So modern math is proven to be incomplete and we cannot prove that it is consistent either. Those 2 words, incomplete and consistent have a very technical meaning here.
The first is that there is a statement in modern mathematics, which is true, but cannot be proven. And even if we expand it, there will always be such a statement. Hence, incomplete.
And the second, we cannot have a system that proves everything as that system will be inconsistent. Basically if a system can prove everything, then we can easily prove 1=1 AND 1 ≠ 1. If both are proven, then we lose meaning since there is no “truth”. But a consistent system cannot prove its self consistency. Ergo, with modern math, we cannot know if math is consistent.
Now, the problem lies in that we use math to model our perceived reality. It means there is a limit to human knowledge, or put simply, there will be something in the universe that we may never know the answer to.
My favorite is the busy beaver function. There exist, at a certain number, that our modern math cannot make any meaningful statement about the function. Here is a great video about it. (youtube link warning). But you can also look at veritasium video for more in depth explanations.
Incompleteness applies to all formal systems of logic, not just maths, which means that the systems we based the scientific method and our best attempts at justice systems and formal argumentation/debate and academia are all subject to incompleteness.
Incomplete systems can also be inconsistent, it’s possible everything we base our collective knowledge on are such systems.
Eli5?
So modern math is proven to be incomplete and we cannot prove that it is consistent either. Those 2 words, incomplete and consistent have a very technical meaning here.
The first is that there is a statement in modern mathematics, which is true, but cannot be proven. And even if we expand it, there will always be such a statement. Hence, incomplete.
And the second, we cannot have a system that proves everything as that system will be inconsistent. Basically if a system can prove everything, then we can easily prove 1=1 AND 1 ≠ 1. If both are proven, then we lose meaning since there is no “truth”. But a consistent system cannot prove its self consistency. Ergo, with modern math, we cannot know if math is consistent.
Now, the problem lies in that we use math to model our perceived reality. It means there is a limit to human knowledge, or put simply, there will be something in the universe that we may never know the answer to.
My favorite is the busy beaver function. There exist, at a certain number, that our modern math cannot make any meaningful statement about the function. Here is a great video about it. (youtube link warning). But you can also look at veritasium video for more in depth explanations.
Two addenda:
Incompleteness applies to all formal systems of logic, not just maths, which means that the systems we based the scientific method and our best attempts at justice systems and formal argumentation/debate and academia are all subject to incompleteness.
Incomplete systems can also be inconsistent, it’s possible everything we base our collective knowledge on are such systems.