[Verse 1]
There's a question that haunts computation's core
Does machine M accept what we're looking for?
Input w goes in, but will it ever halt?
This riddle reveals a fundamental fault
The Halting Problem stands before our eyes
Recognizable truth wrapped in undecidable lies
[Chorus]
Diagonalize, diagonalize
Build a machine that breaks the paradigm
What accepts itself? We'll flip the script
Contradiction caught in logic's grip
Some questions have no algorithmic key
Welcome to undecidability
[Verse 2]
Cantor showed us infinite sets collide
Now we'll use his trick where proofs reside
Assume we solve the acceptance test
Build a clever trap that breaks the rest
Machine D reads its own description clean
Then does the opposite of what it's seen
[Chorus]
Diagonalize, diagonalize
Build a machine that breaks the paradigm
What accepts itself? We'll flip the script
Contradiction caught in logic's grip
Some questions have no algorithmic key
Welcome to undecidability
[Bridge]
Reduce A-T-M to every case
HALT and EMPTY fall from grace
Post Correspondence seeks its match
Rice's theorem springs the trap
Any property that's non-trivial
Becomes computationally mythical
[Verse 3]
From the Halting Problem's seeds we grow
A garden where no algorithms flow
E-Q-T-M asks if languages align
A-L-L-T-M seeks the infinite divine
Each reduction proves another door stays locked
In the mansion where decidability's blocked
[Outro]
Recognition's possible but decision's banned
Some truths slip through our computing hands
The diagonal cuts through every scheme
Undecidability reigns supreme