Comments by "kazedcat" (@kazedcat) on "TED-Ed" channel.

  1. Yes you can add the statement as an axiom but Godel built a constructor that creates new statement that needs to be added as a new axiom. Ultimately you cannot have a complete and finite set of axioms that can prove all true statements.
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  2. Translation is shorthand for very advance meta mathematics. It is like the geometric circle and the algebraic equation x^2+y^2=r^2 is the same thing.
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  3. Godel did prove it. This is the whole idea of the incompleteness theorem. He found a way to prove that there are true statements that cannot be proven.
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  4. Reciprocal only apply to multiplicative inverse of numbers. Mathematicians are able to apply multiplication on other things that are not numbers. Then you have matrix mathematics where multiplication is different between left side multiply and right side multiply and the multiplicative inverse is also different between different sides of operation. Because both addition and multiplication have been extended to other operation that exhibit similar operational property. The more general terms multiplicative inverse and additive inverse has been defined. Inverse property is more general and can be applied to addition, multiplication and other operation that is similar but not exactly the same as addition or multiplication.
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