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We have all heard that if Goldbach's conjecture is independent, then it is true. This is because if GC is false then there is an even number which is not the sum of two primes, and hence a finite proof. What if this number is very large? Maybe beyond our reach?

If one proves that it is independent that eventually every even number is the sum of two primes, can we conclude that eventually every even number is the sum of two primes? Even if we may never know the $e$ from which it is Goldbach's conjecture, all the way, into infinity?

If we know that there is an $e$ from which every even number after and including $e$ is the sum of two primes, is independent, can we conclude that $e$ and every even number beyond, is the sum of two primes?

Even if we may never know what $e$ is?

Let $e$ be the least even integer such that $e$ is the sum of two primes and for every $k=2n$ where $n \in \mathbb{N}$ where $k$ is greater than $e$, then $k$ is also the sum of two primes. Suppose it is independent of ZFC that $e$ exists. Does it follow that $e$ exists?

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    $\begingroup$ When you say "independent", shouldn't the next words be "of XYZ", where XYZ is the name of some axiomatic theory? $\endgroup$
    – S. Carnahan ♦
    Commented Nov 17, 2018 at 6:35
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    $\begingroup$ What other axioms would it be? Aka math $\endgroup$
    – user131478
    Commented Nov 17, 2018 at 6:40
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    $\begingroup$ @Erin ZFC is not the same as "math", I don't understand that comment (and it feels somewhat rude, unfortunately). As shown in the answers, this is already something that PA and much weaker theories address. $\endgroup$
    – Andrés E. Caicedo
    Commented Nov 17, 2018 at 21:18
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    $\begingroup$ @Erin No, sorry; if the question requires editing, it should be edited. If you post on your own blog, you have control of the content. $\endgroup$
    – Andrés E. Caicedo
    Commented Nov 17, 2018 at 21:30
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    $\begingroup$ please don't delete a question which has been answered, the whole point of MO is to have a permanent record of questions and answers. $\endgroup$
    – Carlo Beenakker
    Commented Nov 17, 2018 at 21:34

2 Answers 2

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We can't readily conclude that. Each $\Sigma_1$ statement of arithmetic is provable in PA and hence in ZFC, but not every $\Pi_2$ statement (which is what you would seem to need here) is.

Indeed the $\Pi_1$ statement Con(ZFC) is already not provable (under the assumption that your Eventual Twin Prime Conjecture is independent).

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  • $\begingroup$ Thank you Bjorn. Are you saying that it is not necessarily true that if the existence of $e$ is independent then it exists? $\endgroup$
    – user131478
    Commented Nov 17, 2018 at 8:49
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    $\begingroup$ Yes, "$e$ doesn't exist " is the $\Pi_2$ statement $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Nov 17, 2018 at 12:27
  • $\begingroup$ Cool, thanks! I am trying to accept and upvote the answer. $\endgroup$
    – user131478
    Commented Nov 17, 2018 at 15:16
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Let $S$ be the set of even integers which are not a sum of two primes. Goldbach’s conjecture is that $S$ has no members larger than $2.$.

And, as you say, if we somehow know that it is consistent (with some system) both that it is true and that it is false, then it is true. This since, if it is false, there is a specific counter-example which can be certified by a proof. Perhaps a proof which would take more pages than there are molecules in the known universe , but still a finite proof.

Similarly a statement such as "$S$ has no members greater than $10^{2000000}!$" is true, if independent, for the same reason.

Your hypothetical conjecture is that $S$ is finite. If that is true, that does not force existence of a finite proof.

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