have details of my life explored by becoming notable for being the victim of an asshat, thank you very much.
seems to work ok