Is Alph Naught the smallest infinity? Initially everyone feels the first set of infinity they encounter is the set of natural numbers. Then speculation sets in and the discovery that many infinities are just that, infinity and do not obey the rules of everyday experience. Rational numbers are countable just like natural numbers and therefore, both infinities are deemed equal. Irrational numbers are not countable and therefore are definitely infinitly more of them, so they garner the term Alph Prime.
Now for my question. If Alph Naught is the smallest infinity (and it may not be, and then my question is moot!), why is that? Prime numbers are a smaller subset. Now I am easily convinced that even numbers = the same infinity as all natural numbers or even that rational numbers also = the same as the natural numbers in being equal in number to infinity due to countability.
However, primes are not technically countable (or are they and I am just unaware of any new break through in math. And if they are NOT countable, I have no one to one correspondence with natural numbers. And as we know, primes are definitely a subset of rational numbers. (and I just stated earlier that I accept that even (or odd or any repeating ratio) is equal to the natural numbers in size @ infinity) The number of primes is approximated by the following function #/(ln #) for any number ie 100/ ln 100 = 21 primes between 1 & 100. This is merely an approximation and not an exact formula and this function merely approximates a discreet value and cannot impose a continuous feature to this function. I state this to dispel the circle within a circle argument ie one set is smaller than another set, but still has the one to one correspondence.
Now many people still insist that the number of primes is infinite and equal in size to alph naught. However, the number of primes diminishes to zero at infinity (look at the limit of the approx function, which I admit, may NOT be the actual function and it is often conjectured that NO function does exist). At close to the origin, Primes are thick as flies. However as one chases after infinity, there are billions, then trillions of numbers between each prime (or perhaps a Mersenne prime pair). Admittedly, you have to be in the google range before the primes begin to thin out, but infinity is still infinity larger than a google.
So, how can the number of primes = to Alph Naught (the smallest infinity?)? Or is Alph Naught not the smallest infinity?
Now for my question. If Alph Naught is the smallest infinity (and it may not be, and then my question is moot!), why is that? Prime numbers are a smaller subset. Now I am easily convinced that even numbers = the same infinity as all natural numbers or even that rational numbers also = the same as the natural numbers in being equal in number to infinity due to countability.
However, primes are not technically countable (or are they and I am just unaware of any new break through in math. And if they are NOT countable, I have no one to one correspondence with natural numbers. And as we know, primes are definitely a subset of rational numbers. (and I just stated earlier that I accept that even (or odd or any repeating ratio) is equal to the natural numbers in size @ infinity) The number of primes is approximated by the following function #/(ln #) for any number ie 100/ ln 100 = 21 primes between 1 & 100. This is merely an approximation and not an exact formula and this function merely approximates a discreet value and cannot impose a continuous feature to this function. I state this to dispel the circle within a circle argument ie one set is smaller than another set, but still has the one to one correspondence.
Now many people still insist that the number of primes is infinite and equal in size to alph naught. However, the number of primes diminishes to zero at infinity (look at the limit of the approx function, which I admit, may NOT be the actual function and it is often conjectured that NO function does exist). At close to the origin, Primes are thick as flies. However as one chases after infinity, there are billions, then trillions of numbers between each prime (or perhaps a Mersenne prime pair). Admittedly, you have to be in the google range before the primes begin to thin out, but infinity is still infinity larger than a google.
So, how can the number of primes = to Alph Naught (the smallest infinity?)? Or is Alph Naught not the smallest infinity?
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