[Verse 1] Picture all the rationals scattered on a line Draw a blade and slice them cleanly, left and right divide Every number on the left stays smaller than the right But the left side has no ceiling, no maximum in sight [Chorus] Cut the rationals in two, L and U apart Left side climbs but never peaks, that's the beating heart Gap becomes the number now, boundary made real Dedekind cuts reveal the truth that empty spaces feel [Verse 2] Take all rationals below or equal to the three Right side gets the three and up, partition perfectly But when we hunt for square root two, no rational will do Left gets numbers squared less than, right gets greater through [Chorus] Cut the rationals in two, L and U apart Left side climbs but never peaks, that's the beating heart Gap becomes the number now, boundary made real Dedekind cuts reveal the truth that empty spaces feel [Bridge] No minimum on the right No maximum on the left The chasm holds square root two A treasure from the cleft [Verse 3] Don't search for the missing piece, don't try to fill the hole The absence is the answer here, the void becomes the goal Each real number is a cut, a partition cleanly made The gap itself's the object now, foundation firmly laid [Chorus] Cut the rationals in two, L and U apart Left side climbs but never peaks, that's the beating heart Gap becomes the number now, boundary made real Dedekind cuts reveal the truth that empty spaces feel [Outro] Mathematics builds on voids Where nothing becomes all The cut defines what isn't there And makes the irrational
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