I worked on some code done by someone else, where on massive records, they were always selecting "*" instead of the needed or anticipated values. Big waste when one needs (by ID#) last and first name and selects a whole row instead - then wonders why it's not scaling upwards.
Eh, I wonder if you're overstating the performance implications of that. Those are all row-oriented databases. Unless all of the columns your query needs are found in an index, it's going to have to read the whole row from disk anyway; the extra costs from the * then become (a) memory and CPU usage and (b) network bandwidth. In my experience, network bandwidth is usually not a big problem; memory and CPU usage can be an issue, but the big performance killers tend to be inefficient joins (because they don't scale linearly), while scalar stuff (the which the * would fall into) are usually cheap.
I think that it has a lot to do with the fact that a shockingly large number of people confuse nomenclature with knowledge.
Yes, but who is confusing nomenclature with knowledge here? I think the scientists who pushed for this redefinition are as culpable of that as anybody. We can state all of our knowledge of a celestial object without settling the "question" of whether it's a "planet" or not. Our theories of physics spell out what are the sorts of actual physical facts that we can enumerate about celestial objects (mass, trajectory, composition, chemical and nuclear reactions, etc.). "Is it a planet" is not one of those. Because of this, drawing up some gerrymandered definition does nothing for our knowledge.
We're not the ancient Greeks. They had a distinct concept of "planet" because they saw some "stars" (in the old sense of the term, which means "celestial object") that, compared to others, moved funny. Because of this, they formulated laws of planetary motion distinct from laws of the motion of other "stars." We don't have those because, um, we unified the various theories of celestial and terrestrial motion a few hundred years ago.
In recent decades, we've been finding celestial objects that have all sorts of combinations of composition and trajectory. This is exactly what we should expect from our physical and cosmological theories. The general case is that space has all sorts of stuff floating around in all sorts of trajectories. With better telescopes, since we can see more stuff, we're seeing a lot more random stuff that's unlike what we'd seen before in all sorts of small ways.
Give us an argument why the IAU's definitions of a planet and of a dwarf planet are unreasonable. Please avoid any Appeal to Tradition.
The definitions are unreasonable because no definitions are needed. Nature makes no distinction between "planets" and "dwarf planets." Celestial objects don't split into neat discrete types either by trajectories or composition or any other criterion. They all follow the same laws of physics. There's no separate sets of physical laws for the motion of planets and that of other objects (which is, by the way, the reason the ancients gave the "planets" a distinctive word to distinguish them from other "stars" (in the original sense)).
Basically, "planet" is an obsolete category that no longer does any work in astrophysics. You can exhaustively state everything that we can state about a celestial object without settling the non-issue of whether it's a "planet" or not.
Also, can you craft a definition of a planet that covers Pluto but not Eris and Ceres other than "just what we used to arbitrarily call a planet?"
Of course, since celestial objects differ from each other in degree on a large number of criteria. The only definitions of any interest is the traditional one—on account of being traditional. Stop pretending that we're making some sort of scientific progress by pointlessly redefining a term that's just not useful anymore. Leave the damn definition alone as a reminder of how we got to where we are today.
Re:Yes, yes, for loops!
on
Land of Lisp
·
· Score: 1
Looks familiar, right? C-style languages all but force you to constantly spell out how to iterate over the elements of a collection or sequence and combine their values; the only ways to abstract it are way more verbose than this (e.g., iterator classes, Visitor pattern), or are subject to additional problems (function pointers in C aren't anonymous, and are not lexically scoped).
Hello, and welcome to 2010.
Java has had anonymous inner classes (verbose as they are, they do work as closures) since version 1.1 - that's 13 years now - which can be used to implement map, filter, reduce and the rest of the bunch.
Yes, I know about inner classes, and I've used them before. As you point out, you have to write a freaking class to use them. That's two lines for the class declaration (first line + closing bracket), two lines for the method declaration, and then the code. I.e., map(function, collection) in Java becomes something this:
Also, considering how those soldiers looked, I have to say a war against North Korea would not scare me in the slightest, aside from the nuke thing. I imagine just seeing our well-fed muscular soldiers would make them all want to throw down their arms immediately and become POWs so they could be fed decently.
Two things:
Guns are the great equalizer, and can artillery kills a lot more soldiers than rifles do.
Seoul is the capital of one of the G-20 countries, and a city of 12 million people with a metropolitan area of 24.5 million. It might also be in range of North Korean artillery.
I wonder why China lets that happen, as it would be trivial for them to ban any data coverage in this area and/or report any suspicious activity to the North Korean authorities. Maybe it's a way for them to put some pressure on their North Korean "ally", which has become somewhat of an embarrasment to them lately.
Yeah, it's probably along those lines. My understanding is that the Chinese government isn't very much not crazy about Kimland, but they prefer that to having a US ally state next door like South Korea. More generally, I think they just want NK to be under their sphere of influence first and foremost, so they only support change to the extent that it gives them a favorable position in NK relative to the USA.
So either China relies on this to get intelligence about conditions inside NK, or they think it destabilizes the Kim regime just enough that somebody they like better can take power (without destabilizing it so much that it falls apart and refugees stream across the border even more than now).
When it was first announced (5 years ago now?), I thought the Optimus Maximus [thinkgeek.com] keyboard was going to solve this problem. With a little smarts built into the keyboard I wouldn't mind esoteric key combinations if the result was displayed directly on the keyboard. Something like this might, someday, be the solution but at $1500 dollars it's going to be a while and assuming a direct-brain interface doesn't come first.
Eh, they sell stickers you can stick on your keys.
With a little bit of searching, I come up with about 20 megapixels for a perfect shot on perfect 35mm film, 12 megapixels for a merely "good" shot. The best film scanners can go up to 36 bit color depth per pixel, also.
I've seen so many different numbers given by so many people on this question that I've basically stopped believing all of them. It's a complicated issue; the general opinion, however, is that APS-C digital cameras are as good or better than 35mm film cameras in practice.
One of the reasons the issue is complicated is because the results you get depend on how you perform the comparison. Let's assume that you take two photos of the same scene, using the same lens at the same aperture, but one photo is taken on the film camera and the other on a digital camera with the same frame size. How are you going to compare the photos? Here's three ways you could do it:
You could scan the film on a film scanner, and compare the scan image file to the digital camera's file. But then the problem you have is that the film scanner might fail to reproduce all of the detail on the film. For example, many film scanners have aliasing effects that magnify the appearance of grain in some conditions.
You could make a print from each of the photos, and compare the prints. The problem then is that unless you scan the film (introducing the problems detailed above), you're going to have to use two very different printing methods for the two photos; a digital print for the digital photo, and a traditional darkroom print for the film one. But now the results are a function of the print method as much as the capture type. And, it's a subjective comparison.
You could put the film on a light table and examine it with a magnifier, and compare it by eye with the digital photo displayed on a monitor at the equivalent magnification relative to the sensor size. This is probably the best, but the comparison is subjective again.
And I'm sure that somebody who knows this stuff better than me can pick this apart...
The best DSLRs I can find on newegg today are 21 megapixel cameras in the $6000 range and claim true 14 bits per color channel (which would be 42 bit color), so yes, it seems they've passed 35mm film. The camera tier under that are about 18 megapixels and 22 bit color, for $800-$1300.
You're assuming that the number of megapixels is an accurate representation of the amount of detail (spatial resolution) that the camera can reproduce. It is not; it's an upper bound on the amount of detail that the camera can reproduce, and nearly every digital camera falls significantly short of its sensor's resolution limit, due to the anti-aliasing filters used to eliminate color moiré artifacts, which basically blur the image at the sensor.
But wait, there's more!
The amount of detail captured is as much a function of the lens as it is a function of the sensor. There's a physical upper limit on how much resolution you can get from a lens of a given aperture, and of course, lenses are also imperfect devices. Today's 18 to 24 megapixel digital cameras are actually starting to hit the upper limits on the amount of detail lenses can reproduce.
The 21 and 24 megapixel cameras you cite have a larger sensor than the 18 megapixel models. Larger sensors tend to lead to more detail, because the lenses are larger, and at the same print or display size, they're less demanding on the lens.
When hearing stories like these, you should keep some perspective and remember that the Germans had a superior tank-vs.-tank kill rate over the Allies. More so when you look at well-trained and equipped German tank units, many of which had kill ratios in the order of 10:1.
Also, Allied tankers where not by any means the first to use maneuver by weaker tanks to gain an advantageous position over more powerful tanks. The Germans did this many times in their 1940 invasion of France. The French and British had more powerful tanks (a few Allied medium tanks were more heavily armored than German heavy tanks!), but very few of them had radios. The German tanks were weaker, but most of them had radios. The Germans were much better able to coordinate their tanks, or change orders on the fly. And even then they had a tough time against Allied tanks.
(And no, I don't mean to imply that the Germans were the first to do this sort of trick.)
How can you hear what this sounds like when it's just an article on a web page? You must have synesthesia...
That's a succinct statement of what's wrong with GP's comment.
More generally, I think it's fair to say that people, especially folks who are scientifically educated but not psychologists, are much too prone to seeing human perception according to the following flawed model: human perception consists of five separate, independent sensory systems that feed their input into some sort of central processing unit that integrates them all together. It's because so many people are inclined to assume that that the sort of result obtained in this article is newsworthy for the broad public. I mean, psychologists have known about stuff like this for ages...
There is a great short story about this, I think by Harry Turtledove. One of these alternate history concepts, where the Nazis end up ruling India instead of the British. When Gandhi tries his non-violent resistance the Germans arrest him. After a brief interview with the German commander, who is genuinely curious why Gandhi thinks his methods would have any effect on the German occupation, the German have him shot out of hand. End of story.
...and in the realm of unalternate history, well, we have the 1989 Tiananmen Square demonstrations.
Japan has both apologized and pay reparations numerous times. What more do you want them to do? Resurrect the dead?
The story is much more complicated than that; you're being selective with the facts here. There are a few issues about Japan's attitudes towards its past that still bother the heck out of a lot of people:
In a lot of people's minds, Japan hasn't apologized and paid repairs for everything that they ought to. One of the most famous cases is the WWII comfort women.
The apologies you cite were made by the government of Japan, but there are significant segments of the Japanese public who are much less apologetic about their country's deeds in the first half of the 20th century. There's a lot of folk who would rather pretend Japan's misdeeds never happened, and worse, significant revisionist trends about them. In recent years, these have surfaced as controversial revisions of school history textbooks (yeah, kinda like recently in Texas).
There's the controversial visits by Japanese politicians to Yasukuni Shrine, a Shinto shrine dedicated to fallen soldiers, which many feel it goes to lengths to glorify Japan's war criminals. The shrine is run privately, and the folks who run it are noticeably right-wing and revisionist about Japan's role in WWII.
The trend is pretty clear: there is a significant conservative segment of the Japanese population whose attitudes just piss off the rest of the region, and there are many politicians who pander to them.
Better yet, park your car outside a government building and then call the police saying there is a suspicious device attached to your car. Hey, you did the right, thing, right? How can they fault you?
By demonstrating that you did it with the intent to cause the disruption that you describe. Or in other words: if you truly did notice a suspicious device attached to your car, you're not supposed to exploit the possibility that it might be a bomb in order to fuck with people.
Integers can be bijectively mapped to natural numbers and integer operations can be expressed using first-order predicates and axioms of Peano arithmetic. So for the purpose of my argument replacing "natural number" with "integer" changes exactly nothing.
You're looking at it from the wrong direction. There is a theory about the real numbers that is "weaker" than Peano arithmetic, despite the fact that the real numbers properly contain the naturals. How can this be so? Because that theory does not entail every fact about the natural numbers that Peano arithmetic does. It's axioms are about algebraic properties of the operators (associativity, commutativity, closure, order), not about proving specifically that 2+3=5 or 2x3=6.
Is there an analogous complete theory for the integers? I really do not recall, but I think there might.
Another way of going about this: if I read your reply correctly, you're saying that Peano arithmetic can encode the proof theory of a first-order theory of the integers. That is true, of course, but Peano arithmetic can likewise encode the proof theory of the real numbers. (The fact that the naturals can't be mapped bijectively into the reals is a moot point, because we're talking first-order theories here, and thus, Löwenheim-Skolem applies.) But the relationship can't be inverted--the theory of real closed fields can't encode Peano arithmetic. It's this second direction that I have in mind when I insist equivocating over the integers and the naturals might be a mistake.
And again, I must stress: think of the axioms, not of the domains. The relative cardinality of the naturals and the integers tells us nothing about the completeness of different logical theories--Löwenheim-Skolem is relevant here, again.
A few other people have mentioned UIs that do precisely this, but to add another example: a lot of database administration GUI tools allow you to graphically create tables, indexes and other schema objects, and export the SQL script to recreate them automatically.
What do you mean it changes nothing? Peano arithmetic is a completely different logical theory than real closed fields, and neither of them is about integers; one is about naturals, the other is about reals. If you know that there is no complete algebraic of the integers (I don't), well, just say so.
No, a finite number of is not required. For example, Hilbert-style proof systems for first-order logic have infinitely many axioms, but these all are instances of six axiom schema. IIRC, the real requirement is that it be decidable whether a proposition is an axiom.
Nope. Theory of real numbers works with _real_ numbers, not with integers. To define integers you need to introduce more axioms which will make the resulting theory more powerful (and incomplete).
I believe you're making a mistake here by talking about "integers" instead of naturals. Gödel's theorem is about Peano arithmetic, which defines the natural numbers starting from zero, addition and multiplication.
Anyway, the real point here is that Peano Arithmetic is a very different theory than real closed fields. Read the pages and you'll see, at the very least, that understanding one does little to help you understand the other.
I kinda suspect it comes down to something like this: the theory of real closed fields doesn't allow you prove that, for example, 2 + 3 = 5 or that 2 x 3 = 6, and that the ability to prove this sort of fact is the really critical element. The theory simply assumes that the addition and multiplication operators have certain algebraic properties, and it doesn't concern itself with proving that two specific numbers add up to a specific third one.
I thought that was the entire axiomatic contradiction Godel pointed out. In any system of logic you use, there must necessarily be true statements that you cannot prove.
No, in any system of arithmetic that's at least as powerful as a particular axiomatization called Peano arithmetic.
Fun fact that most people haven't heard about Gödel: he's also the guy who proved the completeness of first-order logic. I.e., if a statement is logically valid in the semantics of first-order logic, then the axioms of first-order logic can prove it.
Eventually, every network gets subdivided at some piece of equipment, be it a transparent bridge or router somewhere. The idea of being a "peer" is an imaginary one really - other than boxes plugged into the exact same switch or router on the same subnet, you're doing a network traversal somewhere.
Yes, and this is precisely the illusion that TCP/IP was designed to provide.
Now, I have to give the gold standard guys some credit, they said it would go up, and it did, bigtime.
Yeah, and just as the "houses only go up" folks, they will be remembered as being right for as long as they did. If you always say the price of X is going to go up, you will be right. At some point.
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I worked on some code done by someone else, where on massive records, they were always selecting "*" instead of the needed or anticipated values. Big waste when one needs (by ID#) last and first name and selects a whole row instead - then wonders why it's not scaling upwards.
Eh, I wonder if you're overstating the performance implications of that. Those are all row-oriented databases. Unless all of the columns your query needs are found in an index, it's going to have to read the whole row from disk anyway; the extra costs from the * then become (a) memory and CPU usage and (b) network bandwidth. In my experience, network bandwidth is usually not a big problem; memory and CPU usage can be an issue, but the big performance killers tend to be inefficient joins (because they don't scale linearly), while scalar stuff (the which the * would fall into) are usually cheap.
I think that it has a lot to do with the fact that a shockingly large number of people confuse nomenclature with knowledge.
Yes, but who is confusing nomenclature with knowledge here? I think the scientists who pushed for this redefinition are as culpable of that as anybody. We can state all of our knowledge of a celestial object without settling the "question" of whether it's a "planet" or not. Our theories of physics spell out what are the sorts of actual physical facts that we can enumerate about celestial objects (mass, trajectory, composition, chemical and nuclear reactions, etc.). "Is it a planet" is not one of those. Because of this, drawing up some gerrymandered definition does nothing for our knowledge.
We're not the ancient Greeks. They had a distinct concept of "planet" because they saw some "stars" (in the old sense of the term, which means "celestial object") that, compared to others, moved funny. Because of this, they formulated laws of planetary motion distinct from laws of the motion of other "stars." We don't have those because, um, we unified the various theories of celestial and terrestrial motion a few hundred years ago.
In recent decades, we've been finding celestial objects that have all sorts of combinations of composition and trajectory. This is exactly what we should expect from our physical and cosmological theories. The general case is that space has all sorts of stuff floating around in all sorts of trajectories. With better telescopes, since we can see more stuff, we're seeing a lot more random stuff that's unlike what we'd seen before in all sorts of small ways.
Give us an argument why the IAU's definitions of a planet and of a dwarf planet are unreasonable. Please avoid any Appeal to Tradition.
The definitions are unreasonable because no definitions are needed. Nature makes no distinction between "planets" and "dwarf planets." Celestial objects don't split into neat discrete types either by trajectories or composition or any other criterion. They all follow the same laws of physics. There's no separate sets of physical laws for the motion of planets and that of other objects (which is, by the way, the reason the ancients gave the "planets" a distinctive word to distinguish them from other "stars" (in the original sense)).
Basically, "planet" is an obsolete category that no longer does any work in astrophysics. You can exhaustively state everything that we can state about a celestial object without settling the non-issue of whether it's a "planet" or not.
Also, can you craft a definition of a planet that covers Pluto but not Eris and Ceres other than "just what we used to arbitrarily call a planet?"
Of course, since celestial objects differ from each other in degree on a large number of criteria. The only definitions of any interest is the traditional one—on account of being traditional. Stop pretending that we're making some sort of scientific progress by pointlessly redefining a term that's just not useful anymore. Leave the damn definition alone as a reminder of how we got to where we are today.
Looks familiar, right? C-style languages all but force you to constantly spell out how to iterate over the elements of a collection or sequence and combine their values; the only ways to abstract it are way more verbose than this (e.g., iterator classes, Visitor pattern), or are subject to additional problems (function pointers in C aren't anonymous, and are not lexically scoped).
Hello, and welcome to 2010.
Java has had anonymous inner classes (verbose as they are, they do work as closures) since version 1.1 - that's 13 years now - which can be used to implement map, filter, reduce and the rest of the bunch.
Yes, I know about inner classes, and I've used them before. As you point out, you have to write a freaking class to use them. That's two lines for the class declaration (first line + closing bracket), two lines for the method declaration, and then the code. I.e., map(function, collection) in Java becomes something this:
.
There's also the fact that hardly anybody writes Java programs like these. Oh, and, anonymous inner classes ain't real closures.
Also, considering how those soldiers looked, I have to say a war against North Korea would not scare me in the slightest, aside from the nuke thing. I imagine just seeing our well-fed muscular soldiers would make them all want to throw down their arms immediately and become POWs so they could be fed decently.
Two things:
I wonder why China lets that happen, as it would be trivial for them to ban any data coverage in this area and/or report any suspicious activity to the North Korean authorities. Maybe it's a way for them to put some pressure on their North Korean "ally", which has become somewhat of an embarrasment to them lately.
Yeah, it's probably along those lines. My understanding is that the Chinese government isn't very much not crazy about Kimland, but they prefer that to having a US ally state next door like South Korea. More generally, I think they just want NK to be under their sphere of influence first and foremost, so they only support change to the extent that it gives them a favorable position in NK relative to the USA.
So either China relies on this to get intelligence about conditions inside NK, or they think it destabilizes the Kim regime just enough that somebody they like better can take power (without destabilizing it so much that it falls apart and refugees stream across the border even more than now).
Eh, they sell stickers you can stick on your keys.
With a little bit of searching, I come up with about 20 megapixels for a perfect shot on perfect 35mm film, 12 megapixels for a merely "good" shot. The best film scanners can go up to 36 bit color depth per pixel, also.
I've seen so many different numbers given by so many people on this question that I've basically stopped believing all of them. It's a complicated issue; the general opinion, however, is that APS-C digital cameras are as good or better than 35mm film cameras in practice.
One of the reasons the issue is complicated is because the results you get depend on how you perform the comparison. Let's assume that you take two photos of the same scene, using the same lens at the same aperture, but one photo is taken on the film camera and the other on a digital camera with the same frame size. How are you going to compare the photos? Here's three ways you could do it:
And I'm sure that somebody who knows this stuff better than me can pick this apart...
The best DSLRs I can find on newegg today are 21 megapixel cameras in the $6000 range and claim true 14 bits per color channel (which would be 42 bit color), so yes, it seems they've passed 35mm film. The camera tier under that are about 18 megapixels and 22 bit color, for $800-$1300.
You're assuming that the number of megapixels is an accurate representation of the amount of detail (spatial resolution) that the camera can reproduce. It is not; it's an upper bound on the amount of detail that the camera can reproduce, and nearly every digital camera falls significantly short of its sensor's resolution limit, due to the anti-aliasing filters used to eliminate color moiré artifacts, which basically blur the image at the sensor.
But wait, there's more!
Um, more like overexposed to hell. The camera basically can't regulate exposure if that's the case.
When hearing stories like these, you should keep some perspective and remember that the Germans had a superior tank-vs.-tank kill rate over the Allies. More so when you look at well-trained and equipped German tank units, many of which had kill ratios in the order of 10:1.
Also, Allied tankers where not by any means the first to use maneuver by weaker tanks to gain an advantageous position over more powerful tanks. The Germans did this many times in their 1940 invasion of France. The French and British had more powerful tanks (a few Allied medium tanks were more heavily armored than German heavy tanks!), but very few of them had radios. The German tanks were weaker, but most of them had radios. The Germans were much better able to coordinate their tanks, or change orders on the fly. And even then they had a tough time against Allied tanks.
(And no, I don't mean to imply that the Germans were the first to do this sort of trick.)
What percentage of Muslim plane travelers hijack planes and crash them into buildings?
That's a succinct statement of what's wrong with GP's comment.
More generally, I think it's fair to say that people, especially folks who are scientifically educated but not psychologists, are much too prone to seeing human perception according to the following flawed model: human perception consists of five separate, independent sensory systems that feed their input into some sort of central processing unit that integrates them all together. It's because so many people are inclined to assume that that the sort of result obtained in this article is newsworthy for the broad public. I mean, psychologists have known about stuff like this for ages...
...and in the realm of unalternate history, well, we have the 1989 Tiananmen Square demonstrations.
The story is much more complicated than that; you're being selective with the facts here. There are a few issues about Japan's attitudes towards its past that still bother the heck out of a lot of people:
The trend is pretty clear: there is a significant conservative segment of the Japanese population whose attitudes just piss off the rest of the region, and there are many politicians who pander to them.
By demonstrating that you did it with the intent to cause the disruption that you describe. Or in other words: if you truly did notice a suspicious device attached to your car, you're not supposed to exploit the possibility that it might be a bomb in order to fuck with people.
You're looking at it from the wrong direction. There is a theory about the real numbers that is "weaker" than Peano arithmetic, despite the fact that the real numbers properly contain the naturals. How can this be so? Because that theory does not entail every fact about the natural numbers that Peano arithmetic does. It's axioms are about algebraic properties of the operators (associativity, commutativity, closure, order), not about proving specifically that 2+3=5 or 2x3=6.
Is there an analogous complete theory for the integers? I really do not recall, but I think there might.
Another way of going about this: if I read your reply correctly, you're saying that Peano arithmetic can encode the proof theory of a first-order theory of the integers. That is true, of course, but Peano arithmetic can likewise encode the proof theory of the real numbers. (The fact that the naturals can't be mapped bijectively into the reals is a moot point, because we're talking first-order theories here, and thus, Löwenheim-Skolem applies.) But the relationship can't be inverted--the theory of real closed fields can't encode Peano arithmetic. It's this second direction that I have in mind when I insist equivocating over the integers and the naturals might be a mistake.
And again, I must stress: think of the axioms, not of the domains. The relative cardinality of the naturals and the integers tells us nothing about the completeness of different logical theories--Löwenheim-Skolem is relevant here, again.
A few other people have mentioned UIs that do precisely this, but to add another example: a lot of database administration GUI tools allow you to graphically create tables, indexes and other schema objects, and export the SQL script to recreate them automatically.
What do you mean it changes nothing? Peano arithmetic is a completely different logical theory than real closed fields, and neither of them is about integers; one is about naturals, the other is about reals. If you know that there is no complete algebraic of the integers (I don't), well, just say so.
No, a finite number of is not required. For example, Hilbert-style proof systems for first-order logic have infinitely many axioms, but these all are instances of six axiom schema. IIRC, the real requirement is that it be decidable whether a proposition is an axiom.
I believe you're making a mistake here by talking about "integers" instead of naturals. Gödel's theorem is about Peano arithmetic, which defines the natural numbers starting from zero, addition and multiplication.
Anyway, the real point here is that Peano Arithmetic is a very different theory than real closed fields. Read the pages and you'll see, at the very least, that understanding one does little to help you understand the other.
I kinda suspect it comes down to something like this: the theory of real closed fields doesn't allow you prove that, for example, 2 + 3 = 5 or that 2 x 3 = 6, and that the ability to prove this sort of fact is the really critical element. The theory simply assumes that the addition and multiplication operators have certain algebraic properties, and it doesn't concern itself with proving that two specific numbers add up to a specific third one.
No, in any system of arithmetic that's at least as powerful as a particular axiomatization called Peano arithmetic.
Fun fact that most people haven't heard about Gödel: he's also the guy who proved the completeness of first-order logic. I.e., if a statement is logically valid in the semantics of first-order logic, then the axioms of first-order logic can prove it.
Yes, and this is precisely the illusion that TCP/IP was designed to provide.
You can have a border firewall without NAT.
Warning: one of these facts may be followed by the other only in terms of time, and not of causal chain...
Yeah, and just as the "houses only go up" folks, they will be remembered as being right for as long as they did. If you always say the price of X is going to go up, you will be right. At some point.