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Most of these entries are Private.
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April 20th - (CJ Invents new math) The Existence Paradox of Irreducibly Infinite Processes
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Hypothesis: The problem of determining if irreducibly infinite processes exist-- is itself an irreducibly infinite process.
Wtf am I talking about?
So Godel incompleteness; any formal axiomatic system will either have true things it cannot prove, or will have paradoxes. BUT ask me this Godel: what about a formal system with an infinite number of axioms? Could that work? Could a theoretical system like this be complete? How infinite would it need to be?
I wonder if an unparadoxical mathematical system could exist only if it started with an infinite number of true assumptions. Is this trivial, or are there deeper things to consider here?
The trick in his proof is reducing everything into Godel numbers, but what about infinitely long Godel numbers?
What kind of uncomputability do you get when you run the halting program on itself, when (when it calls itself) its own filename is infinitely long? (never even gets a chance to be a paradox:/)
But back to infinite godel numbers. For instance, PI, a transcendental number, can be reduced to a generalized continued fraction, or computer algorithm, which could both be expressed in finite godel numbers.
http://en.wikipedia.org/wiki/Pi#Pi_and_continued_fraction
- Can all infinitely long Godel numbers be reduced to finite Godel numbers ?
- I.e. Can all infinite processes be reduced to a set of finite instructions for building them?
- OR do there exist infinite processes which cannot be reduced to finite instructions?
- Does something like this exist: Infinite processes which reduce to infinite instructions for building infinite instructions for building infinite instructions for building infinite instructions....etc?
- If something like that DID exist, would it be impossible to think of it? (thoughts themselves are finite, and so would any reasoning which could approach the existence of one)
My hypothesis: If an irreducible infinite process like this were to exist, not only would computing it be a task for a hypercomputer... but FINDING one in the first place may be a task for a hypercomputer.
(I would love to prove this.)
And if not, then we discover a level of uncomputability above hypercomputers, which is just as cool.
What if we looked to nature for one instead? (((potential idea: What if the search for unifying physical laws is one of these irreducible infinite processes--- Assume all constituents and forces of matter and energy can be broken down further and further, infinitely. A human civilization on a quest to find the unifying physical laws would never find them, but perpetually approach them. This is like a hypertask. You could reduce this hypertask into the finite rules of "humans-> go figure out the universe". But the very existence of humans in the first place is a result of these infinitely compounded laws. You can't create the humans without the laws. You can't create the humans (which never figure out the laws) without figuring out the laws. ??? ))) Would any potential candidate have some absurd amount of self-reference in it?
Or what if we could leverage unsolved problems in mathematics. Connect this to things which may not be able to be proven/cannot be proved or disproved? Like, what if it turns out that we can say something like "IF its impossible to prove if P=NP, then irreducible infinite processes exist".
Okay, so now, IF we assume irreducible infinitely long Godel numbers exist.... what does that say about his proof, his theorem, and of mathematics? What are the implications? Does this still fit nicely with his proof? If so, what does that mean. If not, why and (how) is this a significant idea? Either way, let's rework metamathematics xD
I wonder if this could be proved or disproved. or even more exciting: if it can be proved that it cannot be proved/disproved .. then its like wtf.. or if it can't be proved/disproved whether it can't be proved/disproved whether it can't be proved/disproved whether it can't be proved/disproved whether it can't be proved/disproved... and so in an INFINITE CHAIN!! do such things exist? (is this an extension of the same problem of irreducibly infinite processes? LOL the (dis)proof of itself requires itself)
Tell me what you think, and where/if I slipped in my reasoning. Basically I think it all comes down to:
Hypothesis: The problem of determining if irreducibly infinite processes exist-- is itself an irreducibly infinite process.
I lack the knowledge/ability to formalize these things all fancy academic like. If anyone wants to publish a paper with me, please express your enthusiasm. Or feel free to pirate my ideas and claim them as your own. Just as long as you throw in a  somewhere in there; some sort of allusion to your inspirators, so that we can feel invisibly powerful.
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okay so like, what is finite here? the description of a iip is finite. so that means we can go up to god and be like "God. This is what an Iip is. I need you to look through everything that exists and find me something that matches this criteria, and then start performing it". God then takes these finite instructions, and undergoes his task of looking through EVERYTHING that exists. If after infinitely long, he does not find an Iip, then Iips don't exist, or even God cannot find one. If he finds an Iip and starts performing it, its existence is negated, because the originating instructions were finite. Therefore Iips don't exist.
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does "No one will ever find an x" imply "x doesn't exist" ??
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The next obvious step is to make an ambigram of your absurdist math theory
:P
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LO7
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Okay, now it's legit.
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ltwas a pleasure to follow you on a wit's impulse ^^
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Ambigrammars, is that what they call us? (it evenn looks really legit, all latin like akil uitel lla 'ti6al hllear skool wava) ¿sn llec what teyt 'srewwerbiqwV
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