Original proof of Gödel's completeness theorem
Hey kiddo, do you know what a puzzle is? Imagine we have a big puzzle, and we want to know if every piece of the puzzle can fit together to make a complete picture. That's what Gödel's completeness theorem is all about- but instead of a puzzle, we're trying to figure out if a bunch of math sentences can be put together to make a complete story.
To do this, Gödel came up with a plan. He said, what if we could label every possible math sentence as either being true or false? Then, we could try to put them all together and see if they make sense.
But how do we know if a sentence is true or false? One way is to test it out with numbers. But Gödel didn't want to use just any numbers- he wanted to use special numbers that could represent all other numbers. These special numbers are called Gödel numbers.
Now, Gödel was really clever, so he figured out a way to label every possible math sentence with a unique Gödel number. Then he made something called a proof- just like how we solve a puzzle by matching the pieces, Gödel matched up all the sentences and their Gödel numbers to create a complete story.
And guess what- Gödel was successful! He proved that for any set of math sentences, we can find a complete story that includes them all. It's just like how all the puzzle pieces can fit together to make one beautiful picture.
So, the Gödel's completeness theorem is like a puzzle where we can be sure that all the pieces fit together and make sense. Pretty cool, huh?
To do this, Gödel came up with a plan. He said, what if we could label every possible math sentence as either being true or false? Then, we could try to put them all together and see if they make sense.
But how do we know if a sentence is true or false? One way is to test it out with numbers. But Gödel didn't want to use just any numbers- he wanted to use special numbers that could represent all other numbers. These special numbers are called Gödel numbers.
Now, Gödel was really clever, so he figured out a way to label every possible math sentence with a unique Gödel number. Then he made something called a proof- just like how we solve a puzzle by matching the pieces, Gödel matched up all the sentences and their Gödel numbers to create a complete story.
And guess what- Gödel was successful! He proved that for any set of math sentences, we can find a complete story that includes them all. It's just like how all the puzzle pieces can fit together to make one beautiful picture.
So, the Gödel's completeness theorem is like a puzzle where we can be sure that all the pieces fit together and make sense. Pretty cool, huh?