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From Untyped to Typed Universes

• Untyped universes.
  • Only one type for all objects:
  • Bit strings in computer memory,
  • S-expressions in pure LISP,
  • lambda-expressions in the lambda-calculus,
  • Sets in set theory.
• Organization of objects:
  • Classification of usage/behavior,
  • Characters, integers,
  • Lists, pairs,
  • Functions, programs.
• Type distinction still shallow:
  • Easy to violate type distinctions:
  • Boolean or of character and machine operation?