
Re: infinity
Posted:
Oct 4, 2017 11:46 PM


On 05/10/2017 00:44, FromTheRafters wrote: > 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/anyprime 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
I don't get the point you're trying to make re "other lists of primes". The link points to a YouTube clip of someone running through the standard Euclid's proof, which is fine, but doesn't make the OP's proof "wrong".
I suppose we could criticise the OP for not providing the detailed justification as to why x (in the OP's proof) must be prime, but in fairness I don't think the OP intended to give a complete/formal proof! Most proofs in practice miss out simple steps that the author thinks will be readily filled in by the reader. That's why I supplied an acceptable justification, which assumes we've previously proved the result:
If a number n>1 has no prime factor smaller than n, then n is prime.
With this extra justification the OP's proof is OK, so I wouldn't say the proof was wrong, just that it wasn't complete.
Anyway, that's what I *guessed* the OP would have said if challenged, but perhaps he/she had some other reasoning  having constructed x there are multiple paths we can take to show a contradiction, and they are all OK.
E.g. another approach could be to use a simple result:
Every number n>1 is divisible by some prime number.
(Clearly we can show the OP's x is not divisible by any prime number, since it leaves a remainder of 1 when divided by every prime: contradiction!)
Hmmm, interesting that doing it like this we don't need to have previously proved the uniqueness of prime factorisations, just their *existence*, so perhaps that's an advantage for this approach...
I guess it's also possible that the OP did not have in mind a correct reasoning as to why x must be prime, in which case the OP was making a mistake in claiming to have a proof! :)
Mike.

