- ...four natural numbers .
the natural numbers .
So he set out to compare the natural numbers and the real numbers between zero and one.
As Zermelo indexes each number with the natural numbers , at first it might seem like he'd run into a problem because the natural numbers are only accountably infinite,
equinumerous with a set of natural numbers is just the same thing as to fit into Hilbert's Hotel.
So it would appear there are fewer squares than natural numbers , but Galileo realized he could draw a line matching every natural number with its own square.
So for example, the natural numbers are well ordered, there's a starting point, one, and any subset, say six, seven, eight, also has a clear starting point.
countable if it is equinumerous with a set of natural numbers .
it's a strictly larger infinity than the natural numbers .
It raised a key question, are there more natural numbers or are there more square numbers?
So there are actually just as many square numbers as there are natural numbers .
Ordering them this way is actually what allows us to map the integers to the natural numbers and see that both sets are the same size,
start with zero because for me, the natural numbers start with zero, although that's maybe a point of contention for some mathematicians.
the integer lattice, the grid of points that you get by taking pairs of natural numbers , say, so the upper right quadrant of the integer lattice, yeah?
So we did pairs of natural numbers .
That's why this is called Cantor's Diagonalization Proof and it shows there must be more real numbers between zero and one than there are natural numbers extending out
Some infinities like the set of square numbers, integers or rational numbers can be paired perfectly with the natural numbers .
These infinities like the set of all real numbers, the complex numbers, they can't be matched one-to-one with the natural numbers .
I mean, the Galileo paradox is the idea or the observation that if you think about the natural numbers ,
observing that line segments of different lengths are equinumerous, and the perfect squares are equinumerous with all of the natural numbers , and any two circles are equinumerous, and so
a set is countable if it fits into Hilbert's Hotel, 'cause Hilbert's Hotel basically is the set of natural numbers in terms of the room numbers.
Because every grid point is going to be the Nth point on that path for some N. And that that gives a correspondence between the grid points and the natural numbers
- Okay, so Cantor wants to prove that the infinity of the real numbers is different and strictly larger than the infinity of the natural numbers .
Now, obviously, since the natural numbers are included in the real numbers, we know that the real numbers are at least as large as
So we suppose that the real numbers can be put into one-to-one correspondence with the natural numbers .
So some functions are what's called additive, like if you have a function that must say natural numbers , the natural numbers .
So there's floor zero, floor one, floor two, or room zero, one, two, three, and so on, just like the natural numbers .