Goodstein Sequences
In 1982 Kirby and Paris claimed that it is impossible to prove in Peano arithmetic that Goodstein sequences always terminate. Since then some people believe that such termination can only be proved using transfinite numbers. But there is an easy proof without transfinite numbers, see:
Rationale: Every logical argument must be defined in some language, and every language has limitations. Attempting to construct a logical argument while ignoring how the limitations of language might affect that argument is a bizarre approach. The correct acknowledgment of the interactions of logic and language explains almost all of the paradoxes, and resolves almost all of the contradictions, conundrums, and contentious issues in modern philosophy and mathematics.
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