Hardy field
A Hardy field is like a really special playground for numbers. Think of numbers like swings and slides that you can play with. Now, on this special playground, you can do something really cool. You can add, subtract, multiply, and divide all the swings and slides as much as you want.
But here's the catch. You can only use these swings and slides in certain ways. You can't just make up new rules. That's like trying to play hopscotch with four feet instead of two. It just doesn't work!
So, on this special playground, you can only add, subtract, multiply, and divide the swings and slides in ways that follow certain rules. These rules are called the axioms of a field.
Now, the Hardy field is special because it's a field that has extra, super-duper rules that make it more fun to play with. These extra rules are called the logical axioms.
What do these logical axioms do? They basically make sure that you can play with all the swings and slides you want, without ever getting stuck or lost. You always know where you are, and you're always having fun.
So in a nutshell, a Hardy field is a really special playground for numbers, where you can add, subtract, multiply, and divide all you want, as long as you follow the right rules. And it's special because it has extra rules that let you have even more fun!
But here's the catch. You can only use these swings and slides in certain ways. You can't just make up new rules. That's like trying to play hopscotch with four feet instead of two. It just doesn't work!
So, on this special playground, you can only add, subtract, multiply, and divide the swings and slides in ways that follow certain rules. These rules are called the axioms of a field.
Now, the Hardy field is special because it's a field that has extra, super-duper rules that make it more fun to play with. These extra rules are called the logical axioms.
What do these logical axioms do? They basically make sure that you can play with all the swings and slides you want, without ever getting stuck or lost. You always know where you are, and you're always having fun.
So in a nutshell, a Hardy field is a really special playground for numbers, where you can add, subtract, multiply, and divide all you want, as long as you follow the right rules. And it's special because it has extra rules that let you have even more fun!