Goodstein’s Theorem
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Goodstein’s Theorem claims that every Goodstein Sequence converges to 0, despite each such sequence seemingly growing very large, very quickly – suggesting that perhaps they behave quite opposite to the claim of Goodstein’s Theorem. The theorem is interesting in its own right, and can be proven using arithmetic of transfinite ordinals. What is more interesting though is that it must be proven using arithmetic of transfinite ordinals – as Goodstein’s Theorem is (and can be proven to be) undecidable in Peano Arithmetic if Peano Arithmetic is consistent. This result thus extends to a purely number theoretic testament to Gödel’s Incompleteness Theorems, proving the incompleteness of Peano Arithmetic.
This paper was written during the final year of my Integrated Master of Mathematics degree at the University of East Anglia (UEA), under the supervision of Dr David Asperó. A 15 minute presentation was given to supplement this research report, and can be found here.