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Given any sentence $A$ of ZFC (or any other formal system, really), we have exactly four possibilities:

  1. $A$ is true and not false

  2. $A$ is false and not true

  3. $A$ is true and false at the same time (i.e. ZFC is inconsistent)

  4. $A$ is neither true nor false (i.e. $A$ is independent of ZFC)

Is my understanding correct?

Community
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user132181
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3 Answers3

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No. You are making the classic mistake (that put you in good company with many mathematicians) of confusing true and provable.

Truth is relative to a fixed model of $\sf ZFC$. Either $A$ is true in that model or it's not. Even if $A$ is provable, or if it's not provable from $\sf ZFC$. Given a model of $\sf ZFC$ we can verify that either $A$ is true there or it is not.

So the question should in fact distinguish between:

  1. $A$ is provable from $\sf ZFC$.
  2. $A$ is refutable from $\sf ZFC$ (or disprovable, or $\lnot A$ is provable).
  3. $A$ is provable and refutable from $\sf ZFC$.
  4. $A$ is neither provable, nor refutable from $\sf ZFC$.

Terminological issues aside, we sometime confuse the first and second one, by saying that $A$ is true or false in $\sf ZFC$, since if $A$ is provable, it will be true in all models of $\sf ZFC$.

Asaf Karagila
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    You are like a superhero, really. – user132181 May 17 '15 at 11:19
  • Just a small remark based on the provable/tautology distinction between our answers. Of course, in the safe havens of classical first-order logic we have our lovely soundness and completeness theorems. But shouldn't we maintain this principal distinction for the sake of people who will investigate other realms of logic in their life? – Lord_Farin May 17 '15 at 11:27
  • @Lord_Farin: I'm not sure that I understand your comment. – Asaf Karagila May 17 '15 at 11:30
  • @Asaf I am alluding to the fact that while OP was discussing "truth in every model", i.e. "tautology", your answer instead discusses "provable", which is a distinction that at least I prefer to keep making (probably from my excursions in nonstandard logics). – Lord_Farin May 17 '15 at 11:32
  • @Lord_Farin: I guess this is a personal preference I don't share. I also didn't see the edits to the original question (I started writing my answer with that, but I had to pause for a bit), so I didn't see the "truth in every model". In any case, I am a firm believer of using crutches and scaffolds. First ignore the non-standard stuff (but remember that it's out there), work in the classical well-known world, and when you've reached a certain level of mastery, it is time to explore the outer limits of the map. Tharrrr be Dragons!!! – Asaf Karagila May 17 '15 at 11:36
  • @Asaf Fair enough, I just thought a remark was in order. I guess it's hard to pretend not knowing things :). – Lord_Farin May 17 '15 at 11:39
  • @Lord_Farin: You have much to learn. :-) – Asaf Karagila May 17 '15 at 11:41
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If I understand it correctly, you can only speak about "truthness" in some model. As far as I know, the common way how logicians think is that it is either "true in each model" or "true in some model and false in some other model" (this correcponds to your notion of independence) or "false in each model" or "in each model it is both true and false" (inconsistency of ZFC).

Peter Franek
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What you have done is not but enumerate the "truth table":

$$\begin{array}{c|c} {\sf true} & {\sf false} \\ \hline T & T \\ T&F\\ F&T\\ F&F \end{array}$$

which of course can be done without problem.

However, your terminology is not optimal. I would suggest "tautology" in place of "true" and "contradiction" in place of "false".

Lord_Farin
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    This goes back to the question of what "true" means. I think that it is more common to speak about truthness of formulas in a given model and then you need to be more careful.. but I may be wrong, of course. – Peter Franek May 17 '15 at 11:18