What does Godël’s Incompleteness Theorem actually mean and imply? I just saw Ted-Ed’s video on this topic and didn’t fully understand what it means or what the implications of this are.
In: 742
Basically, any reasonable mathematical system will have some statements that can never be proven true or false.
The easiest way to understand what this means is, ironically, is by listening to children
“Mom, why is the world round?”
“Because we have gravity, that pulls all the dirt and water into a ball.”
“Why?”
“Because anything with mass emits a little bit of gravity.”
“Why?”
“…. Because it just does…”
**”Why Mother?”**
(Side note: Oversimplified Example is Oversimplified)
The reason most parents get infuriated by this incessant “Why? Why? Why? Why?” is because kids don’t understand this principle yet. If you keep asking “Why”, the answer will eventually become “Because it is”
No matter what system of logic you use, some things just **are**. Every system of logic has certain universal principles that are just facts. You can’t really prove why 1+1=2. That’s just how arithmetic works.
It means that any fixed Turing-recognizable model of arithmetic will either fail to prove some factually true things about arithmetic, or else it will be inconsistent, meaning that it will prove everything, including false things. In that sense, every Turing-recognizable and consistent model of arithmetic is “incomplete,” since it will have “blind spots” in terms of true things for which it actually provides a proof. It means that no fixed formal computational model of arithmetic can account for all of the things that are actually true about arithmetic.
For a long time in mathematics the words *true* and *provable* meant the same thing.
Godel showed that they are not. There can be true statements that are unprovable.
This problem’s unsolvable. This problem’s unsolvable. This problem’s unsolvable. This problem’s unsolvable. Here… let radiolab ELI5 this for you! Doesn’t quite sound like it at first, but the part about Godel starts in the beginning.
https://www.wnycstudios.org/podcasts/radiolab/segments/161758-break-cycle
a system can be complete or consistent. not both.
therefore there are unknowable truths of math and logic for instance.
Math is a language that is build by a few rules, for example, x+y = y+x, and with these rules mathmaticians try to proof every statement they make. The dream of mathematics was to have a language that is in itself consistent and complet, in which you can speak about everythink and proof it.
Gödel showed that there can never be such a language. you can for example have a statement about infinity and the proof for this statement takes infinit steps, so it is not provable. The part of math that actualy work without infinit steps is constructive mathematics aka computation. In math you work with lim (limes) which is like saying we take a really big nummer because we cant go to infinity. Or in the limit it will behave very much like infinity, but we dont calculate it because it would take infinit steps.
I like to use the following example.
Let’s say we have a library and are creating useful indexes for it.
We make a red book that lists all the library books that refer to the themselves. Encyclopedias, dictionaries, etc.
We make a green book that lists all the library books that don’t refer to themselves. That will be most of them.
Now, logicians and mathematicians started the 20th century working on the rules of logic that would let us decide for every statement whether it was true or false. Somewhere God should have an answer book listing all the true statements in our system of logic. For any function we would like to be able to determine whether F(x) or not F(x) is in God’s answer book.
Russel’s paradox asks: “Does the green book lists itself?” We can’t answer “yes,” because if it does it shouldn’t. We can’t answer “no” because if it doesn’t, it should. We can’t find either F(x) or not F(x) in God’s answer book. Oops.
And there is another question we could ask. “Does the red book list itself?” If it does…great! “F(red book)” goes in God’s answer book. If it doesn’t…great! It shouldn’t! “Not F(red book)” goes in God’s answer book.
Oops. We don’t want both F(x) and notF(x) to be true.
You probably think “this book refers to itself” sounds like a really weird function. Maybe we can just leave it out of our system of logic? Sweep the problem under the rug.
But Gödel proved that any formal logic with enough tools to include statements like (1+2=3) will have this problem.
Ravaturno’s corollary: You can choose to be consistent or complete but not both at the same time. Example: Science is consistent but will never know everything; religion is “complete” but full of contradictions.
It basically just says that, for certain types of mathematical systems (I can never remember the actual definition, but it ends up being most systems we care about), there will always be theorems that are _independent_ of the axioms of the system and/or that the system itself is _inconsistent_ (i.e. implies a contradiction). An independent statement is a statement that cannot be definitely proven true or false _within_ that system. Or, more precisely, there will be _models_ of the system where the statement is true and others where its false.
The analogy that I typically use is that, for a given work of fiction, there will almost certainly be questions that have no canonical answer because the author simply hasn’t provided sufficient information within the canon to answer. For example, in the Harry Potter universe, there’s no canonical answer to the question of how _exactly_ a horcrux is made because J.K. Rowling never specified. Hence, there can be fan fictions that answer the question one way and others that answer it a different way, and, all else equal, each one would be equally valid, so long as they don’t contradict anything from the original work. Of course, this is only an analogy and certainly an imperfect one, but it really helped me get comfortable with the idea of something being “true in one model but false in another.”
All valid systems of math will have paradoxical problems that can’t be solved by switching to a “superior” system of math. Paradoxes are unavoidable in math systems, so mathematicians can quite trying to come up with “superior” systems (languages).
Recommend reading Godël, Escher Bach by Douglas Hofstadter if you are intrigued enough to pursue this as a lay person. I started reading it in HS well before I was really capable of understanding a lot of it. (I was a musician and mother was an art teacher so Escher and Bach were quite familiar.) But it was still entertaining and gave me an appreciation of a lot of stuff that later became applicable in college while pursuing a math minor.
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