- #1

nomadreid

Gold Member

- 1,456

- 144

Two paradoxes from different domains generate huge number of would-be solutions, and I am not starting this thread in order to promote one solution over the other or to proclaim that I have a new solution. I just wonder whether the techniques used for certain would-be solutions of one could be used for certain would-be solutions of the other. I don't have it worked out yet, but given that it seems a natural idea, I would be surprised if someone had not pursued this line --either successfully or otherwise.

So, here's what I have so far. The two paradoxes concerned are the Liar Paradox and the Grandmother Paradox (from time-travel fantasies) which at least superficially seem to have something in common, both referring to a type of self-reference ; whereby one refers to the code of oneself, and the other refers to a previous stage of oneself.

The techniques I am eyeing are the Diagonal Lemma (Roughly: Given a one-place first-order formula Φ(.) and a means of coding first-order sentences ".", then there is a sentence A such that Φ("A") is true iff A is true.) and the technique from umteen science-fiction movies as well as the more serious suggestion from Kip Thorne in his popular "Black Holes and Time Warps" book, not to mention the "Prisoner of Azkaban" or even ancient Greek ironic self-fulfilling prophecies: that nothing the time-traveller does in the past alters the eventual result. Both of these are essentially fixed-point theorems. However, the devil is in the detail: how best to work this out. Also, since the Grandmother paradox involves stages, I am not sure whether or not another technique which is sometimes evoked in discussions about truth, that of Kripke frames, should be factored in somehow.

Any indications are welcome, even if just to tell me I am being silly.

So, here's what I have so far. The two paradoxes concerned are the Liar Paradox and the Grandmother Paradox (from time-travel fantasies) which at least superficially seem to have something in common, both referring to a type of self-reference ; whereby one refers to the code of oneself, and the other refers to a previous stage of oneself.

The techniques I am eyeing are the Diagonal Lemma (Roughly: Given a one-place first-order formula Φ(.) and a means of coding first-order sentences ".", then there is a sentence A such that Φ("A") is true iff A is true.) and the technique from umteen science-fiction movies as well as the more serious suggestion from Kip Thorne in his popular "Black Holes and Time Warps" book, not to mention the "Prisoner of Azkaban" or even ancient Greek ironic self-fulfilling prophecies: that nothing the time-traveller does in the past alters the eventual result. Both of these are essentially fixed-point theorems. However, the devil is in the detail: how best to work this out. Also, since the Grandmother paradox involves stages, I am not sure whether or not another technique which is sometimes evoked in discussions about truth, that of Kripke frames, should be factored in somehow.

Any indications are welcome, even if just to tell me I am being silly.

Last edited by a moderator: