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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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    $\begingroup$ This reminds me of the haulting problem in computer science: en.wikipedia.org/wiki/Halting_problem $\endgroup$ – Matthaeus Gaius Caesar Aug 12 '20 at 6:33
  • $\begingroup$ @zkutch In comments, you have to make links this way: [text](address), rather than [text][index] followed by [index]: address. $\endgroup$ – Arthur Aug 12 '20 at 7:13
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    $\begingroup$ This is more about logic in philosophy (of science) rather than logic in mathematics. I don't know a word, but the standard phrase would be "falsifiable but not verifiable". See this encyclopedia entry, this blog post, or this youtube video. $\endgroup$ – Mark S. Aug 12 '20 at 11:24
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    $\begingroup$ That said, a new question about how/whether something like this can happen in mathematical logic would be about mathematics. $\endgroup$ – Mark S. Aug 12 '20 at 11:36
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    $\begingroup$ @MarkS. That sounds like an answer. Unless someone provides something better, if you'd be willing to write that up as an answer, I'd be happy to accept it $\endgroup$ – TheEnvironmentalist Aug 12 '20 at 17:12
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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.

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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.

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  • $\begingroup$ Not exactly. I'm not looking for a term for something that is always true, I'm looking for a term for something that may be true and may be false, but can only ever be proven false. The idea is similar to but distinct from that of Gödel's incompleteness theorems, which show that with any axiomatically-derived version of mathematics, there will be postulates that are true, but are impossible to prove to be true even though they are true. I'm looking for a term meaning "can be true or false, but can't be proven true" $\endgroup$ – TheEnvironmentalist Aug 12 '20 at 7:59
  • $\begingroup$ If I correctly understand you would like example of theorem which is independent from some existing axiom system? If so, then en.wikipedia.org/wiki/Continuum_hypothesis is independent from ZFC. Does this make sense? $\endgroup$ – zkutch Aug 12 '20 at 8:04
  • $\begingroup$ I used Gödel's incompleteness to show the difference between objective truth and provable truth. That is, some things are true but there is no way to prove them true. Some things are false but there is no way to prove them false, and philosophy refers to things that can't be proven false (whether or not they are actually false) as unfalsifiable. I'm looking for a word equivalent to unfalsifiable, but for the other case. A word meaning 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 $\endgroup$ – TheEnvironmentalist Aug 12 '20 at 8:10
  • $\begingroup$ Hm.. I'll try to use your terminology: in ZFC axiom system continuum hypothesis is provable truth, while any theorem, which is proved in ZFC is objective truth. As to last sentence let me think little more, please. $\endgroup$ – zkutch Aug 12 '20 at 8:19
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    $\begingroup$ The term matters in the sense that it's easy to construct, or at least plan to construct, proofs for given arguments. However, there are a number of cases wherein proofs are impossible to construct. This concept is used extensively in reference to the idea of decidability, in fact the term I'm looking for could be considered a specific, if odd, instance of semidecidability. The broader concept is used in logic, and extensively in theoretical computer science. It's intriguing to me that this one concept doesn't have a name $\endgroup$ – TheEnvironmentalist Aug 16 '20 at 5:43

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