Why is cantor set uncountable

In binary, you would imagine column headings of , , , , , etc. In ternary, the column headings would be from right to left , , , , …. So, to write a given number, say , in ternary, think what the biggest power of is that is still less than, in this case,.

So that will be. So, in ternary,. Does that make sense? What would be in ternary? For numbers between and , the column headings will be , , , etc.

So would be and would be. Just as another example, in ternary. Can you see how I calculated that? You might think: but what about? And so on, so we can see that in the Cantor set, every number is made up of s and s.

Now, what we have to do is to map every in any number in the Cantor set to a. So, for example, would become. This will give the full set of numbers in the interval in binary. This means there is a mapping which has its image as the whole of the interval.

That is, there is a surjection from the Cantor set to all the real numbers in the interval. Since the real numbers are uncountable, so the Cantor set must be! Theorem The Cantor set has measure. OK, so how can we prove that? Well, how about calculating how much of the interval is removed when forming the Cantor set. Then we could subtract this number from :. So, to form the Cantor set, we started with an interval of measure and subtracted. Then of each of the remaining thirds, we took the middle thirds of each away so we subtracted.

Then there were intervals, each of which had its middle third removed so was subtracted. Then were removed, then and so on. Can you see what the difference between the measure of the and the iteration would be? Well, the size of each interval being removed is a third smaller than in the previous step so the denominator will be. The numerator, on the other hand, doubles with each step as each interval splits into two new intervals.

Asked 11 months ago. Active 11 months ago. Viewed times. CuriousAlpaca CuriousAlpaca 1 1 silver badge 8 8 bronze badges. Note: the rationals are also countable and can not be so ordered either. Add a comment. Active Oldest Votes. Martund Martund 14k 2 2 gold badges 9 9 silver badges 28 28 bronze badges. Martin Argerami Martin Argerami k 14 14 gold badges silver badges bronze badges. Chris Culter Chris Culter As for why the cantor set is uncountable, consider this: At each finite level of the cantor set construction, we "throw out" the middle third of each piece.

HallaSurvivor HallaSurvivor Upcoming Events. Featured on Meta. Now live: A fully responsive profile. Scott Brian M. Scott k 51 51 gold badges silver badges bronze badges. Martin Argerami Martin Argerami k 14 14 gold badges silver badges bronze badges. I didn't know about that. Show 6 more comments. Sign up or log in Sign up using Google.

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