The reals can be ordered, just use x < y. I think you mean that if ZFC is true, we could enumerate unnameable reals (choose one with the axiom of choice, remove it, choose another one, etc.), but you could not enumerate them all. But it is true that you could get a "first" unnameable real.
That ordering is not a well-ordering, which is what the GP specified. A well ordering requires that every non-empty subset has a smallest element. That's not true for the reals ordered by x < y: for example, the set of all reals > 0 has no smallest element.
No one has explicitly shown that the reals can be well ordered, but it's a consequence of the axiom of choice that every set can be well-ordered. So in ZFC there must be a well ordering of the reals, even though no one has found one. Issues like this are why not all mathematicians accept the axiom of choice.
More than that: there is no way to build one (assuming the word "build" means some concrete construction), because it's consistent with ZF that the reals admit no well-ordering. Indeed, you can use forcing to construct a model of R in which there is an infinite but Dedekind-finite subset of R; and you can't well-order such a set, because a well-ordering would turn it into an ordinal, and any Dedekind-finite ordinal is finite. You must use some sort of choice principle to construct a well-ordering. (Of course, it's consistent that they can be well-ordered, too, as you say; or e.g. under the hypothesis V=L, where there's even a canonical well-ordering given by the lexicographic well-ordering L comes with.)
Not a mathematician, so this question may be a bit thick. I see the problem with the set of reals > 0, but is it perhaps that in this case > 0 is the problem and for sets specified as >= 0 it's fine because 0 is a nameable real and the smallest element. Obviously you can't just exclude certain expressions arbitrarily though, so I don't know how you could justify that mathematically.
If there is any non-empty subset that has no smallest element, then the ordering in question is not a well-ordering. You can of course define some subsets of the reals that do have a smallest element in the standard ordering, for example all of the reals that are greater than or equal to 0. But there are also subsets that do not have a smallest element, and that is enough to show that the standard ordering on the reals cannot be a well-ordering.
We just used the standard ordering < to define the set, it has nothing to do with the candidate well-ordering. If that's confusing, consider the set { 10^-x | x \in N } instead. It also has no minimum element in the standard ordering.
A well ordering on a set is a total order such that all non empty subsets have a minimum element with respect to this order. The standard ordering of the reals is not a well ordering, but the axiom of choice is equivalent to the statement that all sets possess a well-ordering. A well-order of the reals would probably look pretty chaotic though.
The title says that the US government /intervened/. As the article explains that happened, but also that that intervention wasn't the only reason why TRON didn't succeed.
The title should have been "Japan tried to build an operating system for the world, the US intervened although it quickly changed its mind, but a Japanese Windows PC seller used that intervention to boycott it". Or maybe just "Japan tried to build an operating system for the world, here's why it failed".
Amazing! I was proud that my OnePlus smartwatch needs recharging only once every 5 days, but 2 years is unbelievable. The difference is partly due to the LED screen of my smartwatch, LCD is much more energy efficient.
On AliExpress I see things like 64 TB (no, not GB!) USB flash drives for less than 10 euro, obviously a scam. I also see less spectacular products but still good looking deals, but without a brand name so you can't search the web for reviews.
On Amazon I never see products like that. That's why I prefer Amazon (and Coolblue and bol.com in Belgium).
I see this kind of fakes on eBay or Amazon all the time. It's even harder to report it than on AliExpress and I never had an obvious report of fakes upheld (AND eBay warns they will forward your identity to the seller. Great thing if the organised crime is involved in this)
Case in point, fake 2TB Sandisk ExtremePro microSD:
If you believe that, I have some USB storage to sell you. Huge capacity, barely used. I keep my inventory in storage under this bridge, which I also happen to have for sale. Interested? ;)
AliExpress is full of fakes and weird things but what it has going for itself is customer support. They'll return you money pretty much in every single case. They're so pro-consumer you could easily scam sellers out of things you buy really. I disputed multiple things in last 5-8 years and I've always either got a new item or a full refund. I don't order that much from there but if I want to buy something cheap I know is made in China anyway, I'd go for Aliexpress
There are people advocating for a new social network that is distributed and not owned by a single company. Turns out it already existed before the modern social networks!
True. But the article mentions that the amplitude of their circadian rhythm is higher (which tells you how much a relevant measure like the amount of melatonin or body temperature varies in a 24 hour cycle), so that is an indication of better sleep. A low amplitude means less difference between sleeping and being awake.
But the article discussed here also points in that direction when it mentions the importance of people on the lookout at night. The article also mentions primitive tribes partying all night, so that is yet another direction.
My boy when very young (just months) went thru a phase where in the middle of the night he would suddenly sit bolt upright, swivel his head to scan the room, and then conk out again. This was not caused by noises.
I found this reassuring and slightly amazing, that he is wired to do his part as a nighttime lookout.
Assuming you are not jesting, I am not certain I would qualify that behavior as any indication of being wired for some sort of nighttime patrol duty based on those details alone.
Fascinating. Also, not that you implied this by any means, but I was not trying to say "you are wrong, I'm right." Rather, who knows? Do all behaviors happen for a reason? Idk.
> Most importantly, when we measured their circadian function directly, small-scale communities had much higher amplitude circadian rhythms than global north populations, which I think is why they report being happy with their sleep even though it’s more fragmented.
This could explain why I feel more rested when I wake up if I take glycine before going to bed, even though I don't sleep more.
Putting aside the sycophantic wording around the orange leader and the total lack of details of how it's going to achieve what they are promising and what is actually going to happen, it's funny to see the distinction between gov officials and the Duke exec:
Gov: valued at $129 million and reinvest that same amount in additional generating capacity
Duke: will reinvest nearly $129 million in additional generating capacity, which may include
> “A big problem is AI is being used a lot by people who aren’t mathematicians, who don’t have a huge mathematical background and are not capable of verifying the output,” Bloom said. “They like to move fast, ask their AI to check it, it grows and grows. We’re seeing a lot more of these 100- to 200-page papers that people are posting. ‘I solved this theorem; I got AI to generate the proof and check the proof and write the paper.’ But no human has read it, and no human is going to read it. It’s a huge challenge now.”
Mathematics has the same problem as open source projects that are overwhelmed with AI slop!
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