Everybody knows what are natural numbers. Or is it so? If I ask, is 2+1 a natural number, everybody (including me) would rush to say it is so - the reason being 3 is a natural number. I agree that 3 is a natural number, but why 2+1? Isn't it silly to question 2+1 = 3? If not, what is 2+1 after all?
The Italian mathematician Giuseppe Peano knew in 1900's that there would be such silly minds like me who will want to question even basic (and trivial) things. So he introduced a set of axioms that would define precisely what those numbers are, in an attempt to capture the notion of a natural number. Here are the axioms:
1. There is a natural number 0.
2. Every natural number a has a natural number successor, denoted by S(a). Intuitively, S(a) is a+1.
3. There is no natural number whose successor is 0.
4. Distinct natural numbers have distinct successors: if a ≠ b, then S(a) ≠ S(b).
5. If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers (This postulate ensures that the proof technique of mathematical induction is valid).
It will be very easy to see that the natural numbers satisfy these axioms. But do they define natural numbers or capture the idea of what natural numbers are? The answer is clearly no. I still haven't got the answer why 2+1 should be 3. None of these axioms say it is so. To take a closer look why it is so, consider an alien coming to the earth who don't have the number 3 in their world. He doesn't know what 3 is and in their system 2+1 equals 4. All they have is the set {1, 2, 4, 5,...}. Now, how does one explain our concept of natural numbers to him. Note that these axioms would still be valid in their system.
So Peano hasn't convinced me yet! If one says that there could be some other set of axioms that would precisely define what natural numbers are - then go please find that and let me know. I will give you million dollars, okay 10 dollars. All one can say is that any other set that satisfies these axioms will be isomorphic to the natural numbers. In other words, the word isomorphism here means, I don't know what natural numbers are, but can we all agree there is one so that if you have a different system of numbers then either agree to our agreed definition or go to hell!
Looks like natural numbers are not so natural at all! More on this later.
The Italian mathematician Giuseppe Peano knew in 1900's that there would be such silly minds like me who will want to question even basic (and trivial) things. So he introduced a set of axioms that would define precisely what those numbers are, in an attempt to capture the notion of a natural number. Here are the axioms:
1. There is a natural number 0.
2. Every natural number a has a natural number successor, denoted by S(a). Intuitively, S(a) is a+1.
3. There is no natural number whose successor is 0.
4. Distinct natural numbers have distinct successors: if a ≠ b, then S(a) ≠ S(b).
5. If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers (This postulate ensures that the proof technique of mathematical induction is valid).
It will be very easy to see that the natural numbers satisfy these axioms. But do they define natural numbers or capture the idea of what natural numbers are? The answer is clearly no. I still haven't got the answer why 2+1 should be 3. None of these axioms say it is so. To take a closer look why it is so, consider an alien coming to the earth who don't have the number 3 in their world. He doesn't know what 3 is and in their system 2+1 equals 4. All they have is the set {1, 2, 4, 5,...}. Now, how does one explain our concept of natural numbers to him. Note that these axioms would still be valid in their system.
So Peano hasn't convinced me yet! If one says that there could be some other set of axioms that would precisely define what natural numbers are - then go please find that and let me know. I will give you million dollars, okay 10 dollars. All one can say is that any other set that satisfies these axioms will be isomorphic to the natural numbers. In other words, the word isomorphism here means, I don't know what natural numbers are, but can we all agree there is one so that if you have a different system of numbers then either agree to our agreed definition or go to hell!
Looks like natural numbers are not so natural at all! More on this later.
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