Set of all decision problems solvable by a Turing machine, i.e. that decide if a string belongs to a recursive language.

Is a decision problem of determining if something belongs to a non-recursive language.

Or in other words: there is no Turing machine that always halts for every input with the yes/no output.

Every undecidable problem must obviously have an infinite number of "possibilities of stuff you can try": if there is only a finite number, then you can brute-force it.

Some undecidable problems are of recursively enumerable language, e.g. the halting problem.

Lists of undecidable problems.

Coolest ones besides the obvious boring halting problem:

A functional shift, also known as a shift in grammatical function or part of speech shift, refers to the process in linguistics where a word changes its function (or part of speech) without any additional morphological change. This means that a word originally belonging to one grammatical category (like noun, verb, adjective, etc.) is used as a word from a different category.

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The carry operator, often denoted as "C" or similar symbols in various contexts, typically relates to arithmetic operations, particularly in binary addition. The carry operator is used to manage the overflow that occurs when the sum of two digits exceeds the base of the numeral system.

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In set theory, "S" is often used as a symbol to represent a set, although it doesn't have a specific meaning on its own. The context in which "S" is used typically defines what set it refers to. For example, "S" might represent the set of all natural numbers, the set of all real numbers, or any other collection of objects defined by certain properties or criteria.

One of the most simple to state undecidable problems.

The reason that it is undecidable is that you can repeat each matrix any number of times, so there isn't a finite number of possibilities to check.