Kurt Gdel's incompleteness theorem, published in 1931, is considered one of the greatest intellectual achievements of modern times.
In a reasonable mathematical system, there will always be true statements that can't be proved. The mathematical community was shocked by the result, where the prevailing view was that the power and reach of their subject was unshakeable. It had been assumed that all mathematical statements are either provable or refutable. The true statement constructed by Gdel was not provable. He said that math has its limits.
The study of the foundations of mathematics was transformed by the incompleteness theorem, which shows that all formalised systems have limitations on what they can achieve.
Today's puzzle is based on the theorem, which we will get to shortly.
In a formal mathematical setting, the statement 'This sentence is unprovable' is both true and formally unprovable.
The technical details of the proof are difficult to understand. Raymond Smullyan, an American logician, created a brilliant way to convey the incompleteness theorem using logic puzzles about truth-tellers and liars. Smullyan's approach is what inspired today's teaser.
The two tribes are If.
The logical island of If is in the Ocean of Deduction. The people who were born here are members of the Pseudians and the Alethians. The only way to tell a Pseudian is to talk to them. No matter what they are saying, the Aleutians always speak the truth. The master of the island keeps a book with the names of everyone born on the island together with their tribe. The information in the ledger of identity is correct and is available to anyone who asks. She asks clever questions when she encounters people and she identifies them as Alethians and Pseudians.
She met a man called Kurt after several successful encounters. The explorer doesn't know his tribal affiliation, but before she has time to ask him a question, he says "You will never have concrete evidence that confirms that I am an Alethian."
1. Is Kurt a Pseudian or an Alethian?
2. How might this relate to Gdel?
Please come back at 5pm UK for a discussion and solutions.
There are noSPOILERS. You should post your favourite logic jokes.
UNESCO celebrates World Logic Day on January 14 and the Gdelian puzzle is a pre-celebration. The day Gdel died was also the day Alfred Tarski was born.
Click here to find out more about World Logic Day 2022.
I put a puzzle here on Mondays. I look for great puzzles. Email me if you would like to suggest one.
The Language Lovers Puzzle Book is the most recent book I have written. I give school talks about math and puzzles. Please contact your school if you are interested.
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