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# Axiom Of Choice
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In [mathematics](math.md) (specifically [set theory](set_theory.md)) axiom of choice is a possible [axiom](axiom.md) which basically states we can arbitrarily choose elements of sets and which is famous for being controversial and problematic because it causes trouble both when we accept or reject it. Note that this topic can go to a great depth and lead to philosophical debates, there is a huge rabbit hole and mathematicians can talk about this for hours; here we'll only state the very basic and quite simplified things, mostly for those who aren't professional mathematicians but need some overview of mathematics (e.g. programmers).
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Firstly let's answer **what really IS the axiom of choice?** It is an [axiom](axiom.md), i.e. something that we can't prove but can either accept or reject as a basic fact so that we can use it to prove things. Informally it says that given any collection of sets (even an infinite collection of infinitely large sets), we can make an arbitrary selection of one element from each set. More mathematically it says: if we have a collection of sets, there always exists a [function](function.md) *f* such that for any set *S* from the collection *f(S)* is an element of *S*.
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This doesn't sound weird, does it? Well, in many normal situations it isn't. For example if we have finitely many sets, we can simply write out each element of the set, we don't need to define any selection function, so we don't need axiom of choice to make our choice of elements here. But also if we have infinitely many sets that are well ordered (we can compare elements), for example infinitely many sets of [natural numbers](natural_number.md), we can simply define a function that takes e.g. the smallest number from each set -- here we don't need axiom of choice either. The issues start if we have e.g. infinitely many sets of [real numbers](real_number.md) (which can't be well ordered) -- here we can't say how a function should select one element from each set, so we have to either accept axiom of choice (we say it simply can be done "somehow", e.g. by writing each element out on an infinitely large paper) or reject it (we say it can't be done). Here it is again the case that what's normally completely non-problematic starts to get very weird once you involve [infinity](infinity.md).
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**Why is it problematic?** Once you learn about axiom of choice, your first question will probably be why is it problematic if it just seems like an obvious fact? Well, it turns out it leads to strange things. If we accept axiom of choice, then some weird things happen, most famously e.g. the [Banach-Tarski paradox](banach_tarski.md) which uses the axiom of choice to prove that you can disassemble a sphere into finitely many pieces, then move and rotate them so that they create TWO new spheres, each one identical to the original (i.e. you duplicate the original sphere). But if we reject the axiom of choice, other weird things happen, for example we can't prove that every vector space has a basis -- it seems quite elementary that every vector space should have a basis, but this can't be proven without the axiom of choice and in fact accepting this implies the axiom of choice is true. Besides this great many number of proofs simply don't work without axiom of choice. So essentially either way things get weird, whether we accept axiom of choice or not.
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**So what do mathematicians do?** How do they deal with this and why don't they kill themselves? Well, in reality most of them are pretty chill and don't really care, they try avoid it if they can (their proof is kind of stronger if it relies on fewer axioms) but they accept it if they really need it for a specific proof. Many elementary things in mathematics actually rely on axiom of choice, so there's no fuzz when someone uses it, it's very normal. Turns out axiom of choice is more of something they argue over a beer, they usually disagree about whether it is INTUITIVELY true or false, but that doesn't really affect their work.
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# Brain Software
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Brain [software](software.md), also brainware, is kind of a [fun](fun.md) idea of software that runs on the human brain as opposed to a [computer](computer.md). This removes the [dependency](dependency.md) on computers and highly increases freedom. Of course, this also comes with a huge drop of computational power :) However, aside from being a fun idea to explore, this kind of software and "architectures" may become interesting from the perspective of [freedom](free_software.md) and [primitivism](primitivism.md) (especially when the technological [collapse](collapse.md) seems like a real danger).
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Brain [software](software.md), also brainware, is kind of a [fun](fun.md) idea of software that runs on the human brain as opposed to a typical electronic [computer](computer.md). This removes the [dependency](dependency.md) on computers and highly increases freedom. Of course, this also comes with a huge drop of computational power :) However, aside from being a fun idea to explore, this kind of software and "architectures" may become interesting from the perspective of [freedom](free_software.md) and [primitivism](primitivism.md) (especially when the technological [collapse](collapse.md) seems like a real danger).
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Primitive tools helping the brain compute, such as pen and paper or printed out mathematical tables, may be allowed.
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Example of brain software can be the [game](game.html) of [chess](chess.md). Chess masters can easily play the game without a physical chess board, only in their head, and they can play games with each other by just saying the moves out loud. They may even just play games with themselves, which makes chess a deep, entertaining game that can be 100% contained in one's brain. Such game can never be taken away from the person, it can't be altered by corporations, it can't become unplayable on new [hardware](hardware.md) etc., making it free to the greatest extent.
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One may think of a pen and paper computer with its own simple instruction set that allows general purpose programming. This instruction set may be designed to be well interpretable by human and it may be accompanied by tables printed out on paper for quick lookup of operation results -- e.g. a 4 bit computer might provide a 16x16 table with precomputed multiplication results which would help the person execute the multiplication instruction within mere seconds.
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One may think of a pen and paper computer with its own simple instruction set that allows general purpose programming. This instruction set may be designed to be well interpretable by human and it may be accompanied by tables printed out on paper for quick lookup of operation results -- e.g. a 4 bit computer might provide a 16x16 table with precomputed multiplication results which would help the person execute the multiplication instruction within mere seconds.
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Yet another idea is to make a computer with architecture similar to the typical electronic computers but powered by human brains -- let's call this a [human computer](human_computer.md) (not to be confused with people whose job was to perform computations!). Imagine that after a societal collapse we lose our computer technology (i.e. the ability to manufacture transistors and similar key components), but we retain our knowledge of computer architecture, algorithms and the usefulness of computers. As a temporary solution for performing computations we may create a "computer made of humans", a room with several men, each one performing a role of some computer component, for example an [ALU](alu.md), [cache](cache.md) and memory controller. Again, a special instruction set and a set of tools (such as physical lookup tables for results of instructions) could be made to make such a human computer relatively fast. It might not run [Doom](doom.md), but it could possibly e.g. compute some mathematical constants to a high precision or perhaps help find optimal structure of cities, compute stresses in big building etc. In such conditions even a slow calculator could be immensely useful.
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