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By Tarski's Theorem, it's complement, the set of true statements in formal arithmetic, is not recursively enumerable. But the set of untrue statements might be. Is it?

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    What do you mean by "untrue"? If "x is untrue" means "NOT x is true", then it is very easy to enumerate true statements given an enumerator for false statements. – 2017-01-19
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    I think (given the "incompleteness" tag) that the author with "untrue" does not mean "false", but "not true". E.g. "This statement is false" is not true, but also not false. – 2017-01-19
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    Tarski's theorem shows that the set of true statements is not arithmetically definable - which is much stronger than just not r.e. If a set is not arithmetically definable, its complement is also not arithmetically definable. – 2017-01-19
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    @CarlMummert you really should give this as an answer, not as a comment :) – 2017-01-19
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    @Anonymous: I forgot why it was a comment, but then I remembered that this is a PSQ. It would be ideal for the OP to improve the question and to respond with the version of Tarski's theorem they are thinking of. – 2017-01-20
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    @Anonymous Note that such statements can't actually be expressed in the language of arithmetic. More generally, in first-order logic, if $M$ is a structure and $\varphi$ is a sentence, either $\varphi$ is true in $M$ or $\varphi$ is false in $M$; there is no middle ground. – 2017-01-20
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    @CarlMummert. What does PSQ stand for? – 2017-01-20
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    @user254665: a problem which is just a statement of a question, exactly in the form that it might be given as an assignment. – 2017-01-21

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If there were an algorithm to enumerate the false statements, just run it, and every time it produces a statement of the form "Not $S$", output $S$. That would enumerate the true statements.