Carrock

I used a notification link and missed your post - sorry. I looked at https://math.stackexchange.com/questions/421580/is-there-a-rational-number-between-any-two-irrationals from your links. First two proofs wrong. I'm not going to refute unlimited incorrect proofs. Less impressive than one link to a wrong proof.

That was just a casual example. I'm not familiar with the subtleties of well-ordering so I may have got it wrong. This statement is true no matter how the set is ordered. There is no way you can use aleph-null rationals to separate all the 2exp(aleph-null) points on a line.

No. That would require that the cardinality of the set of rational numbers be the same as the cardinality of the set of irrational numbers. i.e. 2exp(aleph-null) = aleph-null.

How can it be full? A full set would include rationals (which have been removed). aleph-one - aleph-null = aleph-one. By 'full set' I meant a set of the same size as the set o reals i.e. they can be put in one-to-one correspondence. Analogously, the infinite sets 1,2,3,4..... and 1,1,2,1/2,3,1/3,4,,1/4..... are the same size My point was that there is not a rational number between every two irrational numbers in any real number set. i.e. points are not in general separated when rational numbers are removed. e.g. there is no rational number larger than 2exp(1/2) and smaller than the next highest irrational number.

Surely not. In the real numbers there is a full set of real numbers between each two rational numbers.

It's indeed wrong, but I couldn't see a refutation in the thread. Classically (i.e. if heat was not subject to gravity), if you have a tall column of gas in a vertical gravitational field at equilibrium, the temperature of the gas is constant while its density (and pressure) decreases with height. Otherwise you could run a small perpetual motion machine using the difference in temperature.

"Virtually all Senate Democrats running in Trump states who voted against Brett Kavanaugh were defeated" Why let facts spoil a good headline?

Unless they're hoping Donald Trump will give them a lucrative job, like head of the EPA.

It's standard practice for your password to be encrypted and unavailable even to someone with access to the forum's database. See e.g. http://www.jasypt.org/howtoencryptuserpasswords.html

Luckily I left this topic on my browser and refreshed it or I'd never have seen your edit.... Hard to analyse a video compared to text... Around 5m 20s '"Spooky action at a distance" says that...' At 5min32sec "information in quantum mechanics can travel faster than light" etc followed shortly by 'Nobody understands this, but it's well established and it's a true effect.' "Spooky action at a distance" is well understood mathematically (not by Doctor Don Lincoln) though there is disagreement about its implications, much like 'observing' in quantum mechanics. Action at a distance does not involve the superluminal transfer of information. 'Information' as used by Doctor Don Lincoln has an unspecified different meaning from the normal meaning as in 'information cannot move faster than light.' When I see a false statement justified by 'Nobody understands this' (i.e. the author doesn't understand it) I don't see the point in watching the rest of a video where nothing I don't know already can be trusted. Doctor Don Lincoln does hint that his 'information' is different but anyone trying to learn from this video will likely conclude that physics is really difficult and should be left to clever people like him.

At 5min32sec "information in quantum mechanics can travel faster than light" etc

I watched part of the second video. A claim was that you can send information FTL using quantum entanglement (false) but you can't send a message FTL (true). The sort of video that's worse than a waste of time.

Yes. Both are the Northern Line.

I can't resist providing some local knowledge. Euston Station is two stops away from Camden Town station. This particular statement is true. Euston Station is one stop away from Camden Town station. This particular statement is true. These facts are very important in the rules* of "Mornington Crescent," a simple game often played on the quiz show "I'm sorry I haven't a clue." *See "Mornington Crescent: Rules and Origins" by N. F. Stovold. [/offtopic]

Lack of representation is generally associated with other discriminatory treatment or lack of other rights. I suspect prisoners, in particular, would not vote for politicians who accept money from private prison owners to legislate for more and longer prison sentences for minor offences. In practice, such legislation disproportionately affects those who can't afford to hire a good lawyer. Prisoners would probably make an exception for judges who receive bribes for jailing people for minor offences.

From some earlier post consigned to the outer darkness: An issue in these discussions is that IMO the definition of 'proof' has changed for the worse over the years. From https://www.merriam-webster.com/dictionary/proof archaic : the quality or state of having been tested or tried or a test applied to articles or substances [presumably theories too] to determine whether they are of standard or satisfactory quality So the apparent discovery of superluminal neutrinos proved relativity in the archaic sense - no big deal even for relativity deniers. From some earlier thread, someone said gravity hadn't been proved. I doubt any astronaut would admit to disbelieving in gravitational theory but after a long duration stay on the ISS their actions suggest otherwise. They e.g. release a cup, expecting it just to float and not to fall, or they decide to float downstairs rather than do it the hard way. After a few days, usually without serious injury, their changed behaviour shows they have decided gravity has been sufficiently proved.

But it is enough to prove the bijection ? I don't think so. You said the set is unlimited.. so you must prove your "etc" part for the bijection to be complete.. go ahead.. (take the infinite time in the univers) If you're correct I'm sure you can define the properties of the largest finite number. I presume you checked every a before making that statement (and many others). Or is it only those who oppose you who must check every value?

Maybe a process using unlimited sets would make the (lack of a) problem with infinite sets clearer. You have the unlimited integer set 1,2,3,4... 2*1 is in that set as is 2*2, 2*3, 2*4 etc. The only problem would be if there was a maximum integer n, then 2*n would not be in the set. It's up to you to demonstrate that there is some integer n in that set, but not n+1...

Much earlier, Four D. Jones in the Daily Express had a similar idea. He ended up with an ice cube and a small snowstorm in the Sahara Desert. The only other exploit I remember was when Four moored a rowing boat to Land's End and towed Britain south for the winter. A technical success of course but it didn't end well. A good introduction to science for someone just learning to read....

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I have absolutely no evidence for this, which is proof (see below, or not) that this is an extremely successful conspiracy by science deniers. It can't be a coincidence that as science has increased in importance the definition of 'proof' has changed for the worse over the years. There have been a lot of interminable threads on SF lately about the concept of proof. The success of the conspiracy is such that the old/obscure definitions have rarely been mentioned. From https://www.merriam-webster.com/dictionary/proof archaic : the quality or state of having been tested or tried or a test applied to articles or substances [presumably theories too] to determine whether they are of standard or satisfactory quality So the apparent discovery of superluminal neutrinos proved relativity in the archaic sense - no big deal even for relativity deniers. From some earlier thread, someone said gravity hadn't been proved. I doubt any astronaut would admit to disbelieving in gravitational theory but after a long duration stay on the ISS their actions suggest otherwise. They e.g. release a cup, expecting it just to float and not to fall, or they decide to float downstairs rather than do it the hard way. After a few days, usually without serious injury, their actions show they have decided gravity has been sufficiently proved.

A slight expansion. Unless the files are encrypted I doubt they would be inaccessible. A few computers implement UEFI in such a way that it's impossible to boot from an external device. However it's such a complicated standard that most implementations in practice are insecure. Any errors in my thinking?

There are also various free downloadable rescue disks e.g. https://lifehacker.com/5984707/five-best-system-rescue-discs , which can alternatively be installed on a usb memory stick and used to boot your computer and fix it or copy files.

Earn money from an abstract concept: Permit people, for a fee, to move a parking space from a crowded location to their car's normal location.

This is an interesting article with a poor headline. 'Astronomers do not witness birth of new star from stellar explosion' would be more accurate.