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GEB takes this whole thing to task. If you really want to break a computer, you don\'t worry about things like \
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GEB takes this whole thing to task. If you really want to break a computer, you don\\\'t worry about things like \\\"The set of all sets\\\", or \\\"The set of things that don\\\'t contain themselves\\\". A smart computer that is ZFC aware knows that those are not valid set constructions. Equally, liars paradox has no truth value.
No, if you really want to confuse a computer, you first attempt to see if it understands Godel numbering, and teach it the concept if it does not. Then, long proofs can be reduced to mathematical equations and automatic validation.
Then you introduce substitution, and in no time at all, you have G.
For those that do not know, G is basically the following:<br>
1: Any proposed proof can be turned into a number, with each logic symbol being turned into some digits.<br>
2. You can then create a sentence that asserts that for each number, that number is not a proof of this statement.<br>
3. And it\\\'s true.<br>
0 is not a proof of G.<br>
1 is not a proof of G.<br>
2 is not a proof of G.<br>
...<br>
No \\\"number\\\" is a proof of G.
But that\\\'s what G claims. So what\\\'s the real killer for the computer?
Assume G is true. Then, you have a simple derivation where you conclude If G, then G, and take G as an axiom; this proves G. But this is now a finite proof of G, hence has a number, which means <br>
xyz is a proof of G<br>
which contradicts G.
No problem, right? That means that ~G must be true.
But ~G asserts that there is a \\\"number\\\" that proves G ...
(Of course, \\\"number\\\" isn\\\'t \\\"normal number\\\", and if you\\\'re talking about a computer that uses language to reason rationally ...)
No, if you really want to confuse a computer, you first attempt to see if it understands Godel numbering, and teach it the concept if it does not. Then, long proofs can be reduced to mathematical equations and automatic validation.
Then you introduce substitution, and in no time at all, you have G.
For those that do not know, G is basically the following:<br>
1: Any proposed proof can be turned into a number, with each logic symbol being turned into some digits.<br>
2. You can then create a sentence that asserts that for each number, that number is not a proof of this statement.<br>
3. And it\\\'s true.<br>
0 is not a proof of G.<br>
1 is not a proof of G.<br>
2 is not a proof of G.<br>
...<br>
No \\\"number\\\" is a proof of G.
But that\\\'s what G claims. So what\\\'s the real killer for the computer?
Assume G is true. Then, you have a simple derivation where you conclude If G, then G, and take G as an axiom; this proves G. But this is now a finite proof of G, hence has a number, which means <br>
xyz is a proof of G<br>
which contradicts G.
No problem, right? That means that ~G must be true.
But ~G asserts that there is a \\\"number\\\" that proves G ...
(Of course, \\\"number\\\" isn\\\'t \\\"normal number\\\", and if you\\\'re talking about a computer that uses language to reason rationally ...)
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You can say that the \'\'people\'\' themselves \'\'may\'\' be NotSoDifferent, whatever they do or do not believe in. To make a sweeping generalization of atheism and religion being exactly similar except in that one has a god and one doesn\'t, however, is as faulty as a lot of sweeping generalizations are.
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You can say that certain \\\'\\\'people\\\'\\\' themselves \\\'\\\'may\\\'\\\' be NotSoDifferent, whatever they do or do not believe in. To make a sweeping generalization of atheism and religion being exactly similar except in that one has a god and one doesn\\\'t, however, is as faulty as a lot of sweeping generalizations are.
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You can say that the \'\'people\'\' themselves \'\'may\'\' be NotSoDifferent, whatever they do or do not believe in. To make a sweeping generalization of atheism and religion being similar, however, is as faulty as a lot of sweeping generalizations are.
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You can say that the \\\'\\\'people\\\'\\\' themselves \\\'\\\'may\\\'\\\' be NotSoDifferent, whatever they do or do not believe in. To make a sweeping generalization of atheism and religion being exactly similar except in that one has a god and one doesn\\\'t, however, is as faulty as a lot of sweeping generalizations are.
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You can say that the \'\'people\'\' themselves may be NotSoDifferent, whatever they do or do not believe in. To make a sweeping generalization of atheism and religion being similar, however, is as faulty as a lot of sweeping generalizations are.
to:
You can say that the \\\'\\\'people\\\'\\\' themselves \\\'\\\'may\\\'\\\' be NotSoDifferent, whatever they do or do not believe in. To make a sweeping generalization of atheism and religion being similar, however, is as faulty as a lot of sweeping generalizations are.
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You can say that the \'\'people\'\' are NotSoDifferent, whatever they do or do not believe in. To make a sweeping generalization of atheism and religion being similar, however, is as faulty as a lot of sweeping generalizations are.
to:
You can say that the \\\'\\\'people\\\'\\\' themselves may be NotSoDifferent, whatever they do or do not believe in. To make a sweeping generalization of atheism and religion being similar, however, is as faulty as a lot of sweeping generalizations are.
