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algebra angle answer arbitrarily small Archimedes assume axiomatic system BODY AXIOMS calculate called cardinality clock arithmetic colors complex number continuum hypothesis contradiction countable set counting numbers cube David Hilbert decimal defined definition denial difference elements equal equation Euclidean Euler example EXERCISE false Fermat's Last Theorem fifth postulate finite number formula fractions Gauss given Godel Hilbert HINT imaginary numbers imaginary unit indirect argument indirect proof infinite sets infinity integer integral solutions interval irrational numbers length look mathematicians mathematics means measure zero multiple negative Non-Euclidean geometry number line odd number paradox parallel particular premiss positive integers positive number prime factors prime numbers primitive terms prove quadratic Ramanujan rational numbers real number rectangle repeating block rule SAMPLE PROBLEM Show sides solve square statement Step straight line subsets Suppose Theorem triangle true twin primes whole number write