I've been told True Arithmetic is syntactically complete(either $φ$ or $¬φ$ is provable for every $φ$), yet one cannot decide which of the two. This makes no sense to me. How can one say they proved either $φ $ or $¬φ$, without knowing which?

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    $\begingroup$ You probably believe that the twin prime conjecture is either true or false. So if you take "all true facts about $\mathbb{N}$" as your axioms, then you know that either the twin prime conjecture or its negation is an axiom! So you know you can prove one of them (with a very short proof too), but you have no idea which one you can prove! $\endgroup$ Sep 22 at 17:16
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    $\begingroup$ 'provable' is not the same as 'proved' $\endgroup$
    – Bram28
    Sep 22 at 17:27
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    $\begingroup$ @Bram28 Thanks, I now see where I misunderstood. $\endgroup$
    – setblack7
    Sep 22 at 17:41
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    $\begingroup$ This is actually exactly the sort of thing that led Brouwer to reject the law of the excluded middle! See plato.stanford.edu/entries/intuitionism . $\endgroup$ Sep 22 at 19:10


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