# Can We Create a Sentence With Infinite States?

misc

There’s a classic story of a person who approaches two doors, each flanked by a guardian. The person is told that they may ask one question to ascertain which door is the correct one to continue on their way. One guardian speaks only the truth, while the other only lies. The solution? To ask the following:

If I asked the other guard which door is the correct one, what would they say?

It’s a brilliant little sentence; a true statement and a lie gives a lie. This is akin to multiplication in math; a positive times a negative gives a negative. If the person asked the guardian who always tells the truth, the guardian would answer honestly that the other would lie, and would thus recommend the wrong door. If the person asked the guardian who always lies, then the guardian would lie, and say that the other would recommend the wrong door. Either way, the result is the same; the guardian will tell the person which door they shouldn’t go through, leaving them to go through the other.

Another fun bit of word play is the Pinocchio paradox, in which Pinocchio says something along the lines of:

My nose grows now

If his nose is not currently growing, then he is lying, so his nose will grow. But wait! Since his nose is now growing, he was right and isn’t lying! It’s a fun little example of a circular sentence where it causes its opposite, then itself in an endless cycle. There’s a Disney-less version of the same paradox:

This sentence is a lie

If the sentence is true, it must be a lie. But that then makes it true. Again, an endless cycle flip-flopping between a state of truth and lie.

All this to get to my pondering; is there a sentence which could be constructed that would spawn in infinite number of states? While the Pinocchio paradox is technically never-ending, it is circular, and so each return to a previously visited state “crushes” the last one. Can a sentence be constructed which expands infinitely outwards, each new paradox or contradiction introducing a new, novel state, or at least one parallel to an existing one? I would imagine it taking a form similiar to a recursive function with no root. The exercise, alas, seems beyond my imagination or capabilities. True and false are binary states; are there alternative states a sentence can possess?