This, alongside the second incompleteness theorem, showed me that any formal system cannot prove itself, and any system deemed "consistent" can only be considered as such from outside that system.
A great way that I visualize it (thanks to Hofstadter) is like Escher's never ending staircase painting. If you look at each step, it looks just fine. In fact, you can say that each step on its own is "consistent". But it is only when you take a step back, outside the system of each isolated step, that you see the paradox.
Similarly, the theorem applies this to mathematics, and by extent, any formal system. Just as we can see each "step" going from one to the next, in mathematics we must hold an axiom as truth without proof, so that subsequent theorems may use it as the next "step". It is, in a sense, a paradox which we agree to use.
In practical terms, I may extrapolate that further and more philosophically than was intended (or that I have commonly seen), but the subsequent Halting Problem combined with the Curry-Howard isomorphism shows me that Godel's Theorem has major real world effects. The idea that there can be something that is true but not provable rings true to me in many regards. I, like many, spend nearly every moment of the day unconsciously putting faith into the "truth" that I am not alone in the universe, thereby fending off solipsism.
I cannot prove "I" exist. But it is true. What are the other truths that cannot be proven? That question guides my thinking.
Essentially, all logically valid formulae have proofs in first-order logics.
Remember, logically valid formulae means that if you enumerate all possible interpretations (boolean values for the variables), the formulae remains true.
So you could prove everything that ever exists by creating a set of all possible proofs using the given deduction rules.
I think going through The Little Schemer by Daniel Friedman and Matthias Felleisen helped me a lot. It builds up your knowledge in a conversational way to achieve understanding of quite complex ideas in simple terms. In that case, it builds up to an understanding of the halting problem, and then to the Y Combinator.
The click felt to me similar to understanding recursion for the first time--it doesn't fit into your head naturally. Once it does "click", even then you have to re-remember it later, and trace your logic on how you "realized" recursion.
Godel's Theorem brings that "feeling" of recursion and takes it to the extreme, in that its subject matter is "meta"-mathematical. So in the same way that recursion can't be grokked in terms of only loops, Godel's Theorem can't be grokked only in terms of equations.
I don't know if that made sense, I've also seen some links in this thread which I think may help as well
Von Neumann said: "In mathematics you don't understand things. You just get used to them."
For practical purpose of getting the completeness theorem to click, my advice would be to stop thinking about it and instead just use it in a bunch of proofs. The way you use it in the a proof is you say something like "...and since, as we've just shown, this sentence phi is true in all structures for the language, it follows that there is a proof P of phi, and therefore..."
If writing such proofs isn't the sort of thing you want to do, then you almost certainly don't actually need to know Goedel's Completeness Theorem, which is a highly technical theorem whose philosophical, epistemological, metaphysical, mystical, and magical applications are all completely overblown by people who want to sell books.
pkulak|6 years ago
vga805|6 years ago
theWheez|6 years ago
A great way that I visualize it (thanks to Hofstadter) is like Escher's never ending staircase painting. If you look at each step, it looks just fine. In fact, you can say that each step on its own is "consistent". But it is only when you take a step back, outside the system of each isolated step, that you see the paradox.
Similarly, the theorem applies this to mathematics, and by extent, any formal system. Just as we can see each "step" going from one to the next, in mathematics we must hold an axiom as truth without proof, so that subsequent theorems may use it as the next "step". It is, in a sense, a paradox which we agree to use.
In practical terms, I may extrapolate that further and more philosophically than was intended (or that I have commonly seen), but the subsequent Halting Problem combined with the Curry-Howard isomorphism shows me that Godel's Theorem has major real world effects. The idea that there can be something that is true but not provable rings true to me in many regards. I, like many, spend nearly every moment of the day unconsciously putting faith into the "truth" that I am not alone in the universe, thereby fending off solipsism.
I cannot prove "I" exist. But it is true. What are the other truths that cannot be proven? That question guides my thinking.
meowface|6 years ago
solinent|6 years ago
Remember, logically valid formulae means that if you enumerate all possible interpretations (boolean values for the variables), the formulae remains true.
So you could prove everything that ever exists by creating a set of all possible proofs using the given deduction rules.
theWheez|6 years ago
The click felt to me similar to understanding recursion for the first time--it doesn't fit into your head naturally. Once it does "click", even then you have to re-remember it later, and trace your logic on how you "realized" recursion.
Godel's Theorem brings that "feeling" of recursion and takes it to the extreme, in that its subject matter is "meta"-mathematical. So in the same way that recursion can't be grokked in terms of only loops, Godel's Theorem can't be grokked only in terms of equations.
I don't know if that made sense, I've also seen some links in this thread which I think may help as well
xamuel|6 years ago
For practical purpose of getting the completeness theorem to click, my advice would be to stop thinking about it and instead just use it in a bunch of proofs. The way you use it in the a proof is you say something like "...and since, as we've just shown, this sentence phi is true in all structures for the language, it follows that there is a proof P of phi, and therefore..."
If writing such proofs isn't the sort of thing you want to do, then you almost certainly don't actually need to know Goedel's Completeness Theorem, which is a highly technical theorem whose philosophical, epistemological, metaphysical, mystical, and magical applications are all completely overblown by people who want to sell books.
disease|6 years ago