```Date: Oct 5, 2017 3:30 AM
Author: FromTheRafters
Subject: Re: infinity

Mike Terry expressed precisely :> On 05/10/2017 04:03, FromTheRafters wrote:>> 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).">>>> From:>>>> https://primes.utm.edu/notes/proofs/infinite/euclids.html>> That's all well and good, but again this is a comment specifically on > Euclid's proof, or some specific variation of that proof.  The link you give > is not talking about the OP's proof, and certainly does not know what the OP > was thinking for the justification for claiming that x (in the OP's proof) is > prime.  The links are NOT saying the OP's proof is "wrong", which seems to be > what you're suggesting.>> Do you accept:>> a) that IF the OP had managed to supply a valid reasoning>     for x being prime, then the OP's proof would be valid.>     (You may think no such reasoning exists, but that's not>      the question here)>> b) that there are a number of such valid reasonings available,>     although (we agree) the OP did not supply any of them>> ?>> Mike.Okay, I guess. Still, it seems to me that stipulating a 'lastprime' rather than an n^th prime where n is a natural (finite) number is not valid. It is like saying you have an actual last natural number as an index. This leaves out the 'headroom' needed to make my assertion correct.That seems to be the only difference between this and Euclid's proof, hence my comment. Maybe the OP will chime in and make this clearer.
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