Yablo’s paradox cannot be formulated in ZF because it uses the idea of truth. If you reformulate it so that every sentence says that the rest of the sentences can be disproven, all of the sentences will be false, but there will be no proof in ZF of this for any of them. This simply shows that ZF is not (and cannot be) complete, not that there is anything wrong with it.
It is not necessary that the YP is “formulated in ZF”.
It’s enough that ZF yields some byproduct, like countably infinite sets, which are used to make (in this case a semantic) paradox.
Then something must be wrong either with ZF, either with the semantics which allows YP formulation. Possibly with both, but at least with one of them.
If the list of Yablo’s sentences is a finite one, then the last statement of the list is just true. And all those before the last are false and no paradox there.
This doesn’t mean, that something is necessary wrong with ZF, only likely. There might be a problem solely with the semantics which permits Yablo’s sequence. Still possible.
But the YP formulation is a very elementary one. There might be others, equally elementary. I didn’t say they are, but that they might be. The “infinite geometry” looks suspicious to me.
There are countably infinite lists in ZF. That doesn’t make the general fact that in some situations you can produce a paradox with a countably infinite list, a reason to think you can do that in ZF. You might as well argue, “We can produce paradoxes in natural language. So maybe we can do it in mathematics too.”
And maybe you could have made that argument, before people tried it. As others have pointed out, many people have looked for contradictions in ZF for a very long time, and none have been found. There is no reason to think there are any.
Yablo’s paradox cannot be formulated in ZF because it uses the idea of truth. If you reformulate it so that every sentence says that the rest of the sentences can be disproven, all of the sentences will be false, but there will be no proof in ZF of this for any of them. This simply shows that ZF is not (and cannot be) complete, not that there is anything wrong with it.
It is not necessary that the YP is “formulated in ZF”.
It’s enough that ZF yields some byproduct, like countably infinite sets, which are used to make (in this case a semantic) paradox.
Then something must be wrong either with ZF, either with the semantics which allows YP formulation. Possibly with both, but at least with one of them.
If the list of Yablo’s sentences is a finite one, then the last statement of the list is just true. And all those before the last are false and no paradox there.
This doesn’t mean, that something is necessary wrong with ZF, only likely. There might be a problem solely with the semantics which permits Yablo’s sequence. Still possible.
But the YP formulation is a very elementary one. There might be others, equally elementary. I didn’t say they are, but that they might be. The “infinite geometry” looks suspicious to me.
Once again, as others have already told you, Yablo’s paradox cannot generate any sort of paradox whatsoever in ZF.
I told the others, that the countably infinite sets MIGHT be infected, since a finite list of Yablo sentences DOESN’T yield a paradox.
While a countably infinite list of Yablo sentences—DOES yield the mentioned paradox.
AFAIK, the infinities come out of ZF. Don’t they?
There are countably infinite lists in ZF. That doesn’t make the general fact that in some situations you can produce a paradox with a countably infinite list, a reason to think you can do that in ZF. You might as well argue, “We can produce paradoxes in natural language. So maybe we can do it in mathematics too.”
And maybe you could have made that argument, before people tried it. As others have pointed out, many people have looked for contradictions in ZF for a very long time, and none have been found. There is no reason to think there are any.