Kleene's recursion theorem
Kleene's recursion theorem is like making a machine that can copy itself. Imagine you have a machine (let's call it Machine A) that makes other machines. Machine A can make different types of machines, but it can also make another machine just like itself. This means that if you let Machine A make another machine, the new machine can now make machines just like itself, and so on.
But how can you do this in maths? Say you have a function (let's call it F) that takes a mathematical expression as an input, and returns another mathematical expression as an output. And, let's say you want to make a new function (let's call it G) that takes a number as an input, and returns the same expression as F, but with the input number inserted somewhere.
Kleene's recursion theorem tells you that you can make a function H that uses both F and G. When you give H a number as an input, it uses G to create a new expression (by inserting the input number), and then applies F to this new expression. So basically, H is a function that "copies" or "repeats" F with different inputs, using G to create these new inputs.
It's like taking a Lego block and putting it into a machine that makes more Lego blocks. The new Lego blocks are the same as the original one, but it's like they've been "copied" or "repeated" because the machine made them using the original block as a guide. Kleene's recursion theorem allows you to do the same kind of "copying" or "repeating" with mathematical functions.
But how can you do this in maths? Say you have a function (let's call it F) that takes a mathematical expression as an input, and returns another mathematical expression as an output. And, let's say you want to make a new function (let's call it G) that takes a number as an input, and returns the same expression as F, but with the input number inserted somewhere.
Kleene's recursion theorem tells you that you can make a function H that uses both F and G. When you give H a number as an input, it uses G to create a new expression (by inserting the input number), and then applies F to this new expression. So basically, H is a function that "copies" or "repeats" F with different inputs, using G to create these new inputs.
It's like taking a Lego block and putting it into a machine that makes more Lego blocks. The new Lego blocks are the same as the original one, but it's like they've been "copied" or "repeated" because the machine made them using the original block as a guide. Kleene's recursion theorem allows you to do the same kind of "copying" or "repeating" with mathematical functions.
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