Proof

• Introduction to Theoretical Computer Science: Between Finite and Infinite - From Mathematical Theory to AI and Autonomous Driving
  • It is generally not possible to prove statements about Infinity with a finite number of verifications.
  • Instead, we use logic to prove them, relying on the computational power of human thinking.
  • Infinity cannot be proven.
    • When we trace the source of the infinite elements in a theorem, we end up with either the universal quantifier ($\forall$) or axioms that define the existence of an infinite number of elements.
      • Even techniques like ε-δ proof are essentially logical formulas that use the universal quantifier ($\forall$).
    • Infinity is an artificial construct introduced by humans as a starting point for discussions.
    • On the other hand, it is also possible to have discussions that exclude infinity.
    • If you want to know more about this, I recommend researching it yourself.
  • Mathematical Induction provides a clear understanding of both finite and infinite concepts.
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  • Systems like Testing involve testing a finite number of elements from an infinite set, but they do not provide mathematical guarantees.
  • Mathematical Guarantees
    • Hiring smart people to perform these guarantees every time is difficult and expensive.
    • Therefore, we want to use automated methods on computers.
      • One such method is verifying the inclusion relationship of infinite sets of inputs that produce a specific output in automata.
    • There are also research efforts to apply these concepts to the physical world.
      • This approach involves generalizing concepts like Automaton using category theory and applying them to other fields.
      • For example, proving the correctness of Autonomous Driving. #math#mathematical foundations