Goldbach's Conjecture and 1-1 correspondence

Question

I know math only (somewhat) as a recreation, so I know this is a naive and ignorant question, but I don't have the mathematical terminology or experience to figure out why it has to be incorrect. I am not claiming a proof of Goldbach's Conjecture, just trying to determine what I'm missing in what the desired proof is supposed to show. I know the StackExchange moderation is tough and this kind of question might get closed off the bat, but I hope someone will indulge me in a little help on this one.

Here goes: If it is true that the prime numbers can be put into a 1-1 correspondence with the even numbers, and if doubling any prime number yields an even number, why doesn't that prove Goldbach's Conjecture?

Again, this sounds so trivial that of course it can't be the answer. But I'm curious about why it isn't.

Thank you for any help.

Answer 1

Suppose that the prime number 7 is put in correspondence with the even number 12 (or any other even number you like).

Then it's true that $7 + 7 = 14$ is even, and hence the even number 14 is the sum of two primes, but this doesn't say anything about the even number $12$ that was the "correspondent" for 7.

Answer 2

This has nothing to do with the question, unfortunately. The conjecture is that every even number is the sum of two primes. Your observations, though undoubtedly true, don't say anything about sums.

Answer 3

@JohnHughes answer is really about as elementary as possible. But I'll try to write a version that isn't over your head.

Yes, you can make a list of the primes, and then match the list to a list of the even numbers. Suppose (just to make my argument clear) that the lists are

primes 2  3  5  7  11  13  17 ...
evens  2  4  8 10  12  13  16 ...

That happens to match them in their natural order, but any matching would do.

Now your argument doubles every prime, which will give you the list dp (for "double primes")

  dp   4  6  10  14  22  26  34 ...

All those numbers are even, but there are clearly plenty of other even numbers, like, say, $8$, not on the list dp. Those other even numbers may or may not be the sum of two primes ($8 = 3 + 5$ is). We hope they all are, which would make Goldbach prescient.

Note that carefully reading what you wrote shows that you never used anything about the list evens .