> Are there any uses or applications for Gödel's theorems (other than their mere implications)?
Theorems are implications, so I'm not quite sure what you mean.
There are practical applications of Gödel's theorems, if that's what you mean.
His incompleteness theorems are useful for determining when to give up on a certain class of approaches toward a problem. I think this is what you meant by "implication", but the importance of this knowledge is difficult to over-state.
His completeness theorems form the groundwork for first-order theorem proving, which has plenty of practical applications.
The Liar Paradox is famous because it seems to break the foundational tenant of classical that all statements are either true or false.
Your example is also interesting, but doesn't really break things in the same way.
I don't necessarily agree with this, I think there are plenty of important things to consider with both, but that's why one became famous without the other.
It does break things in the same way, because the tenet is that all statements are either true exclusive or false.
If we can follow sound reasoning such that some statement can be either true or false, based on the exact steps we take, that is exactly the same problem: a contradiction somewhere.
"This sentence is true" doesn't contradict itself, but two different ways of evaluating it contradict each other.
Researcher A assumes the sentence is true and comes to the conclusion that it is.
Researcher B assumes that it is false, substitutes in the value and also concludes that it is correct.
Neither has found a contradiction in isolation, but A and B's findings contradict each other.
In terms of electronics-based intuition, "this sentence is false" is like an unstable circuit: e.g. an inverter feeding its own output to its output. This might result in oscillation. "This sentence is true" is like bistable feedback: it gives us a flip-flop which "latches" whichever value we inject.
The problem is much simpler: logical systems are undecidable, and the Liar's Paradox is the logic analogue of infinite loops. But this does require you understand logic as a process (something people do) rather than having weird dualistic notions about natural laws of logic.
At a minimum, you can say of statements that are self-contradictory that they are self-refuting. Whatever paradoxical class you wish to put them in, they at least are not true.
By the way, the Liar Paradox is not just an academic curiosity. You run into it in pop philosophy a lot. "There are no absolute truths" is self-contradictory in the same way the Liar Paradox is. So is "Science is the only way we can really know what's true".
Something can be colorless, yet green metaphorically (immature, new, inexperienced). Ideas can certainly be green in this sense, and lacking physical form they are colorless.
Something can sleep metaphorically if it is not being put to use.
Sleep can be furious; I've seen it. Just perhaps not the kind of sleep that is slept by ideas that have fallen by the wayside.
That may be the irresolvable crux of the problem with that sentence: finding a semantics for "sleep" which is compatible with some interpretation of "furiously", where the sleeping subject can be "ideas".
Perhaps the modern grammatical transitive sense of "to sleep" can rescue it. These days we can "sleep a device" meaning "to put it to sleep". "Bob furiously slept his tablet computer, turned out the lights and himself went to sleep".
Maybe some ideas furiously sleep us.
This entire comment sleeps furiously nearly every reader.
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Theorems are implications, so I'm not quite sure what you mean.
There are practical applications of Gödel's theorems, if that's what you mean.
His incompleteness theorems are useful for determining when to give up on a certain class of approaches toward a problem. I think this is what you meant by "implication", but the importance of this knowledge is difficult to over-state.
His completeness theorems form the groundwork for first-order theorem proving, which has plenty of practical applications.
Unprovable isn't false. If any axiom could talk, it could say that about itself. "I'm true, but unprovable; I'm an axiom, darn it!"
It's consistent; but the problem is that it's that way whether we assume it to be true or false, oops.
For some reason, "this sentence is true" isn't famous, like its cousing, the Liar Paradox, even though it has something significant to add.
Your example is also interesting, but doesn't really break things in the same way.
I don't necessarily agree with this, I think there are plenty of important things to consider with both, but that's why one became famous without the other.
If we can follow sound reasoning such that some statement can be either true or false, based on the exact steps we take, that is exactly the same problem: a contradiction somewhere.
"This sentence is true" doesn't contradict itself, but two different ways of evaluating it contradict each other.
Researcher A assumes the sentence is true and comes to the conclusion that it is.
Researcher B assumes that it is false, substitutes in the value and also concludes that it is correct.
Neither has found a contradiction in isolation, but A and B's findings contradict each other.
In terms of electronics-based intuition, "this sentence is false" is like an unstable circuit: e.g. an inverter feeding its own output to its output. This might result in oscillation. "This sentence is true" is like bistable feedback: it gives us a flip-flop which "latches" whichever value we inject.
By the way, the Liar Paradox is not just an academic curiosity. You run into it in pop philosophy a lot. "There are no absolute truths" is self-contradictory in the same way the Liar Paradox is. So is "Science is the only way we can really know what's true".
[Edit: Ah, I see. You're saying that it can be simply false without self-refutation entering the picture at all. Yes, it can.]
Something can sleep metaphorically if it is not being put to use.
Sleep can be furious; I've seen it. Just perhaps not the kind of sleep that is slept by ideas that have fallen by the wayside.
That may be the irresolvable crux of the problem with that sentence: finding a semantics for "sleep" which is compatible with some interpretation of "furiously", where the sleeping subject can be "ideas".
Perhaps the modern grammatical transitive sense of "to sleep" can rescue it. These days we can "sleep a device" meaning "to put it to sleep". "Bob furiously slept his tablet computer, turned out the lights and himself went to sleep".
Maybe some ideas furiously sleep us.
This entire comment sleeps furiously nearly every reader.