Depending on what you allow as a "formal system", there's always one where the number is definable.
Well, the formal system itself must be definable with a finite amount of information (or else it's not definable in practice; we cannot read an infinite definition in order to know all of its axioms. A formal system with an infinite definition would not be very useful.)
So if we agree that every formal system definition must be finite, that means that the amount of possible formal systems is countable.
Thus couldn't we define our set as "all the real numbers definable by a finite statement in any possible (finite) formal system"?
In other words: If you can define the number using a finite amount of information, it belongs to our set.
Not all real numbers can be defined like this (because else it would mean that the set of real numbers is countable.) There have to be some numbers that are impossible to define using a finite amount of information.
So the (mostly philosophical) question is: What purpose do these numbers serve, other than being a curiosity? We can define them to conceptually exist, but they will never be the answer to any problem or formulation.
(Some interesting properties can be defined for them, though. For example multiplying two such numbers together may result in a number that's in our countable set. The other way around obviously can't happen.)
Thanks for the explanation.
I was thinking earlier today the following, and it's probably constructivist thinking:
Let's define a subset of the real numbers which consists of all numbers that are expressible with a finite definition. I understand that this is (most probably) an ill-defined set, but let's just assume that it can be well-defined, just for the sake of argument.
That set is countable (basically by definition).
The rest of the real numbers, thus, cannot be expressed with any finite expression whatsoever. Are these numbers useful in any way, shape or form?
Those numbers cannot be the answer to any problem (because, by definition, any number that's the answer to a problem is defined by a finite statement, ie. the problem itself, and thus belongs to the countable set defined above.) If those numbers cannot be the answer to any problem, and they cannot be expressed in any way, what's their use? Do they even "exist" (by some definition of "exist")?
I suppose this is more a question of philosophy than math...
It's interesting that we can define a set of numbers which contains elements that cannot be expressed.
I suppose this is starting to delve into the question of how the set real numbers is defined, which apparently isn't actually something that's as trivial or easy to define as one would hastily think.
I just find it fascinating that there exist real numbers that cannot be expressed in any way.
The set of all real numbers that can be expressed with a finite definition is countable (pretty much by definition). Since this doesn't cover all real numbers (because they are uncountable), this means that there are real numbers that cannot be expressed in any way (because you can't literally write an infinite expression to define it).
In fact, the set of real numbers that cannot be expressed in any way is larger than the set of numbers that can.
I think that's fascinating.
It was new and surprising information for me that the set of algebraic numbers is countable.
In fact, I think this could be extended further.
I don't know if there exists a name for the set of numbers expressible using analytical expressions, so I'll call them "analytical numbers" (in the same vein as "algebraic number" means a number that can be expressed with an algebraic expression.)
Analytical expressions not only include algebraic expressions, but also closed-form expressions as well as infinite series and continued fractions.
"But wait a minute", you would object, "for them to be countable, they would have to consist only of finite expressions; if we allow infinite expressions, they become uncountable."
That's true, except that all those infinite series and continued fractions need to be expressible with a finite notation, or else you could not define them. Thus, although the series may be infinite, it can be expressed with a finite expression (in the same way as eg. the decimal expansion of 1/3 is infinite, yet you can express it with a finite expression.)
So I postulate that the set of analytical numbers is countable.
This set contains many irrational and transcendental numbers (eg. pi = 4*atan(1)).
This got me thinking that this whole idea implies that there are real numbers that cannot be expressed with any kind of finite expression (because if all real numbers were expressible with finite expressions, they would be countable, which we know is not the case.)
So does that mean that the following is impossible to answer? "Give me an example of a real number that's not expressible with a finite expression."
It was new and surprising information for me that the set of algebraic numbers is countable.
However, thinking about it, it actually makes sense. Each algebraic number can be represented with a finite polynomial. This fact, all by itself, pretty much makes them countable by definition.
You learn new things every day...
Been watching the series once again, and I'm noticing tiny tidbits that hadn't noticed before (doesn't that tell something about a TV series?), like in the previous post, and like:
In Call of the Cutie Rainbow Dash says that she was the first in her class who got their cutie mark, and Applejack says that she was the last in her class.
In The Cutie Mark Chronicles it's established that the getting of their cutie marks is linked (all of them got the final nudge because of a single event, ie. Dash's first sonic rainboom). It can be directly deduced that from all the main six ponies, Dash got her cutie mark first (she got it immediately when she made the sonic rainboom) and Applejack got hers last (she had to travel all the way from Manehattan to Ponyville after seeing the rainbow before she got her mark).
The two things above probably aren't related, and it's probably coincidence (ie. the writers weren't explicitly thinking about this when they wrote the scripts), but I find it a curious tidbit.
Another thing I noticed is a slight inconsistency: In Stare Master the CMC break Fluttershy's table, and try to fix it. They make a mess of it and deduce that carpentry is not their talent. However, immediately in the next episode (The Show Stoppers) it's heavily implied that building things is Apple Bloom's talent. While it might not be carpentry per se, it's pretty close, and she should have been at least somewhat capable.
(One could argue that she was distracted by the other so much that she didn't get to show her talent in that instance. However, it still feels a bit inconsistent that she didn't say anything at all that showed competence.)
Was watching some old episodes, and this struck me as... something, in hindsight.
"Can I help? How about if I help clear out the clouds? ... Right, no wings..."
It's not that scary. Some people just seem to have this... I don't know... obsession with allocating objects individually with 'new' (which is what causes memory management issues in C++). But the thing is, in a good portion of cases you'll never need it. (Naturally it's good to know, but it's not something that comes up all the time.)
For the fractional part, we can map terminating numbers like this, by taking the first bit and XORing it with the rest:
0 -> 00000000…
0.125 -> 01000000…
0.25 -> 10000000…
0.375 -> 11000000…
0.5 -> 11111111…
0.625 -> 10111111…
0.75 -> 01111111…
0.875 -> 00111111…
Whereas nonterminating numbers are written normally:
0.2 -> 00110011…
0.4 -> 01100110…
0.6 -> 10011001…
0.8 -> 11001100…
This is a bit embarrassing to admit, but I can't understand anything of what you are saying there.
I mean, I understand binary, and I understand how integers and reals can be represented with them (after all, I'm a programmer), but I still have no idea what you are saying there (or how it solves the problem).
It used to be in the stone age that it was a bad idea to just put the original huge image on a web page and tell the browser to scale it down, because it made the page needlessly heavy to load. This was the time of dialup modems and slow landline connections (like eg. 32kB/s). Fortunately those times are long past, and thus it doesn't matter anymore, and thus it's ok.
Lets say you're playing pool on a frictionless surface, and you hit the cue ball from the bottom left hand pocket at a 45° angle. What would be the formula for figuring out how many cushions the cue ball bounces off of before it goes into one of the corner pockets? Remember, the pool table can be any dimension. Whether it be 2x3 or 5435x4356.
I think that would be impossible to express in closed form, and even with a recursive function it would become amazingly complicated, depending on how accurately you want to model the physics of the scenario.
Normally the table causes the ball to rotate (just moving on the table causes it to start to rotate in that direction, and then when it bounces off a wall, its rotation changes, and all this makes the exiting angle different from the incoming angle; also the speed of the ball comes into play due to all this). Even if you consider the table so frictionless that it doesn't cause the ball to rotate at all, ie. its orientation never changes, and the walls 100% elastic, then there's still the question of the size and shape of the pockets.
Ensuring object lifetimes does not mean remembering to free what you allocate, but rather that your objects do not disappear before you are done with them.
This seems to assume that you need to allocate objects individually with 'new'. That's not even nearly always the case.
The way to program optimally in C++ is also quite different from other languages, even when those languages resemble it (such as Java or C#).
In Java, for instance, you just carelessly allocate every single object with 'new'. And with that language it's justified: You don't have any other option, but it's nevertheless rather efficient. In fact, Java is extraordinarily faster at allocating objects dynamically than C++ is (if you allocate like 100 thousand individual objects with 'new' in Java and in C++, you'll probably notice that Java does it like a hundred times faster or something like that.) Unfortunately C++ (and C, and any language that uses the C runtime) is still embarrassingly slow at allocating memory. (Also C/C++ suffers from memory fragmentation more easily than a competent Java/C# runtime.)
However, in C++ it's seldom a good idea to be allocating hundreds of thousands objects individually with 'new'. Not only is it slow, it creates a memory management nightmare. Your program usually becomes needlessly complex and inefficient. (There are contexts in which this is inevitable, or at the very least the best solution design-wise, but that's not even nearly always the case.)
When you do it "right" in C++, you'll actually notice that your program not only becomes more efficient, but safer, simpler and easier to understand and follow, with less chances of memory management errors.
Knowing and understanding all the good principles, however, requires studying and experience. It's not a trivial path.
Apparently you missed the part where I said, "fully fleshed out," AKA not a simple flash game.
I was referring to the game mechanics, rather than that particular implementation of it.
That's not to say that that type of game mechanics couldn't be implemented as an AAA game. For example Beyond Good&Evil has sections that use this type of mechanic. However, I'm not sure this would translate well into a MLP game. (I really think that something like the Naruto game I mentioned above would be much more suitable.)
Somehow Korean sounds a lot like Japanese. I mean, at some points I could swear it's Japanese (I know how it sounds, even though I don't understand the language, except for a half dozen individual words), but then it doesn't... It's strange.
Link to video
Derpy as a filly in the background. How cute...
Wait, what? O_o
