Closures

can someone explain them to me.. I'm thinken they might be simple but my pee sized brain can't comprehend most definitions off of google

Infinity is like a circular road with no starting point. When you reach the end(since its circle, impossible) you win the game.

>>120
You should really pick an actual radical ideology. You'll make a much better schizophrenic that way.

>>119
If you ignore Cantor's work, you can say that a sequence of numbers that does not end is infinite. For example, there is no largest natural number - if you say you found one, just take its successfor to show otherwise. That's the simplest kind of infinity. A much stranger type of infinity is that of real numbers - any number has an infinite amount of digits (there is no last digit).

If you take Cantor's work into consideration, he basically takes the set theory (a bunch of axioms that define it, look at http://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory), and 2 axioms of particular importance (which are rather artificial, but are needed to define 'infinity' as a concrete value as far as ordinals are concerned), the axiom of infinity ( http://en.wikipedia.org/wiki/Axiom_of_infinity ) and the axiom of power set ( http://en.wikipedia.org/wiki/Axiom_of_power_set ). Within set theory, you can find the cardinality of some set (the amount of elements in the set). He then defines the cardinality of N (or of the infinite set defined in that axiom before this), and that value is denoted by N₀, or aleph-null. What he did here is say that the cardinality of an infinite set is defined as a concrete value, the ordinal N₀, instead of merely left as some undefined infinite. That is what I meant by accepting the concrete infinity - you can say that the number of natural numbers is potentially infinite, but what Cantor did was consider this infinite value something concrete, after that he shows that the cardinality of R (real numbers) is greater than that of natural numbers (N₀), and this greater value is N₁. By the axiom of power set, he then shows that for any such ordinal (N_x), there is always a set whose cardinality is a greater infinite ordinal. He also set forth a conjecture which states that the next ordinal after N₀ is N₁ and there is nothing inbetween, this is called the continuum hypothesis. The very strange thing about this hypothesis is that it was proven that it's impossible to prove and disprove, which is a very weird result which likely shows that set theory as defined by ZF+Infinity+PowerSet might be inconsistent in some ways (along with some other paradoxes that plague it). Regardless of what you think of wether real numbers exist or not, or wether you can treat infinities as something concrete like he does, it does show you want happens when you do, and math is done by taking some axiomatic system and running with it to obtain results - if the results show that it's inconsistent, it was still a worthy exercise, I do know that I learned something by reading some of his work.

>>123
You're too smart to be on here.

>>124
You realise he's just regurgitating the contents of a mathematical logic course, right? This is standard fare for a COMPUTER SCIENTIST.

>>121
Circular Road would be a simple recursion.

>>122
I'm already a subjective idealist -- an extreme position on a bottom-up vs top-down scale. Jews force `top-down`, as they love communism, set theory and type classes -- which are oposites of anarchism, justificationalism and prototypes.

>>123
you can say that a sequence of numbers that does not end
define "does not end"

>>125
You realise he's just regurgitating the contents of a mathematical logic course, right?
Nope. I also hate "infinitesimals" buzzword, which is standard for engineers.

This is standard fare for a COMPUTER SCIENTIST.
Commuter Scientists love Set Theory.

>>125
He still is on a different level.

>>127
Infinitesimals aren't a buzzword at all. They're an alternative approach to calculus based on the concept of defining the smallest possible number. All practiced calculus was based on them until the late 19th Century, when limits were invented and applied to make math more logically rigorous.

Although we've never been able to prove that infinitesimals are sound reasoning, there's nothing you can do with limits that you can't do with infinitesimals, and vice versa. It's like Turing machines and lambda calculus.

>>129
proof?

>>129
define "smallest possible number"
define "limits"
define "sound reasoning"

>>129
Turing machines and lambda calculus.
Pseudoscience. IRL there're only Finite State Automations, no Turings.

>>126
define "does not end"
A sequence that does not have a last element. If you assume it's the last element, you'll be able to find another one, and if you assume that other one is the last element, you'll be able to find another one and so on.

>>130 -> http://en.wikipedia.org/wiki/Infinitesimal

>>133
you'll be able to find another
What if don't have enough memory?

>>126
(defvar xs '(1))
(rplacd xs xs)
xs
(mapcar #'print xs)
Wait until it stops printing xs. That means does not end.

>>134
so, where is the proof?

>>136
It's just a simple recursion. Like 1/3

>>137
There isn't one. Having a proof that infintesimal-based calculus is equivalent to limit-based calculus would be proof that they're sound. It's only the case that we have yet to find a counter-example to the claim that they're equivalent.

>>139
>>131

>>127
infinitesimals
While I can't say they sit too well with me, they are well defined, at least within set theory.
Here's an example:
x² = 2, where x > 0 => x = √2.
It can be shown that x is not a rational number (that is, you can prove that there is no p/q = x, with (p,q)=1). The proof is elementary and is shown in middleschool textbooks, so I'll refrain from showing it here.

One way of defining x is to use 2 infinite sets in ZFC:
{{y∈Q|y² < 2},{y∈Q|y² ≥ 2}}
This is called a Dedekind cut. You're defining all the rational numbers to the left (smaller) and to the right (greater or equal) to this irrational number. By doing this you can approximate this irrational number with ever-increasing accuracy. Would any side ever reach this number, despite approaching it ever-so-closely? No, because that would require infinite number of digits in the fraction, which is not something a rational number can be, however the cardinality of both of those sets is aleph-null (infinite, same as N) as they are an interval of Q.

>>141
within set theory
define "set"

>>141
>{{y∈Q|y2 < 2},{y∈Q|y2 ≥ 2}}
Looks like haskell, but why do you use {} instead of [] for list comprehension?

Also, Q[y] is much nicer than your y∈Q and doesn't require off-keyboard characters.

>>143
FUCK FUCK FUUUUUUUUUUCK GOD DAMN HOW CAN YOU BE THIS STUPID FUUUUUUUUUUUUUUUCK I FUCKING HATE YOU GO KILL YOURSELF

>>143
It's just standard mathematical notation. It's a convention. There is no reason for syntax or language despitate that that's how it was written by its inventors and that's how it stayed. Of course, if you'd prefer other syntax, you're always free to define one.
>>142
http://en.wikipedia.org/wiki/Set_(mathematics)

>>144
Inventing notation requires you to define that notation. I thought that was clear to you when you started posting ``in LISP'' code and nobody understood it. Only when you posted the implementation did people stop criticizing your posts merely based on intelligibility of the syntax.

s/despitate/despite//

>>146
It's just standard mathematical notation.
Your standard looks ugly as Haskell. Are there alternatives? Even in D&D games there are alternative classes. I want to play fucking Kensai/Mage!

http://en.wikipedia.org/wiki/Set_(mathematics)
Can't find defintion of "set" there.

>>145
Please, be more serious. Mathematics is a serious subject, /c/ is serious board.

>>147
Inventing notation requires you to define that notation.
I know. But there're already much better notations. Why can't we use these instead of math one? They also easy to OCR, parse and copy-paste.

>>150
I know. But there're already much better notations. Why can't we use these instead of math one?
I'm actually with you on this one. The mathematical notation is full of ambiguities and special cases, though the set notation is pretty well defined and consistent when they don't go ``look I use English in formulae''.

>>151
set notation is pretty well defined and consistent when they don't go ``look I use English in formulae''.
(keep f xs) is much more consistent than {x∈xs|f(x)} craziness.

>>152
Sometimes they even omit explicit `xs`, writing {x|f(x)} and you have to guess real `xs`.

/prog/ can be such a weird place at times.

>>152-153
I've seen worse.

>>155
Because Set Theory cannot into unit tests.

>>150
Point to one and we can use it. I suppose some theorem-prover's syntaxes would be less ambiguous than the standard notation used in set theory and in math in general which at times are too close to natural language despite needing to be more exact/concrete. Something like http://en.wikipedia.org/wiki/Nqthm might work.

Point to one and we can use it
http://en.wikipedia.org/wiki/LISP

>>158
That's a family of languages. You need specific semantics and symbol names to talk about more concrete things.

>>159
Sorry, I've other work to do, Set Theory (together with Java) being just a major inconvenience, getting a university diploma. Nothing I can do there, except following Unabomber's way and sending packages to major Universities.