[Verse 1] When two sets share the very same size There's a bridge between them we can devise A bijection maps each element through One-to-one correspondence, perfectly true Count the naturals, one two three four But integers seem like so much more [Chorus] Countable or not, that's the question we face Can we match it with naturals in infinite space Bijection's the key, the golden thread Some infinities larger than we ever dreamed Countable sets dance with naturals in line Uncountable sets break that design [Verse 2] Integers zigzag, zero one minus-one two Cantor proved rationals countable too Through his clever grid, fractions align Diagonally swept in organized time Even pairs of naturals bow to the count Dovetailing through each infinite amount [Chorus] Countable or not, that's the question we face Can we match it with naturals in infinite space Bijection's the key, the golden thread Some infinities larger than we ever dreamed Countable sets dance with naturals in line Uncountable sets break that design [Bridge] But reals resist the counting game Cantor's diagonal brings them shame Binary sequences stretch beyond reach Power sets tower past what we can teach No bijection spans that mighty gulf Uncountable infinities, proud and full [Verse 3] Algebraic numbers still play along Countable unions keep the song But characteristic functions reveal The power set's uncountable deal Zero-one sequences infinite and wild Mathematics' most rebellious child [Chorus] Countable or not, that's the question we face Can we match it with naturals in infinite space Bijection's the key, the golden thread Some infinities larger than we ever dreamed Countable sets dance with naturals in line Uncountable sets break that design [Outro] Cardinality shows us infinity's face Some countable, some beyond natural's embrace
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