Hi everyone!
I want to explain cantor's proof on showing the set of real numbers is bigger than the natural numbers.
There were a lot of questions on the part when cantor is working his way down the diagonal and changing the digits. if there are other parts you want explanation on please do tell me and I'll try to to my best in answering them.
Ok back to the proof! If our digit is no greater than 5 we increase it by one and else decrease it by one.
in the first line(see the picture to get a better idea), we will change our digit in the way I just mentioned. Again second line second digit, third line third digit and .... after all the lines are done, make a new number with those changed numbers. we know it's not the same as our first number since it has different number in the first position and we know it is not the same as the n'th number since it has different digit in the n'th position! there we have it! we made a new number that was not in the list.
I want to explain cantor's proof on showing the set of real numbers is bigger than the natural numbers.
There were a lot of questions on the part when cantor is working his way down the diagonal and changing the digits. if there are other parts you want explanation on please do tell me and I'll try to to my best in answering them.
Ok back to the proof! If our digit is no greater than 5 we increase it by one and else decrease it by one.
in the first line(see the picture to get a better idea), we will change our digit in the way I just mentioned. Again second line second digit, third line third digit and .... after all the lines are done, make a new number with those changed numbers. we know it's not the same as our first number since it has different number in the first position and we know it is not the same as the n'th number since it has different digit in the n'th position! there we have it! we made a new number that was not in the list.