0

I don't quite understand the concept of undecidability/independence. Okay, a sentence is not provable in a certain axiomatic system, but can it be true nonetheless? Or is it "truth value-less"? In the first case, what determines the truth or falsity of a statement?

Say the Goldbach conjecture is undecidable. What if I find a counterexample and prove it false anyway? Does this mean that undecidability doesn't have to do with truth value, but only with provability of such truth value? Intuitively, prove it or not, the counterexample either exists or it doesn't. Law of excluded middle should agree with me, right? That means that a $\Pi_1^0$ sentence either is true or isn't, independently on its undecidability in whatever axiomatic system.

Am I correct with this line of reasoning?

Elvis
  • 1,543
  • 2
    A model determines the truth value of a sentence. Undecidable means there are models (satisfying the chosen axioms) where the sentence is true and models where it is false. The completeness theorem establishes a connection between "provable" and "true in every model". – Karl Apr 07 '24 at 00:40
  • See also this recent question and answer related to undecidability and excluded middle. – Karl Apr 07 '24 at 00:43
  • 2
    "Say the Goldbach conjecture is undecidable. What if I find a counterexample and prove it false anyway?" Then it wouldn't be undecidable, now would it? – spaceisdarkgreen Apr 07 '24 at 04:08
  • Goedel's undecidable sentence is the prototypical example of a sentence of arithmetic that is true and unprovable, IOW that sentence is a theorem of arithmetic. -- Whence, adding its negation to the theory does not make the theory inconsistent, but it does make it a false/unsound theory of arithmetic. Namely, I disagree on philosophical/foundational grounds that "a model determines the truth value of a sentence": a model really is just yet another syntactic construct with no semantics attached. Which is of course my criticism to the whole model-theoretic approach, FWIW... –  Apr 08 '24 at 07:22

1 Answers1

1

Let us illustrate the notion of decidability by a passage from Jon Barwise (in Handbook of Mathematical Logic edited by him). After he has given the axioms of group,

(1) $\forall x\forall y\forall z[ x + ( y + z ) = ( x + y ) + z ]$,

(2) $\forall x [ x + 0 = x]$,

(3) $\forall x\exists y [ x + y = 0]$,

and for an abelian group,

(4) $\forall x\forall y [ x + y = y + x]$,

he notes (bold font added)

The theory of abelian groups is a decidable theory, whereas the theory of groups is undecidable. That is, one can give an effective procedure which will tell of an arbitrary sentence $\psi$ involving $+$ and $0$ whether or not $\psi$ is a logical consequence of (1)-(4), i.e., whether or not is true in all abelian groups. There can be no such procedure for the theory of groups.

Hence, the issue about decidability is not whether a statement $\psi$ (or a theory) has a determinate truth-value or not, but whether there is an effective (mechanical) procedure to determine whether the statement follows from a given set of axioms or not.

So, an undecidable sentence can have a determinate truth-value. In logical systems that force determinate truth-values onto statements (not necessarily a system must be bivalent), it has definitely.

Beyond the conception of formal systems, it is a matter of philosophical dispute. To have an idea of the questions involved, see, for example, Øystein Linnebo's SEoP article “Platonism in the Philosophy of Mathematics.”

Tankut Beygu
  • 4,412