送交者: steven 于 2005-6-20, 17:17:48:
回答: it's like what xinku said 由 eng 于 2005-6-20, 13:52:12:
And you don't know what Peano's axioms really are. Number system exists, naturally as stars and planet. Long before nameing Sun, it is there. Long before the discovry of Earth orbiting the Sun, it is so. Same as number system, we give it a name, and notations so that we can record it and communicate with others. We didn't invent those properties, there are there ever since. We only take a subset of those properties naturally exist, and give them notation for us to understand and use them, just like Newton's law of gravity. Peano's Axioms are the description of a subset of properties of number. The description is a human invention, but not the properties themselve. What Godel told us was: by using the way we process information, namely reasoning, we will never be able to process all those properties which exist naturally.
You know 10 + 1 = 11, how do you know it is correct? Without the properties described by Peano Axioms, there is no way to know if these answer is correct. You can count 10 + 1, but you cannot count 10^100. In fact, without those properties, you can even get to 1 from 0.
A simple example, given a set of all people since 100 years ago, let say, you are the first element, who is next? However, given a set of all the people of your family tree day 100 years, we can find the first one, and start from that person we can trace it to you, and that parent-childern relationship is the axiom of this set, which naturally exists, and we use it to put an constrain to the set we are interested in, so that we can do something about it.