FromTheRafters wrote on 10/4/2017 : > Mike Terry brought next idea : >> On 04/10/2017 22:13, FromTheRafters wrote: >>> Doctor Allan explained : >>>> does infinity exists and if so how can i prove it by math ? >>>> >>>> Infinity is a concept based on observation, in the same way as there >>>> is a concept of "stone"--imagine "stone"; you know what this concept >>>> means, because you can find examples of it, but no single example is >>>> the actual gestalt. Just like the concept of "one", "infinity" is not >>>> a standalone object, but a descriptive concept. >>>> >>>> What does it mean? Well, there are many meanings, even in >>>> mathematics, as another poster has recounted. A simple one is a proof >>>> of the infinity of the number of prime numbers. 2,3,5,7.... Let's >>>> see what happens if we assume the number of primes is finite. Let's >>>> see what we get when we multiply them all together and add one. >>>> >>>> x=2*3*5*7*...*lastprime+1. >>>> >>>> x is clearly larger than all the primes. Is x a prime? Let's try >>>> factoring it! x/2 leaves remainder 1, x/3 leaves remainder 1... in >>>> fact, x/any-prime leaves a remainder of 1, so x must be prime! >>> >>> As I understand the proof, this is wrong. It means x is *either* prime >>> or is a composite number with at least one of its prime factors not on >>> the list. >>> >> >> The list contained ALL the prime numbers, on the assumption that there were >> only finitely many of them. So x has no prime factors less than itself, >> and so must be prime. This establishes a contradition, as it is greater >> than all the numbers in the list, and hence also is not a prime. So we >> conclude there are infinitely many primes...) >> >> Regards, >> Mike. > > I understand that, but there are other lists of primes possible which one can > assume to be *all* of them. Those interested might enjoy this whole lecture, > but I copied it starting at the relevant part. > > https://youtu.be/lzyWL1LTlq4?t=479
Here, replying to myself because I can provide a quote from another source:
"It is a common mistake to think that this proof says the product p1p2...pr+1 is prime. The proof actually only uses the fact that there is a prime dividing this product (see primorial primes)."