KJW

I looked at the video but the part I was most interested in was a secret. Even before I watched the video, I was expecting the liquid glass to be a silylating agent. There is nothing new about silylating agents in general. For example, they are used to render laboratory glassware hydrophobic. However, I was curious about the specific silylating agent in this case, which might be quite novel. I anticipate that the silylating agent would be more "glasslike" than typical silylating agents (which have organic groups attached).

I doubt that Newton's laws or Kepler's laws were named by Newton or Kepler after themselves. More likely, they were named by other people somewhat later. Actually, it is quite common for a law to be given the name of someone who is not the person who originally discovered it. From Wikipedia article "Stigler's law of eponymy": Stigler's law of eponymy, proposed by University of Chicago statistics professor Stephen Stigler in 1980, states that no scientific discovery is named after its original discoverer. Examples include Hubble's law, which was derived by Georges Lemaître two years before Edwin Hubble; the Pythagorean theorem, which was known to Babylonian mathematicians and to Indian mathematicians before Pythagoras; and Halley's Comet, which was observed by astronomers since at least 240 BC (although its official designation is due to the first ever mathematical prediction of such astronomical phenomenon in the sky, not to its discovery). Stigler attributed the discovery of Stigler's law to sociologist Robert K. Merton. (From htps://en.wikipedia.org/wiki/Stigler%27s_law_of_eponymy)

I assume that. However, it is reasonable to assume that a downvote came from the one antagonistic person who did post in the thread.

some bigotries which very common in this forum. step 1: one demonstrates or proposes an opinion. step 2: a well known member attempts to disagree to that opinion. step3 : there is occuring of existence of many members downvoting that opinion (regardless the reality in that opinion, in fact this is a weakness of opinionating). And this is bigotry, isn't it? I don't understand who is who in the above. In the "Today I Learned in Mathematics" thread, only Genady and studiot were downvoted, and they were presumably by you (as the only person with the motive to downvote these two posters).

[math]R{}_{p}^{\mskip{0.05 cm}·}{}_{t}^{\mskip{0.05 cm}·}{}_{p}^{\mskip{0.05 cm}·}{}^{p}_{\mskip{0.05 cm}·}{}^{t}_{\mskip{0.05 cm}·}{}_{p}^{\mskip{0.05 cm}·}{}_{p}^{\mskip{0.05 cm}·}[/math] [math]R{}_l{}_q{}^r{}_s{}_t{}^g{}^v{}^h{}_x[/math]

That's it!! Just to clarify, the definition says, "for each y in S", which includes x0, whereas y must not be equal to x0.

I think I see it: y has to be not equal to x0.

@Genady, is the mistake you see that subset S requires at least two elements and not merely be non-empty?

[math]\dfrac{\partial \star}{\partial \overline{x}^{\mu}}[/math]

Q: What's the difference between Iran and Vietnam? A: Trump had a plan to get out of Vietnam!

Have you considered ammonia solution? You could also try (if you can obtain it) ethylenediaminetetraacetic acid (EDTA).

I think one thing that says they lean to the right is if they have a national flag in their front lawn. (This probably isn't limited to the US.)

I have to admit that this actually took me by surprise. After taking some time to think about this, I realise that it is ironic that what I said about two different types of axioms, the "deeper stuff" that you said was "not even wrong", appears to be key to my misunderstanding of the notion of completeness. Yes, there are two different types of axioms: one that defines a mathematical universe, and another that constrains that universe. It would seem that I neglected the mathematical universe. That would be because of the way I view mathematics, which is that everything exists unless proven otherwise. That means, for example, I assume the existence of multiplication even if it has not been explicitly defined. My mathematical universe contains multiplication, contains infinity, contains transfinite numbers, contains the axiom of choice, the continuum hypothesis has a definite answer, etc. But of course, that's not how this subject in mathematics is done. The mathematical universe is defined explicitly by the axioms, and notions such as completeness are based on it. So, I actually can see how a system with few axioms can be complete. And I can see that Gödel's incompleteness theorem is not trivial.

Surely this statement cannot be right. Say there are G axioms and axiom G is found to be provable from the other axioms A to F. Unless the proof of G is independent of one or more axioms, say B, how does this lead to an inconsistent set ? The point is that if an axiom is proven or disproven, then it is no longer an axiom. If the "axiom" is proven, it is a theorem. If the "axiom" is disproven, the set of axioms is inconsistent. Sometimes this is difficult to avoid. For example, in group theory, the existence of inverse elements can be expressed as: For each element [math]g[/math], there exists an inverse element [math]g^{-1}[/math] such that [math]g^{-1} g = g g^{-1} = e[/math] (the identity element). However, only one of the statements [math]g^{-1} g = e[/math] or [math]g g^{-1} = e[/math] is actually an axiom as the other statement can be proven. Which one is the axiom, and which one is the theorem is an arbitrary choice, so it seems natural to include both in a single statement so as to not arbitrarily break the symmetry.

This understanding is wrong. Well, it certainly is true that "any finite set of axioms is limited in what can be proven (or disproven) from them", so why is this not what (in)completeness is about? However, this is what axioms are. The problem is that it doesn't actually differentiate between statements that are axioms and statements that are theorems.