Solving Theseus’s Ship Using Set Theory

(https://shitwhole.com/miscellaneous/ship-of-theseus/)

Theseus’s Ship is one of the most interesting paradoxes, with many explanations regarding its solutions. However, I think there is a more solid and logical solution or explanation for this paradox—using Set Theory.

But first, we must explain the paradox in simple terms: If you replace each original/old material of the ship one by one with new materials, would it still be the same ship?

Set Theory Explanation

To start, we will assume that the new materials will be the exact same, just “other” materials.

Set Theory is the study of sets, and in each set there are elements that make up the set. The number of elements in a set are named “cardinalities.” We can now draw some parallels into this: Think of Theseus’s Ship as a kind of “set” that contains its own elements (wood planks, other materials, etc.)

To use as comparison, let us say Set X contains all the colors of the rainbow: {red, orange, yellow, green, blue, indigo, violet} and Set Y also contains all the colors of the rainbow: {red, orange, yellow, green, blue, indigo, violet} they are equal as they contain the same elements and cardinalities (|7|).

Now let us use the same logic and principle for Theseus’s Ship, the elements of Ship A have {wood, metal, flag} and the elements of Ship B that will replace Ship A will be {flag, wood, metal}. According to the principles Set Theory, Ship A and Ship B are equal sets. As the definition of an equal set is “exact same elements and exact same cardinalities.” With the assumption we have, that is, the ship is just merely being replaced with the exact same materials, we can use this explanation as a solution for Theseus’s Ship, and our answer would be: Yes, it would still be the same ship.

Conclusion

It is important to note that this is just one of many explanations regarding Theseus’s Ship, as there are also other more interesting variations to this paradox and also more interesting responses and solutions.

It is also important to note that for this explanation to work, we must agree on the problem at hand; the materials for the ship have to be the exact same thing, as our answer depends on the fact that it follows the concept of an equal set.