That is what you get, when programming immitates math. Even Java and PHP looks beautiful compared to that. Mathematics should be banned as a harmful and obfuscated teaching.
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Anonymous2012-01-09 1:21
Pattern matching in Erlang is sexy
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Anonymous2012-01-09 1:23
>>1 classified subexpressions according to the informal usage relation to the accumulation parameter and results of recursions.
Even the text sounds like some pseudoscience, abusing vague buzzwords.
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Anonymous2012-01-09 1:30
I can't tell if I'm horrified or I love it.
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Anonymous2012-01-09 1:43
treat it as a natural language filled with symbols. It's something that is meant to communicate to people rather than machines, but it is also supposed to be definitive, unambiguous, and easy to manipulate symbolically.
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Anonymous2012-01-09 2:05
>>5 treat it as a natural language filled with symbols
There is no fucking language, it's an impenetrable stream of symbols whiсh blur into a single opaque black square, when you state at them for enough time.
It's something that is meant to communicate to people rather than machines
Doubt anyone, except author, could read this condensed pile of hebrew symbols.
but it is also supposed to be definitive, unambiguous, and easy to manipulate symbolically.
Why wont they use Lisp instead of math? Lisp is definitive, unambiguous and pretty easy to manipulate.
Your problem isn't with math, but with its colloquial syntax, or more generally with natural language.
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Anonymous2012-01-09 2:17
>>7 Your problem isn't with math, but with its colloquial syntax
Sorry, but to learn math, you've to learn it's syntax first, which is where most students completely abandond science in favour of easier subjects (like liberal arts).
more generally with natural language.
Doubt anyone understands natural language in every case, because any natural sentence could be interpreted in several ways.
>>6
I could say the same thing about chinese characters, but that is only because I have not yet learned chinese. If I learned chinese, I'm sure I could read it as easily as anyone else.
>>8
You are missing his point. Your problem isn't with math itself, but with not taking the time to understand what the symbols mean. I'd have the same problem with chinese above. My inability to understand chinese writing isn't due to anything specific about chinese, but due to my lack of understanding of what the symbols mean and how the grammar works. It is an encoding that people use to communicate ideas to each other. Once you understand the encoding, you can read and construct in your mind the ideas they may be trying to communicate.
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Anonymous2012-01-09 2:37
>>9 If I learned chinese, I'm sure I could read it as easily as anyone else.
How do you know? How do we know? How do they know?
only because I have not yet learned chinese.
Learning chinese would take years of practice, while living in China. You can learn Chinese and it's usage using dead textbook.
You are missing his point. Your problem isn't with math itself, but with not taking the time to understand what the symbols mean.
I can learn Lisp by reading the first chapters of SICP. Doubt there is an quick article which will teach you how to translate http://www.cse.unt.edu/~idl99/Proceedings/ahn/img50.gif into plain english.
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Anonymous2012-01-09 2:38
>>10 You cant fully learn Chinese and it's usage using dead textbook.
self fix
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Anonymous2012-01-09 2:40
>>9
In order to understand what the symbols mean you have to have the meaning first.
Words and symbols are abstract, irrelevant objects. It's useless to create a pointer to something if you can't create that concept in your language. What was it in SICP? You can't make arrays of arrays in Basic or Fortran or something because they aren't closed?
That's the ultimate problem: A symbol is MEANINGLESS without MEANING.
>>13
What's kind of sick is because the image is about static analysis of expressions something, you could use the paper itself as the mathematical model for divising a program to translate it to english.
>>12
You have to parse them first. Parsing even a simple infix expression is very hard: there are binding powers, parens, precedencies, left-to-right, bottom and top-down orders, missing operand, unary/binary ambiguities (`-` is used as both binary and unary op).
Well it is readable to some people. The symbols either get their meaning from the some kind of standard usage, or they need to be defined by the author at some point in the text. Certain symbols get used for different things for convenience, so you have to look it up.
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Anonymous2012-01-09 2:47
>>14
Strongly doubt, a laymen will understand loaded pseudoscientific sentences in a style of "classified subexpressions according to the informal usage relation to the accumulation parameter and results of recursions."
In the days when Sussman was a novice, Minsky once came to him as he sat hacking at the PDP-6.
"What are you doing?", asked Minsky.
"I am training a randomly wired neural net to play Tic-tac-toe", Sussman replied.
"Why is the net wired randomly?", asked Minsky.
"I do not want it to have any preconceptions of how to play", Sussman said.
Minsky then shut his eyes.
"Why do you close your eyes?" Sussman asked his teacher.
"So that the room will be empty."
At that moment, Sussman was enlightened.
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Anonymous2012-01-09 2:49
>>15
Except the symbols and orders are irrelevant without explanation as to what the symbols are. You could do symbolic logic, you would even set up truth tables like 5+5=10 and come out with incredible proofs as to why they should be used that way, but ultimately, they're meaningless without putting them to something. I can say 6+6 = 14, 6+6 = 12, 6+6 = C. I tell you these statements are all true, and very simple ones at that. They don't differ in symbolics, except more or less on the truth table. I could use any symbols I want. Similarly, I could come up with any order for operations I want. Denominators over numerators? Why not. Negative exponential systems? Sure. The difference is in how those numbers are applied. That is mathematics, not conjuring up cryptic analyses and methodologies and purporting them as "self evident" truths.
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Anonymous2012-01-09 2:51
>>16 Well it is readable to some people.
By very limited circle of people, who have enough time, courage and will to learn all that mathematics shit.
The symbols either get their meaning from the some kind of standard usage
That is irrelevant. The point is: symbols look scary and hard to read.
>>21
anyways, math wasn't always done symbolically. I heard about a some historical middle eastern mathematicians that used long complete sentences to describe formula, and algebraic manipulation. But if you read one of their sentences, you'd understand the need for symbols.
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Anonymous2012-01-09 2:59
>>22
A good alphabet maps onto sounds. You then can translate words back and forth by translating sounds to letters. Not so with Chinese (it's letters have no sounds and there are thousands of them).
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Anonymous2012-01-09 3:00
>>22
Not sure about your language, but there's only what, 7 parts of speech in english? On top of that, modifiers always get evaluated left->right until part of speech is reached (we don't see dogs red, but I will concede we do see shit like hastily running/running hastily). On top of that, the letters don't modify the behavior of the order of processing - this is an incredible difference (excluding that math is numerical processing).
You even have to admit, many, many people in the world prefer english because although it's a shitty, obtuse language, it's better than theirs. (50000 pictographs? Honorifics? Gendered nouns? Horrible mouthgore of pronounciations?)
Even better, we teach people to abstract words. We don't teach people how to abstract numerical relationships, we just says "parentheses go here or there or whatever" (only in a book from ~1947 did I read the term "algebraic expression")
anyways, the paper looks interesting. Thanks for the link. I've been meaning to keep up with stuff like this. If I find out what it is about, I'll bump the thread I guess.
In testing one must find a way to repeat the erroneous behavior so that the cause can be identified. I am an expert [i][o][u][b]BBCODE QUALITY ASSURANCE ENGINEER[i][o][u][b]
>>12 That's the ultimate problem: A symbol is MEANINGLESS without MEANING.
I think that's what >>1 (or yours?) problem is: you're forgetting that a string of symbols can be interpreted and has a meaning, or to say it more clearly: semantics. Math talks about relations and behavior of very abstract objects, that is, of truth about general objects and structures - if you added real-world characteristics to them, the truth would remain unchanged, the only thing that sets them apart are very simple general properties and the rest is irrelevant. To put it another way, finite numbers will always have the same properties given the standard interpretation of arithmetic, regardless of whatever other properties you ascribe to those numbers (such as beings rocks, bits, atoms, whatever) - math can be seen to describe these general timeless relations, the syntax is less important as long as it can be learned and someone can interpret it, if you wanted to, you could do all your inferences and syntax in lisp, despite that most mathematicians tend to use more 'natural' styles that they're now used to (I'm not claiming that the syntax they use is that good, just that it works).
>>40
I know that you're probably an ultrafinitist, but theorem provers/formal systems can easily talk about things they can't reach. Infinity for me is just a way of talking about uncomputable properties, which computable objects(such as programs) can have, for example, wether a program halts or not, or more general convergence-related properties. Infinity is not a meaningless concept, even if you never directly touch it, but it's always needed when you want to go meta(such as about 'all' programs or 'all' functions). Since you seem very pragmatic and don't like to talk about such general things, maybe math isn't for you (not talking about syntax here, but about semantics), however these 'unreachable' concepts are important and they lead to deeper truths which will nevertheless hold, regardless of what you think about 'infinity' or wether you want to contemplate it at all.
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Anonymous2012-01-09 7:33
>>44 it's always needed when you want to go meta(such as about 'all' programs or 'all' functions).
There is no such thing as "all". It's a buzzword.
>>51
So you don't accept any abstract concepts, fine then, then you don't need to talk about math or computer science, as they don't study the behavior of actual programs, but the behavior of ideal programs. If your actual programs match the behavior of ideal programs, that's only because the physical implementation is close enough. However, how if you insist on not giving any privileges to the abstract and only giving privileges to physically existing things, what do you think this universe is? You can't even make the hypothesis that it exists as a coherent structure, or if you do, you can only infer it indirectly through your observations, thus you're making use of the abstract. If you don't make any such hypothesis, the only thing that you know are your perceptions at the moment (here and now) and your possibly unreliable memories. Sounds like a bad way to waste one's intelligence.
What does /prog/ think of the lambda calculus? No doubt you guys looked into it.
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Anonymous2012-01-09 10:19
>>53
It has ugly syntax. (/\crap.crap /\crap.crap) /\crap.crap looks like crap.
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Anonymous2012-01-09 11:01
Mathematics is purely functional, symbolically evaluated, has unbounded recursion, and allows new ad hoc operators, functions, constants, variables, and orders of evaluation to be defined. APL is a formal language defined in terms of mathematics, described in terms of mathematical operations as a notation used in the formal description of the System/360, making APL functions "overloaded operators" of mathematical notation.
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Anonymous2012-01-09 13:59
>>44
The symbol _never_ has meaning, the way it is used and related to by other symbols defines its meaning.
Mathematics /is/ about real-world structures, that is what mathematicians do, they define a universe and put the objects into it and declare themselves to have seen some godlike properties despite declaring in advance what'll happen.
There is a simple reason why lambda calculas took so mind-boggling long to develop, and why lisp is still confusing to 90% of programmers:
They can't CONCEIVE of a thought that has no label, no name, no representation other than it's relations. In NO mathematical axiomatic methods, proofs, anything, do you have things like 1. An unbounded space 2. there is NOT, AND, NOR, and they can feed into eachother. 3. There is time.
ALL proofs start with defining names, and only afterwards do they care of relations. It is of no use to anyone if someone can memorize every language in the world if they don't know what any of the words mean, and this is what the discussion has been in the thread, so good job taking one quote of someone out of context and being retarded to the rest of the thread.
Mathematics /is/ about real-world structures, that is what mathematicians do, they define a universe and put the objects into it and declare themselves to have seen some godlike properties despite declaring in advance what'll happen.
They are idealized structures which soon end up being far divorced from what is physically computable (but not divorced from what is effectively computable, or Turing computable). They can't CONCEIVE of a thought that has no label, no name, no representation other than it's relations.
Use gensyms, graphs or whatever you wish. In NO mathematical axiomatic methods, proofs, anything, do you have things like 1. An unbounded space 2. there is NOT, AND, NOR, and they can feed into eachother. 3. There is time.
1. Most proofs that I've seen have unbounded spaces or domains. Most proofs in arithmetic are done over natural numbers, which are unbounded.
2. You can define in relations in terms of each other. As far as logic is concerned, you don't really need all 3 of those. A NAND or NOR is sufficient.
3. Time is useful as far as describing computations, but it can always be abstracted away as a relationship between states. f:N->N, f computable, f(time)=state. Possible recursive definition: f(0)=initial_state. g:N->N, g computable, f(n)=g(f(n-1)), where g just describes how to compute the next state given the current state. Want to know what g's implementation would be like? Take a look at a Turing machine or Primitive Recursion Functions/Primitive Recursive Arithmetic. good job taking one quote of someone out of context and being retarded to the rest of the thread.
I only answered it once, well, twice now, counting your answer, probably again out-of-context.
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Anonymous2012-01-09 14:32
>>57
You're still a moron with no possible future as a computer programmer.
>>58
Back to /bankruptcy court, kodak_gallery_programmer/
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Anonymous2012-01-09 14:42
>>59 Most proofs in arithmetic are done over natural numbers, which are unbounded.
Hey dipshit, if natural numbers were unbounded, then they couldn't be countable. I believe the correct term that you're looking for is "countably infinite". But I could be wrong since I'm only a programmer and you're only a toilet scrubber.
Unbounded just means not limited by any upper bound. There is no finite maximum number k which is greater than all finit natural numbers. Bounded means that there is some upper bound which is greater than all elements in some domain. Countably finite is a correct way to refer to naturals as their cardinality is aleph null. Another way is calling them enumerable.
>>60
There are several definitions of bounded and unbounded and none of them involve countability. Given the absolute value as the metric the natural numbers are uncountable.
>>65
Yes, naturals are countable. Reals are not. Cantor's diagonalization proof shows that. If you're a classical finitist, you may not think reals can exist in nature, but I do think considering computable reals should be fine(computable reals are countably infinite though, have a measure 0 within the whole of real numbers). However that shouldn't matter, even if they turn out to be useful fictions instead of something stronger: they are a good bet which allows speedup, and so far the bet has been giving useful results, even though we don't know if it's really a correct bet (if ZFC or some other infinitary system is consistent).
>>66
If k is maximum, then k+1 exists, which contradicts your upper bound assumption. Learn to read. No.
You're stupid and you know nothing of computability/recursion theory. Just get out of here or read a book already.
>>64 Natural numbers are unbounded, they have no upper bound. That's why they're (countably) infinite.
It depends you idiot. What happens if I have some function called f defined over a set of natural numbers that doesn't satisfy either the onto on one-to-one condition that is necessary for countability?
Exactly. Now shut your pie hole and get ready to go scrub another toilet.
>>68 You're stupid and you know nothing of computability/recursion theory. Just get out of here or read a book already.
I never said nor implied recursion theory you idiot. Good lord. Have you ever done a formal math proof in your entire life?
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Anonymous2012-01-09 15:00
The natural numbers are countably infinite but also unbounded given a metric that makes geometrical sense when used on the real numbers. There is no sense in talking about boundedness without a metric, I think you could easily construct a metric in which an open ball of radius 1+epsilon centered on any natural number would extend to every other natural number for every epsilon > 0 and so the set would be bounded with respect to that metric.
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Anonymous2012-01-09 15:04
>>69 It depends you idiot. What happens if I have some function called f defined over a set of natural numbers that doesn't satisfy either the onto on one-to-one condition that is necessary for countability?
Then you have said nothing about the countability of that set, you may always create a function with those properties from any set to another set.
>>69 What happens if I have some function called f defined over a set of natural numbers that doesn't satisfy either the onto on one-to-one condition that is necessary for countability?
We were talking about the set of naturals, not on functions defined on them.
As for the other posts: most of the posts were talking about naturals, not reals. An upper bound for a set of naturals just means the set has a finite cardinality.
>>72
Are you really this stupid? Having a set defined by the natural numbers is a way to define a set. This is something you learn in the first week of set theory you dumbass.
>>75 By the way any subset of a countable set is countable
That is incorrect you mental midget. What happens if I have a set of real numbers in the interval [0,1]. The interval, which is defined on the set of natural numbers is countable. However, the subset of this interval isn't.
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Anonymous2012-01-09 15:12
>>74 Having a set defined by the natural numbers is a way to define a set.
I never stated otherwise you fucking moron, you made a false statement that a set consisting solely of natural numbers might not be countable, they always are and the proof is trivial but apparently your monkey ass is probably too retarded to pass basic mathematics.
>>73
I was thinking of some kind of enumerated list that maps to a set of natural numbers.
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Anonymous2012-01-09 15:15
>>76
Listen you piece of shit retard, [0,1] isn't a subset of the natural numbers you fucking moron, do you sincerely believe that 0.5 is a natural number?
Go read any basic book on mathematics you fucking moron, how are you even able to operate a fucking computer is beyond me.
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Anonymous2012-01-09 15:15
>>77
Stop projecting and go scrub another toilet you fucking idiot. Again, you have no possible future as a computer programming.
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Anonymous2012-01-09 15:17
>>80
That's rich coming from someone who believes that the interval [0, 1] is a subset of the natural numbers. Go fuck yourself you fucking retard, you're wrong and you're a fucking moron. This is basic mathematics.
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Anonymous2012-01-09 15:17
faggots everywhere, get a fucking room already and spread your AIDS amongst you
>>79 Listen you piece of shit retard, [0,1] isn't a subset of the natural numbers you fucking moron,
I have several math proofs done by some well known professors at UC Berkeley that say otherwise.
do you sincerely believe that 0.5 is a natural number?
I said the subset was the real numbers. In other words, I never implied that 0.5 is a natural number. Again, you're stupid. And again, you have no possible future as a computer programmer.
>>81
It satifies the definition of a subset. I have a lot of my math proofs from grad school at UC Berkeley to support this fact. Now shut up and go scrub another toilet.
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Anonymous2012-01-09 15:21
>>83
You don't even know what a subset is you fucking piece of shit retard. For a set A to be a subset of a set B every element in A has to be an element of B, 0.5 is in [0,1] but 0.5 isn't a part of the natural numbers, so you're wrong you god damned fucking retard. You don't know what you're talking about and you're a god damned retard.
I never implied that 0.5 is a natural number.
You just did by stating that [0,1] is a subset of the natural numbers you fucking retard.
Again, you don't know what you're talking about, again you're a fucking retard.
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Anonymous2012-01-09 15:21
I used to think kodak was somewhat intelligent, now I don't anymore.
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Anonymous2012-01-09 15:22
Silly Kodak-san, the natural numbers are a subset of the real numbers, not the other way around, silly Kodak-san.
>>84
By the way you fucking moron you have yet to make a single correct statement in this thread beyond that the natural numbers are countable, congratulations, you're at the level of a challenged high school student you piece of shit.
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Anonymous2012-01-09 15:25
>>85 You don't even know what a subset is you fucking piece of shit retard. For a set A to be a subset of a set B every element in A has to be an element of B
No. What happens if the set includes the empty set?
You just did by stating that [0,1] is a subset of the natural numbers you fucking retard.
Yes, and from this subset, I can construct set of real numbers such that each value in [0,1] maps to a real number.
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Anonymous2012-01-09 15:26
>>88
Huh? You have no clue what you're talking about. Again, have you ever done a formal math proof in your entire life?
The set S defined by the interval [0,1] is:
- Finite, if ∀x∈S.x∈Z.
- Countably infinite, if ∀x∈S.x∈Q.
- Uncountably infinite, if ∀x∈S.x∈R.
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Anonymous2012-01-09 15:38
>>69 It depends you idiot. What happens if I have some function called f defined over a set of natural numbers that doesn't satisfy either the onto on one-to-one condition that is necessary for countability?
For your statement to make even a shred of sense it needs to be a function defined over some other set to the natural numbers, not over some subset of the natural numbers. And even then you have proved nothing, you may always construct such a function from any set to another set, consider a set X now let f(x) = 0 for each x in X, this is a function which is neither bijective or surjective while X might be countable or uncountable.
If I'm generous I'm guessing what you really mean is that if there does not exist any function f that is an isomorphism from the set to the natural numbers then the set isn't countable.
>>71
Yeah the metric you're looking for is d(n, m) = if n != m then 1 else 0, the naturals are bounded with respect to this metric.
>>75
Is correct and the proof is even in the post.
>>83
You seem a bit confused here, [0, 1] is a real interval, it is defined as the set { x in the Reals : 0 <= x <= 1}, and that is not a subset of the natural numbers.
If I'm being generous I suppose what you're trying to state is that {0, 1} is a subset of the natural numbers, which might or might not be true, most definitions of the natural numbers don't contain 0 but some do, let's be generous and assume that 0 is a member of the natural numbers.
>>92
0 is natural, rational and real; 0.5 is not natural, it is rational and real; pi is not natural nor rational, but it is real. Thus, N is a subset of Q is a subset of R.
Seriously, go back to high school.
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Anonymous2012-01-09 15:40
>>92
No for example 0.5 is in [0,1] but not in the natural numbers, so [0,1] is not a subset of the natural numbers.
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Anonymous2012-01-09 15:43
Taken from http://en.wikipedia.org/wiki/Countable_set Proposition: Any subset of a countable set is countable. Proof: The restriction of an injective function to a subset of its domain is still injective.
There you go, it's simple and a single sentence long, I suggest that you revisit some basic high school mathematics before you run your mouth again you fucking retard.
>>96
Take the interval [3,4]: ∀x∈{3,4}.x∈N ∀x∈{3,...,3.001,...,3.9,3.99,...,4}.x∈Q ∀x∈{3,3.1,...,3.14,...,π≈3.1415926535897932384626433,...,3.9,3.99,...,4}.x∈R
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Anonymous2012-01-09 15:47
Smart people get mathematics right, unintelligent people get mathematics wrong or think mathematics is scary.
>>99
What are you trying to prove? Again, [0,1] isn't a subset of the natural numbers, if it were then every element in [0,1] would also be in the natural numbers, 0.5 isn't a member of the natural numbers so [0,1] isn't a subset of the natural numbers. It's that fucking simple you god damned piece of shit retard, go back to special ed and try to learn something this time you fucking moron.
>>94 You seem a bit confused here, [0, 1] is a real interval, it is defined as the set { x in the Reals : 0 <= x <= 1}, and that is not a subset of the natural numbers.
No, you're confused. Let's say I have the interval [0,1] defined on the set N. I can construct a function k* such N maps to a set real numbers, R, in [0,1].
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Anonymous2012-01-09 15:56
>>104
If you see >>76 he clearly states that it's the real number interval, but let us be generous, from now on, let [0,1] be the set {0,1} and please, I'd very much like you to attempt to continue with the argument in >>76 with this new piece of information, I'm sure that the result will be very humorous.
Natural numbers are defined rather simply by 0 and all finite applications of the successor function. Natural numbers are provably enumerable/countably finite.
Real numbers are defined by dedekind cuts, not always accessible in a 'constructive' way, which is why one would need the "Axiom of Choice"(when talking about set theory) if they want to quantify over reals. Real numbers are provably non-enumerable/uncountably finite. The reals as a set/structure do contain the naturals, although one needs to consider what axiomatic system, model and interpretation is being used when talking about something. Some concepts don't exist or mean different things in different interpretations, such as various things being argued in this thread.
It would do well for people to specify exactly which systems they are talking about, but civilized discussion has become hard, mostly due to a certain kodak or kodak-like person who insists on insulting everyone he responds to and forcing those replying to him to insult back. I'll take my leave and try not to reply any further, at least until the discussion has returned to more civilized levels.
>>106
I said the set of real numbers *in* the interval [0,1]. That implies there function that defines a mapping from N to R. Where N is the set of natural numbers and R is the set of real numbers.
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Anonymous2012-01-09 16:01
>>105 Let's say I have the interval [0,1] defined on the set N
All right so [0,1] means {0,1} now, that is fine. I can construct a function k* such N maps to a set real numbers, R, in [0,1]
So now there is a function which maps from N to R, and it only has values in [0,1]? ([0,1] I presume means {x : 0 <= x <= 1} now, correct me if I'm wrong).
>>106 >>76 is:
- claiming that the real interval [0,1] is a subset of the natural numbers.
- using that claim to prove that subsets of countable sets can be uncountable.
The fallacy here is that the real interval [0,1] is not a subset of the natural numbers, because 0.5 nor pi are natural numbers, but are contained in [0,1].
Let's use the integer interval [0,1] instead, which is the set {0,1}. It has 2 elements: 0, 1. It is countable. It is a subset of the natural numbers, because 0 and 1 are natural numbers.
We proved that kodak-kun is retarded. QED
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Anonymous2012-01-09 16:07
>>110
Well mister Kodak, I am not impressed, both of these still hold true the real interval [0,1] is not a subset of the natural numbers any subset of a countable set is also countable
Go on, prove them wrong now.
>>113
He's just another programmer without any mathematical ability and no mathematical talent, just another code monkey. He should stick to the stuff he actually knows which is programming.
>>116 And what happens if I have two ore more subsets that map to the same set?
Not anything that makes that statement false.
Let me find the proof.
No need to, 0.5 is in the interval [0,1] while 0.5 isn't in the natural numbers, therefore the interval [0,1] is not a subset of the natural numbers.
>>120
Congratulations, you've proved that rational numbers are countably infinite.
Now, where's the proof for real numbers?
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Anonymous2012-01-09 16:26
>>118
No, but saying any subset of a countable set is also countable is sloppy. The general way to assert something like this is to say that it is both one to one and onto. That way you can be assured that any two given subsets won't map to the same set.
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Anonymous2012-01-09 16:27
>>121
And this is a subset because r* maps from N to R.
>>120
A function can not be the subset of anything, if you are talking about the set {k*} then that is also not a subset of the natural numbers as there are no functions in set of natural numbers. Do continue however, I am finding this quite humorous.
>>125
Did you even make it beyond the 10th grade you idiot? Alao, I never said the function itself was a subset. Cripes you are on stupid SOB. I guess I should have seen it coming when you tried to assert that
I can't believe what I'm reading. No wonder Kodak is worth shit nowadays. They employed programmers who don't have the level of Mathematics required in seventh grade. Let's see the "lol I did Berkly" proof.
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Anonymous2012-01-09 16:32
>>127
And again, this breaks down if I have two sets that map to the same set.
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Anonymous2012-01-09 16:33
>>122
There is no need for any extra clauses, any subset of a countable set is countable is a true proposition, there is no extra condition that makes this proposition false. It's not a sloppy statement at all, it's quite pristine in fact.
It's odd to me that you as a programmer has had so little training in mathematical thinking.
Your spelling is as bad as your reading comprehension. It's "Berkeley". Not "Berkly" you idiot.
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Anonymous2012-01-09 16:35
>>129 this breaks down if I have two sets that map to the same set
No it doesn't you god damned piece of shit moron, it's always true. Didn't they teach you how proofs work in special ed you fucking retard?
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Anonymous2012-01-09 16:36
>>130
Again, what happens if the there are two or more subsets that map to the same set? It's uncountable because you don't know how many subsets are getting mapped.
>>134
You have zero clue what you're talking about. Again, what happens if I have two more more subsets that map to the same set? How do I count it? I can't because it doesn't satisfy one of the conditions for countability.
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Anonymous2012-01-09 16:39
>>133
Nothing happens, any subset of a countable set is countable continues to be true, the proof is quite simple and two versions of it has been posted in this thread.
You can't evade mathematical proofs, it's not a matter of who shouts loudest and who spams the most insults. If it's been proven then it's true, you can't undo that I don't understand why you're trying.
>>133
You seem to be awfully confused by the way, you're using map and set as interchangeable objects while they are quite different.
What are you attempting to count? The elements of the subset?
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Anonymous2012-01-09 16:42
>>138 Nothing happens, any subset of a countable set is countable continues to be true
Yes, something happens. It becomes uncountable because I can't determine the exact number of duplicate subsets that are getting mapped.
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Anonymous2012-01-09 16:44
>>137
Well, you can count the sets if they are subsets of a countable set. What they map to is completely irrelevant and you're quite frankly a fucking moron if you think otherwise.
>>142 It becomes uncountable because I can't determine the exact number of duplicate subsets that are getting mapped
And why do you feel you need this information to determine if any of the subsets are countable?
>>143 What they map to is completely irrelevant and you're quite frankly a fucking moron if you think otherwise.
And what happens if this involves some kind of password scheme? Or what happens if this some kind of credit card transaction? Exactly, you don't know because you don't work as a programmer.
>>146
Oh, I don't know. Maybe it's because we handle customers credit card transactions. Or maybe it's because we user accounts that span over several thousands of servers.
>>147
You still can't change the mathematical truth that any subset of a countable set is also countable, what I work as or you work as is completely irrelevant, mathematics don't work that way kid.
I'm going to hold this over you Kodak, I don't understand why you don't yield when faced with mathematical truth but I quite frankly don't care. It's quite embarrassing that you have so little training in mathematical and logical thought, but I guess some companies settle for lesser programmers.
You don't even know what a subset is, you're a less capable mathematician than your average challenged high school student.
I suppose you're just incapable of logical thought, I guess this is a result of your mental status as a fucking retard.
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Anonymous2012-01-09 16:54
>>137
Wanna state that a touch more formally? Your language is ambiguous.
>>137
Technically the set of all subsets of a set A, the so called powerset: P(A), has greater cardinality than the set A, to be more precise |P(A)| = 2|A|. The following does hold true: |N| = |Z| = |Q| = \aleph_0 - or naturals, integers and rationals are countably infinite.
Through Cantor's diagonalization argument, it can be shown that reals have a greater cardinality than naturals, called the cardinality of the continuum c, and assuming the Axiom of Choice, it's \aleph_1. An interesting thing about the powersets of infinite sets is that they will always give you the next cardinal, so the powerset of naturals has the same cardinality as that of reals (assuming said axiom).
However, I don't think that's what you are talking about at all. I think most people here are merely talking about the cardinality of well-known sets such as naturals and reals, while it doesn't seem to me clear what you're talking about, although if I had to give it a try, I'll say this:
there are functions f from N to N (f:N->N). There are a countable infinity of such enumerable functions which are computable, although they are not all total functions, some are partial (to say it in a more clear language: some are undefined on some inputs, or they never halt). The actual set f:N->N of total functions is not enumerable (including non-computable): this is non-trivial, but it can be proven by diagonalization, the same technique used to prove the uncountability/unenumerability of the reals.
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Anonymous2012-01-09 16:57
>>152
His language is that of your typical unintelligent person, he likes to mix in words he heard in the math classes he failed so he can act intelligent in front of others. He has no clue what he's talking about, he's just another programmer with no mathematical ability or talent, just another code monkey.
He has showed to us today that he's completely devoid of anything that may be called intelligence and all that is left is the pathetic shell of a man that spends his time on the internet trying to argue against mathematical truth.
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Anonymous2012-01-09 16:57
>>151 You still can't change the mathematical truth that any subset of a countable set is also countable,
Yes, there are cases where this is true. However, there a just as many cases where this isn't.
You don't even know what a subset is, you're a less capable mathematician than your average challenged high school student.
You still never answered my question about the empty set you stupid shit.
I suppose you're just incapable of logical thought, I guess this is a result of your mental status as a fucking retard.
Stop projecting and go scrub another toilet.
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Anonymous2012-01-09 16:59
>>154
This is coming from someone that works as a general laborer.
>>156
I have no idea what >>154's job is, nor do you.1 However, I do think that ad hominem's or various appeals to one's social status are rather irrelevant, it doesn't matter if someone is a bum or the president of a country: either they are speaking the truth or they are not and their current status is irrelevant - maybe a richer man has a higher chance of getting a better education, but that doesn't mean that whoever it is you're speaking is wrong merely based on their social status, instead you should judge content merely by its truth value and comprehensibility.
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Anonymous2012-01-09 17:04
>>155 Yes, there are cases where this is true. However, there a just as many cases where this isn't.
No it is always true you fucking retard, don't you know what a fucking proof is?
See >>75 and >>97 you fucking retard, those are only two of the possible proofs, here let me get you some more since you're apparently too fucking dumb to operate Google, http://lmgtfy.com/?q=The+subset+of+a+countable+set+is+countable here you can choose any one of those, the proposition is always true and never fails. You still never answered my question about the empty set you stupid shit.
Please restate your question, it must have gotten lost in all the other retarded shit you were saying.
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Anonymous2012-01-09 17:05
>>158
I never said otherwise you fucking idiot. Man you're hella dumb.
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Alpha Male!gD3Op2fhHs2012-01-09 17:08
>>157
Trust me brah he's not rich or intelligent, he's a poor unintelligent general laborer and every post he has made in this thread shows it. He's hella dumb, and he's mad jelly that I'm smarter than him, more academically and athletically accomplished than him and that I'm way better looking than him. Judging from his posts he's a 4/10 at best.
Just check out /prog/-scrape to see some of the instances where I demolished his scrawny ass.
However, there a just as many cases where this isn't.
mean in that context? Because it kind of looks to me that you think it's false in some cases.
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Anonymous2012-01-09 17:12
So to recap, statements that are still true
any subset of a countable set is also countable the real interval [0,1] is not a subset of the real numbers
And just a note at the end, Kodak thinks both of these are false despite mathematical evidence.
>>162
Also, he contradicted himself multiple times, why do you still take him seriously?
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Anonymous2012-01-09 17:17
I don't think Kodak is bright enough to scrub my toilet.
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Alpha Male!gD3Op2fhHs2012-01-09 17:18
Kodak is a jelly little manlet who doesn't know that he should shut up when he's facing his betters, he's a pathetic omega male who constantly get owned in real life by alpha males such as myself, that's why he acts out so much on the internet because it's the only time he can feel like a man.
Hey manlet, I mean, hey Kodak, how tall are you again? Oh you're still 5 feet tall? Tough luck brah, I'm sure you can get those leg extensions you want to get when you're done scrubbing enough toilets, at least that way a 3/10 woman might consider you if she's drunk.
>>162 [0,1]∩ N = {0,1}. |{0,1}| = 2, countable. |[0,1]|=c=\aleph_1, uncountable. [0,1]/ N = (0,1). |(0,1)|=c, uncountable
Of course [0,1] as real interval isn't a subset of naturals, and I don't know why would anyone even bother talking about sillyness.
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Anonymous2012-01-09 17:20
>>163
I never said [0,1] itself was the real interval.
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Alpha Male!gD3Op2fhHs2012-01-09 17:23
>>168 |[0,1]|=c=\aleph_1
You can't assume the continuum hypothesis and still use ZFC brah.
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Anonymous2012-01-09 17:25
>>168 [0,1]∩ N = {0,1}. |{0,1}| = 2, countable. |[0,1]|=c=\aleph_1, uncountable. [0,1]/ N = (0,1). |(0,1)|=c, uncountable
What's this supposed to prove?
>>170 http://en.wikipedia.org/wiki/Continuum_hypothesis CH is independent of ZFC. Like CH, GCH is also independent of ZFC, but Sierpiński proved that ZF + GCH implies the axiom of choice (AC), so choice and GCH are not independent in ZF; there are no models of ZF in which GCH holds and AC fails. Kurt Gödel showed that GCH is a consequence of ZF + V=L (the axiom that every set is constructible relative to the ordinals), and is consistent with ZFC. As GCH implies CH, Cohen's model in which CH fails is a model in which GCH fails, and thus GCH is not provable from ZFC
http://en.wikipedia.org/wiki/Axiom_of_choice#Independence Assuming ZF is consistent, Kurt Gödel showed that the negation of the axiom of choice is not a theorem of ZF by constructing an inner model (the constructible universe) which satisfies ZFC and thus showing that ZFC is consistent. Assuming ZF is consistent, Paul Cohen employed the technique of forcing, developed for this purpose, to show that the axiom of choice itself is not a theorem of ZF by constructing a much more complex model which satisfies ZF¬C (ZF with the negation of AC added as axiom) and thus showing that ZF¬C is consistent. Together these results establish that the axiom of choice is logically independent of ZF.
>>171
Nothing, just showing the obvious: that the real interval [0,1] isn't a subset of naturals, and that it contains only 2 naturals within it, 0 and 1.
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Alpha Male!gD3Op2fhHs2012-01-09 17:33
>>172
Exactly brah, there is no point in assuming the continuum hypothesis if you're only using constructs from ZFC.
>>173
You mirin? Don't be mad because you're unaesthetic, I can't help my genetics.
>>179
Oh well, that's ok, everyone can make a mistake every now and then.
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Alpha Male!gD3Op2fhHs2012-01-09 17:41
>>177
Usually he just cries alone in his dirty apartment, then after a while he starts cross dressing in his late sisters dirty clothes and after that he usually feels better.
He'll be back brah, and more furious than ever because some alpha stole the girl he thought was cute back in high school, omegas are funny like that.
Kodak-san! The way everyone attacked you in this thread shows how insecure you've made them! Continue your work Kodak-san! It's working!
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Anonymous2012-01-10 15:43
>>148
Strange I didn't notice this thread before now, I understand why Kodak were at odds with everyone though, he mistook his ability to immediately count some user accounts with the mathematical definition of countability.
So, do you have mathematics with nicer Lisp-like syntax and without infinite faggotry? Cuz you know, we are living in 21st century, there should be more anarchy and subcultures, more opinions, and single monotheistic mathematics just sucks here. Why cant we have separate mathematics for women and for buddhists? Autocracy must be stopped!
>>76 That is incorrect you mental midget. What happens if I have a set of real numbers in the interval [0,1]. The interval, which is defined on the set of natural numbers is countable. However, the subset of this interval isn't.
ahaha kodak_faggot_jobless sucks at math
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Anonymous2012-04-22 15:06
>>189 Why cant we have separate mathematics for women and for buddhists?
feminist positive-action faggot detected