# Term for a logical statement that can only be proven false

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?

• This reminds me of the haulting problem in computer science: en.wikipedia.org/wiki/Halting_problem – Matthaeus Gaius Caesar Aug 12 '20 at 6:33
• @zkutch In comments, you have to make links this way: [text](address), rather than [text][index] followed by [index]: address. – Arthur Aug 12 '20 at 7:13
• 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. – Mark S. Aug 12 '20 at 11:24
• That said, a new question about how/whether something like this can happen in mathematical logic would be about mathematics. – Mark S. Aug 12 '20 at 11:36
• @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 – TheEnvironmentalist Aug 12 '20 at 17:12

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.

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.