Tarski's axiomatization of the reals
Hey there kiddo! So, do you remember what numbers are? Yes, they are like the labels we put on things so we can count and measure stuff, right?
Now, we can imagine that we have numbers that have little dots (called decimals) after them - like 4.5 or 3.14159. These are called "real numbers."
The mathematician Alfred Tarski wanted to come up with a way to describe all the different ways we can use real numbers. He gave us a set of rules, called "axioms," that help us do just that. These axioms are like a checklist we can use to make sure we're doing math correctly.
One of Tarski's axioms says that real numbers have something called "order." This means we can say that one number is bigger or smaller than another number. For example, we know that 10 is bigger than 5, right?
Another of Tarski's axioms says that real numbers are "complete." This means that no matter how close two numbers are to each other, there is always another real number in between them. This might seem a bit weird, but it's actually really important in math!
Finally, Tarski's axioms say that real numbers work nicely with addition and multiplication. This means that if we add or multiply two real numbers together, we always get another real number.
So, to sum up: Tarski's axiomatization of the reals gives us a set of rules we can use to play around with real numbers - to add them, subtract them, and compare them. These rules help us make sure our math is always correct!
Now, we can imagine that we have numbers that have little dots (called decimals) after them - like 4.5 or 3.14159. These are called "real numbers."
The mathematician Alfred Tarski wanted to come up with a way to describe all the different ways we can use real numbers. He gave us a set of rules, called "axioms," that help us do just that. These axioms are like a checklist we can use to make sure we're doing math correctly.
One of Tarski's axioms says that real numbers have something called "order." This means we can say that one number is bigger or smaller than another number. For example, we know that 10 is bigger than 5, right?
Another of Tarski's axioms says that real numbers are "complete." This means that no matter how close two numbers are to each other, there is always another real number in between them. This might seem a bit weird, but it's actually really important in math!
Finally, Tarski's axioms say that real numbers work nicely with addition and multiplication. This means that if we add or multiply two real numbers together, we always get another real number.
So, to sum up: Tarski's axiomatization of the reals gives us a set of rules we can use to play around with real numbers - to add them, subtract them, and compare them. These rules help us make sure our math is always correct!