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Does every mathematical question have an unambiguous answer?

For example, suppose I were to assert "In the decimal expansion of pi, does there occur in at least one location a billion 1's in a row?" Now, that is a purely mathematical question, so does it have an unambiguous answer?

Perhaps that was too easy of an example. What about something like the Continuum Hypothesis? I know it is independent of the ZFC axioms of set theory, but could it still be that the Continuum Hypothesis is true in the "real" universe of set theory? My belief is that mathematics is a black and white subject with no ambiguity. But is that really true?

Basically, my question is whether mathematical questions are completely unambiguous and precise and admit only "yes" or "no" as an answer.

Roger Vadim
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user107952
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    Realists, such as Quine and Putnam, have argued that mathematical statements have an objective truth value independent of minds and language. Anti-realists (formalists) consider undecidable statements to have no objective truth value. – nwr Nov 21 '22 at 21:15
  • this is essentially an open question, answering one way or the other would decide the debate between the constructivists and the platonists – emesupap Nov 21 '22 at 21:34
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    I disagree with the comments above. As asked, the answer to your question is a definitive no. It is easy to ask an ambiguous "purely mathematical question" with no unambiguous answer, e.g. "are groups commutative?". It may seem that we could fix that by demanding that the question itself be "unambiguous". The problem with that is that there is no general way to tell in advance when this is so. CH was once thought to be an unambiguous question, it turned out otherwise. Even if the ephemeral "real universe of set theory" saves this one there are plenty more where it came from. – Conifold Nov 22 '22 at 09:19

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Even whether 00 = 1 or is better left "undefined" is not unambiguous (or "unambivalent" might be a tad more technically fitting an adjective, but I get what you mean). Some mathematicians favor interpreting the exponential as referring to a function from 0 to itself, which "should" compute to 1 then (I believe John Conway was one of these, or his argument also depends on some sort of "usefulness" for 00 = 1, like holding to the equation lets us compute some other thing that must be left more free-floating otherwise).

Or is 2 := {{0}} or {0, {0}} (or something else)? You might think it irrelevant, or trivial, which definitional equation (reduction) you accept, and indeed there are only so-called "junk theorems" that depend on the difference.

Or consider, "How many (strongly) inaccessible cardinals are there?" Is this a "real" question? How would you go about figuring out the answer? You can get, "At least 2 or 3," in a more-or-less normal version of set theory—0, ℵ0, and V—although in that last case, that depends on if you suppress the powerset axiom so that V can be a universal set (without the pesky contradictory quality of having a larger powerset, then), or if you hold fast the replacement scheme so that V ≠ ℵω (otherwise you have cf(V) = ℵ0, which makes V accessible after all), or something else along those lines (you can also warp V into an accessible if you use a Corazza embedding to avoid the Kunen barrier, for example).

So I only know set theory well enough to comment this much, and so again, the only example of dependably unclear equations from outside set theory, that I'm familiar with, is the self-exponential for 0. See Saharon Shelah, "Logical Dreams" and his follow-up for some speculative remarks about further ambiguously possible options. (So note that your question itself might turn out to be one possible answer to itself!)

Kristian Berry
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