An Example of Induction: Fibonacci Numbers
... Theorem 2. The Fibonacci number F5k is a multiple of 5, for all integers k ≥ 1. Proof. Proof by induction on k. Since this is a proof by induction, we start with the base case of k = 1. That means, in this case, we need to compute F5×1 = F5 . But, it is easy to compute that F5 = 5, which is a multi ...
... Theorem 2. The Fibonacci number F5k is a multiple of 5, for all integers k ≥ 1. Proof. Proof by induction on k. Since this is a proof by induction, we start with the base case of k = 1. That means, in this case, we need to compute F5×1 = F5 . But, it is easy to compute that F5 = 5, which is a multi ...
mplications of Cantorian Transfinite Set Theory
... David Hilbert described Cantor's work as:“...the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.” "I see it but I don't believe it.” Georg Cantor on his own theory. “…the infinite is nowhere to be found in reality” David Hilbert. ...
... David Hilbert described Cantor's work as:“...the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.” "I see it but I don't believe it.” Georg Cantor on his own theory. “…the infinite is nowhere to be found in reality” David Hilbert. ...
Square roots
... There are many ways to see this fact, already accessible to us after the first lecture! (In case you are wondering why this might be interesting, it is rumoured that Hippassus of Metapontum was killed over his discovery of this fact, so this was a very astonishing and abrasive discovery at one time. ...
... There are many ways to see this fact, already accessible to us after the first lecture! (In case you are wondering why this might be interesting, it is rumoured that Hippassus of Metapontum was killed over his discovery of this fact, so this was a very astonishing and abrasive discovery at one time. ...
Mathematical Ideas that Shaped the World
... For every whole number n, there is a next whole number n+1. ...
... For every whole number n, there is a next whole number n+1. ...
Godel incompleteness
... a complete system, and this system is called Euclidian geometry... Another example of complete system is the Presburger arithmetic, which is a first order theory of the natural numbers with addition... Even though it is not as powerful as the Peano axioms because it does not have multiplication, it ...
... a complete system, and this system is called Euclidian geometry... Another example of complete system is the Presburger arithmetic, which is a first order theory of the natural numbers with addition... Even though it is not as powerful as the Peano axioms because it does not have multiplication, it ...
Notes
... One of the modern mathematicians to have a major impact on our understanding of the continuum and the real numbers is the topologist L.E.J. Brouwer famous for his fixed point theorem and for an approach to mathematics called intuitionism. He was not only a brilliant mathematician, he was good at “pr ...
... One of the modern mathematicians to have a major impact on our understanding of the continuum and the real numbers is the topologist L.E.J. Brouwer famous for his fixed point theorem and for an approach to mathematics called intuitionism. He was not only a brilliant mathematician, he was good at “pr ...
MATH 350: HOMEWORK #3 1. G.C.D.`s 1. Write the g.c.d. of 666 and
... 1. G.C.D.’s 1. Write the g.c.d. of 666 and 1414 as an integral combination of 666 and 1414. 2. The improved division algorithm states that given integers a and b 6= 0, there are always integers q and r such that a = qb + r and |r| ≤ |b|/2. In class we discussed improving the Euclidean algorithm by u ...
... 1. G.C.D.’s 1. Write the g.c.d. of 666 and 1414 as an integral combination of 666 and 1414. 2. The improved division algorithm states that given integers a and b 6= 0, there are always integers q and r such that a = qb + r and |r| ≤ |b|/2. In class we discussed improving the Euclidean algorithm by u ...
Complexity
... 1. Zero is a number. 2. If a is a number, the successor of a is a number. 3. zero is not the successor of a number. 4. Two numbers of which the successors are equal are themselves equal. 5. (induction axiom.) If a set S of numbers contains zero and also the successor of every number in S, then every ...
... 1. Zero is a number. 2. If a is a number, the successor of a is a number. 3. zero is not the successor of a number. 4. Two numbers of which the successors are equal are themselves equal. 5. (induction axiom.) If a set S of numbers contains zero and also the successor of every number in S, then every ...
Kurt Gödel and His Theorems
... Hilbert program • Solution to the foundational crisis of mathematics • Ground all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent ...
... Hilbert program • Solution to the foundational crisis of mathematics • Ground all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent ...
hilbert systems - CSA
... S U {~X} is also Consistent If not, S U {~X} |- X S |- (~X > X) { Deduction Theorem } (~X > X) > X { Theorem } S |- X { Modus Ponens } S U {~X} is satisfiable { Model Existence Theorem } S U {X} is not valid. ...
... S U {~X} is also Consistent If not, S U {~X} |- X S |- (~X > X) { Deduction Theorem } (~X > X) > X { Theorem } S |- X { Modus Ponens } S U {~X} is satisfiable { Model Existence Theorem } S U {X} is not valid. ...