Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Re: infinity
Replies: 27   Last Post: Dec 17, 2017 2:31 AM

 Messages: [ Previous | Next ]
 Mike Terry Posts: 767 Registered: 12/6/04
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/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

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:

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.