Gentzen's consistency proof
Okay, so let's say you have a set of building blocks, like Legos. You can build lots of different things with them, like a house or a car or a spaceship.
Now imagine you have a rulebook that tells you how you can put the Legos together. For example, it might say that you can't put a block on top of another block if it's bigger than the block underneath it.
Gentzen's consistency proof is kind of like a rulebook for building with ideas instead of Legos. But instead of just saying what you can and can't do with the ideas, it also makes sure that if you follow those rules, you won't end up with any contradictions.
What's a contradiction? Well, let's go back to the Legos. Imagine you built a tower that's three blocks high, but then you decide to take out the middle block. What happens? The tower falls down, right? That's because taking out that block made the tower unstable and contradictory.
The same thing can happen with ideas. If you're trying to make an argument, and you say something that contradicts something else you said earlier, or something someone else said, then your argument falls apart.
Gentzen's consistency proof helps make sure that doesn't happen. It lays out the rules for building with ideas (like, you can't use circular reasoning, where you use your conclusion to prove your premises), and then it proves that if you follow those rules, you won't end up with any contradictions.
Why is this important? Well, if there are no contradictions in your argument, then you can be more confident that it's true. And that's a really important thing to have when you're trying to prove something!
Now imagine you have a rulebook that tells you how you can put the Legos together. For example, it might say that you can't put a block on top of another block if it's bigger than the block underneath it.
Gentzen's consistency proof is kind of like a rulebook for building with ideas instead of Legos. But instead of just saying what you can and can't do with the ideas, it also makes sure that if you follow those rules, you won't end up with any contradictions.
What's a contradiction? Well, let's go back to the Legos. Imagine you built a tower that's three blocks high, but then you decide to take out the middle block. What happens? The tower falls down, right? That's because taking out that block made the tower unstable and contradictory.
The same thing can happen with ideas. If you're trying to make an argument, and you say something that contradicts something else you said earlier, or something someone else said, then your argument falls apart.
Gentzen's consistency proof helps make sure that doesn't happen. It lays out the rules for building with ideas (like, you can't use circular reasoning, where you use your conclusion to prove your premises), and then it proves that if you follow those rules, you won't end up with any contradictions.
Why is this important? Well, if there are no contradictions in your argument, then you can be more confident that it's true. And that's a really important thing to have when you're trying to prove something!