In logic, some statements can't be proven true, only proven false.
For example, the statement "the universe is infinite" can be disproven by discovering its bounds, say by launching a rocket that crashes into the all-encompassing, mysterious wall near a galaxy far, far away; but cannot be proven true, as the case of an infinite universe is observationally indistinguishable from the case of a universe "so large we haven't found its bounds yet".
The term I'm looking for could be viewed as equal and opposite to "unfalsifiable", in that I'm looking for a term roughly equivalent to "untruthable". Does such a term exist? Also, does the term "unfalsifiable" hold this strict interpretation of "provably true or unknowable" in logic, as I've only ever heard it used in the context of philosophy?
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$\begingroup$I think this is more traditionally a topic in philosophy (of science) rather than logic in mathematics. I don't know of a single word for it, but the standard phrase would be "falsifiable but not verifiable". For some references for this usage, see this encyclopedia entry, this blog post, or this youtube video.
$\endgroup$ $\begingroup$As I understand from our dialog in comments main phrase is
"something that may be true, or may be false, and can be proven false if it is false, but is impossible to prove true even if it is true"
Let me make light analysis of this sentence. Firstly we are speaking about some relation/predicate, denote it by $\boldsymbol{\mathfrak{A}}$, because we want to characterize it with false or true. Then we need some proving mechanism so, generally, some other relations, axioms, predicates and logical scheme(s), denote it by $\boldsymbol{\mathfrak{M}}$, using which we create proof(s), objective truth, provable truth. Now, in environment $\boldsymbol{\mathfrak{M}}$ is possibility to forbid proof relation $\boldsymbol{\mathfrak{A}}$ only when it is true.
Firstly I am interesting is this that one about which we want to speak? And second - suppose it is possible to create such $\boldsymbol{\mathfrak{M}}$ and $\boldsymbol{\mathfrak{A}}$, why we need them? Sorry, if some ideas are outside of strange.
$\endgroup$ 8 $\begingroup$The computational complexity class of such problems is defined as RE. Any problem in RE can be proven by a Turing machine to be true in finite time if it is true, but can not necessarily be proven false, even if it is false. A Turing machine is a universal machine that can solve any logical or mathematical problem that any finite algorithm or computer can solve.
For example, whether or not a Turing machine program halts (finishes its computation) or not is an example of a problem in RE. If a program is eventually computed to halt, you know the answer is yes, but if it never halts for as long as you run it, that doesn't mean it's impossible it won't halt in the future, and indeed it is impossible to prove that it eventually will halt, which is the halting problem.
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