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Full-Text Articles in Logic and Foundations
Gödel's Incompleteness Theorems, Derick Swarey
Gödel's Incompleteness Theorems, Derick Swarey
Senior Honors Theses
The Incompleteness Theorems of Kurt Godel are very famous both within and outside of mathematics. They focus on independence and consistency within mathematics and hence a more thorough understanding of these is beneficial to their study. The proofs of the theorems involve many ideas which may be unfamiliar to many, including those of formal systems, Godel numbering, and recursive functions and relations. The arguments themselves mirror the Liar’s Paradox in that Godel constructs a statement asserting its own unprovability and then shows that such a statement and its negation must both be independent of the system, otherwise the system is …
The Life Of Evariste Galois And His Theory Of Field Extension, Felicia N. Adams
The Life Of Evariste Galois And His Theory Of Field Extension, Felicia N. Adams
Senior Honors Theses
Evariste Galois made many important mathematical discoveries in his short lifetime, yet perhaps the most important are his studies in the realm of field extensions. Through his discoveries in field extensions, Galois determined the solvability of polynomials. Namely, given a polynomial P with coefficients is in the field F and such that the equation P(x) = 0 has no solution, one can extend F into a field L with α in L, such that P(α) = 0. Whereas Galois Theory has numerous practical applications, this thesis will conclude with the examination and proof of the fact that it is impossible …