• I
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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.

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andrewkirk
Homework Helper
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It's helpful to distinguish between veridical and falsidical paradoxes. The former are conclusions that are surprising and anti-intuitive, but not logical contradictions. The latter are logical contradictions.

The grandmother paradox, like in Azkaban or in Heinlein's 'All you zombies', is veridical. It doesn't seem to make sense, yet it does. The 'rule' that it seems to contradict is something like 'every sequence of events must have a cause outside the sequence'. But that is just a natural expectation based on our observation of the everyday, non-time-travelling world. It is not a rule of logic. Lay people often describe quantum mechanics as paradoxical. What they mean is that it contradicts their intuitions, not that it generates logical contradictions. The 'paradoxes' of QM are veridical.

The liar paradox is, depending on how presented, either meaningless or a falsidical contradiction. Most presentations of it are in natural language, and are meaningless. They defy attempts to formalise them. A successful formalisation of it is Russell's Paradox, formulated in the context of naïve set theory. The paradox is falsidical, meaning it generates a contradiction, and that contradiction tells us that naïve set theory is inconsistent, which is why it was replaced by Zermelo-Frankel and its successors. Other formalisations are possible using second-order and higher-order logic. Those formalisations lead to formal contradictions, which demonstrate that the particular version of logic being used is inconsistent.

Quine has a third sort of paradox, called an antinomy. Kant also played around with antinomies. According to wiki, there's a fourth type of thing that is sometimes called paradox, that has been discussed since Quine. The wiki article covering this is quite good.
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Thank you, Andrewkirk, for your extensive response about paradoxes. However, it appears that I did not express myself very well, because your answer misses the point of my question. To be clearer, I am not interested in the paradoxes per se (er, well, I am of course interested in them, but not in this thread), but rather in two (possibly three) results that, while they come up in the discussion of paradoxes, are themselves not paradoxes, and appear to be equiconsistent with a standard number theory. That is, the Diagonal Theorem is not a paradox. The concept of a fixed point in recursive functions is also not paradoxical, even though Kip Thorne's solutions (to the matricide paradox) to which he referred (and briefly explained) on pages 511-515 of "Wormholes and Time Machines: Einstein's Outrageous Legacy" are curious and not simple, and I guess you could call them paradoxical (veridical) if you wish to call all counter-intuitive concepts paradoxical (thereby placing much of modern physics in this category). But that is a side point, and I am not concerned about terminology. Rather, I wish to figure out how to use the diagonal theorem or another fixed-point theorem and/or possibly Kripke frames to formalize Thorne's solution.

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Ah, I did not search correctly the first time. As I surmised, I am not the first to think of doing this, and a good discussion is to be found in the following :
https://plato.stanford.edu/entries/time-travel-phys/
But I am not sure that this is the last word.

stevendaryl
Staff Emeritus