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** Personality Constructs seem pretty close to being human, so it could be that even though GLaDOS knows there is no answer, she subconsciously tries to work it out anyway and gets trapped.
** I'm not a mathematician, but there are statements which can't be proven. There are also statements that can't be proven to be impossible to prove, etc. It may be that logically proving certain paradoxes are, in fact, paradoxes is impossible. An AI, no matter how complex, has to be built on logic. If it's impossible to determine a paradox is paradoxical, then the program tasked to determine which queries are worth considering will also enter an infinite loop - because it can't determine that it's indeterminate. "This statement is false" is not an example of such a paradox, since simply using the routine "If A=> not A and not A => A, quit" would resolve the issue, but they may exist.
*** This, I think, is a variation on the [http://en.[wikipedia.org/wiki/:Entscheidungsproblem |Entscheidungsproblem]], which basically says that there exists no algorithm that, given the description of a formal language (e.g. arithmetic or boolean logic) and a statement in that language, can determine the truth of the statement.
*** But on the other hand, computer programs nowadays do have safeguards against paradoxes in the form of specifications (like treating a logical contradiction as a boolean false), and things like any given variable only holding one value at any point of time (so that variable A can't be both true and false at the same step in the algorithm). A kind of fork bomb—i.e. a process that can duplicate itself or create new running processes infinitely—would probably be a better choice.
*** If I can math geek a bit: even more subtle, there are mathematical statements that ARE true, but which can't be proven true. [http://en.[wikipedia.org/wiki/:G%C3%B6del%27s_incompleteness_theoremsB6delchr(27)s incompleteness theorems|One particularly famous example]] involves a specific function f(x, y) and a specific number n where we can easily prove "f(1, n) does not equal 0", "f(2, n) does not equal 0", etc. for any particular integer but there's no way to prove that "for all x, f(x, n) does not equal 0" short of an infinitely long proof that goes through every single integer individually. So a computer trying to prove "for all x, f(x, n) does not equal 0" would never reach a contradiction (since the statement is true), but would also never finish the proof. Bonus amazing fact: the function f in question can be interpretted as "this function is zero if and only if the statement with number x is a valid proof of the statement with number y" and n just happens to be the number for the statement "for all x, f(x, n) does not equal 0". In otherwords, the statement is asserting it has no proof, or more generally... "THIS! SENTENCE! IS! FALSE!"
** I just added this to the WMG page, but: Can't you just imagine Cave Johnson saying "Whaddaya ''mean'' paradoxes don't harm our AIs!? I want you to make a special paradox-detector that'll fry every circuit in its brain, and I want you to put it in every single one of our AIs, on the double, or you're fired!"
*** Actually, I can imagine that quite vividly, and for a moment I even wondered to myself if he ever actually said "I want you to make a special paradox-detector that'll fry every circuit in its brain".
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** Gravity?
** You might note that this has been commented on many, many times. In fact, I don't even see the point in asking, given [[MST3K Mantra]].
** Creating portals changes the [http[wikipedia://en.wikipedia.org/wiki/Topology |topology]] of space around the them. You could come up with a physics-y handwave-y solution involving [http://en.[wikipedia.org/wiki/Noethers_theorem:Noethers theorem|Noether's theorem]] (which says that [http://en.[wikipedia.org/wiki/Conservation_law:Conservation law|conservation laws]], such as the conservation of energy, are caused by symmetry, and breaking this symmetry destroys your conservation law) if you were so inclined.
** It's not really free energy, merely a very efficient method of using gravity to transfer kinetic energy from the planet to the falling object. If you placed two portals inside an airless tube, and set an object in an "infinite" fall, the object would accelerate indefinitely, and eventually reach relativistic speeds requiring enough force to accelerate further that the earth would start pulling itself out of its orbit with its own gravity via the "falling" object. This would only be possible if the airless tube was constructed exactly at the geographic north or south pole, however, or the earth's rotation would make falling at relativistic speeds impossible. Factor in air resistance and all the kinetic energy is dissipated in a closed loop that doesn't affect the earth's trajectory through space... I think... Someone should do that math on that.
** The problem is moving upwards in a gravitational field, you are gaining gravitational potential energy. This happens from moving from a low-placed portal to a high-placed one. It is unavoidable. Of course, we are assuming it does take zero energy to move from a low-placed portal to a high-placed one. Maybe the portal gun has a built in store of energy for this purpose.
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