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Topic: Re: infinity
Replies: 21   Last Post: Oct 6, 2017 1:38 AM

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FromTheRafters

Posts: 193
Registered: 12/20/15
Re: infinity
Posted: Oct 4, 2017 11:03 PM
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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



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