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Is it possible to know whether any and all statements are true, false, or undecidable under standard mathematical axioms, e.g. ZF?

Xander Henderson
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D J Sims
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    I think there's an interesting question here (and probably with a simple answer) if you state more precisely exactly what you're asking. I'm surprised at the downvotes, though. –  Dec 25 '16 at 00:48
  • No, because in many cases, proving something is undecidable proves it is true (example : the Goldbach conjecture). – reuns Dec 25 '16 at 01:58
  • This is an excellent question, but it is still an open question AFAIK. – Hank Igoe Dec 26 '20 at 21:40

3 Answers3

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I will prove:

There does not exist an algorithm that, for any statement P in the language of set theory, classifies P as being provable, disprovable, or independent of the axioms of ZF

Suppose you had an oracle $M$ that solves this problem. You can use this to create an algorithm that enumerates the theorems of a complete extension of ZF:

  • Let $A$ be a tautology
  • For every statement $P$ in the language of set theory:
    • Use $M$ to classify the status of $A \Rightarrow P$
    • If $A \Rightarrow P$ is provable in ZF, output $P$
    • If $A \Rightarrow P$ is independent of ZF, then set $A = A \wedge \neg P$

The idea is that the algorithm iteratively builds a list $A$ of new axioms to add to ZF to make it complete; the oracle $M$ is used to ensure we never select an inconsistent set of axioms.

Gödel's incompleteness theorem says that there does not exist an algorithm that can do this; consequently there is no algorithm for determining the output of $M$.

  • Cool${}{}{}{}{}$ – GFauxPas Dec 25 '16 at 01:27
  • So the answer is no – D J Sims Dec 25 '16 at 03:06
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    You prove that no algorithm exists. Is this equivalent to the question asked? How so? – Conor O'Brien Dec 25 '16 at 06:03
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    @Cᴏɴᴏʀ O'Bʀɪᴇɴ: The question asked is rather ambiguous; the statement I proved is one of the ways of turning it into something precise. Note, incidentally, that there is an algorithm capable of recognizing whether any particular statement (e.g. the claim "$P$ is independent of ZF") is provable or disprovable: simply enumerate all proofs. (p.s. your user name is difficult to ping) –  Dec 25 '16 at 06:13
  • @Hurkyl I see, thank you. About the username : I had thought I had changed it on all sites, but apparently it only updated on one site. – Conor O'Brien Dec 25 '16 at 06:17
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No, because every axiomatic system has it's own limits (after that limit you have to postulate something to continue developing the layers of complexity), because in fact to construct a complete axiomatic system you have to postulate an infinity of infinity axioms, because you can construct an infinity of infinities of statements and theroms .... and the notion of infinity is transcendental (countable, uncountable..... and more bigger cardinal sets), so it's so far to reach any statement, because you can construct bigger and bigger sets which have larger cardinality, for example you have naturals countable, real number not countable, super real number which is the power set of real numbers and you can keep on forever, so you can index by this power sets much larger number of statement that you cant reach by any means, so you can't prove them so you can't decide if it's true or false or undetermined. Axioms are not all you need to construct a theory, because there is an infinity of axioms within us, including your brain mechanisms for examples, or the laws governing the matter whom you are constituted of, all that it's an intrinsic axioms, because simply to construct something for the pure void, you have to postulate in infinity of axioms because you have to construct every thing from scratch, and the existence it self is infinite compared to the existence is the reason why $\infty\times 0 $ maybe equal to some thing like $1$ or $2$ ... but $0\times$ any number $=0$, you can't postulate a finite number of axioms and construct really something, every finite axiomatic system will be incomplete at certain range of it's development in much higher layers of complexity.

kuilin
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Abdelmajid Khadari
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  • The theory of the arithmetic of real numbers is complete, and the axioms enumerable by algorithm. (this theory is more commonly called the theory of real closed fields) –  Dec 25 '16 at 01:43
  • i give real number as example, i tell you that the power set of intgers is the real number and the power set of real number is Surreal numbers, and you can construct a bigger and bigger cardinalities, the point her is not the real number the point is that you can construct bigger and bigger cardinalities that you can use to index bigger and bigger cardinalities of statments. – Abdelmajid Khadari Dec 25 '16 at 01:50
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No, in general this is not possible to know for all statements. For example, it is conceivable that the Riemann Hypothesis might forever remain "false or undecidable or true". (Also, in the particular example of the Riemann Hypothesis, the states "undecidable"/"true" are not mutually exclusive.)

Alex
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