In cantors diagonalization he basicly states that we cannot put the set of natural numbers and the set of real numbers into a 1 to 1 match up because their are more real numbers then natural numbers, he proves this by showing that if you have a set of real numbers, and you move diagnally adding 1 to the numbers to create a new number, this new real number will not be numbered by our natural numbers. my confusing is cant we do they same process to the natural numbers in order to create some number that can name this new real number?
unless ive missunderstood something
#Cantors diagnalization proof
errant cove
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proud field
In cantors diagonalization he basicly states that we cannot put the set of natur...
problem is, you can apply the diagonalization again and get another number which has no mapping from the naturals. essentially no matter how many times you add in that new number from the diagonalization, you will always get some new real number in the reals thats not mapped from the domain.
elder epoch
His diagonalization argument, proves that the set of infinite sequences is not enumerable. And by constructing a one-to-one map from the set of infinite sequences to the reals, it is proven that R is also not enumerable.
elder epoch
And by definition the natural numbers are enumerable, the identity function is a one-to-one correspondence. You can't represent a natural number as an infinite sequence of digits, every number has a finite number of digits. So you also can't apply the diagonal argument.