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Mathematical Logic

Mathematical Logic. “The point of philosophy is to start with something so simple as not to seem worth stating and to end with something so paradoxical that no one will believe it.” Bertrand Russell. Colleen Duffy Advisor: Dr. Jeff McLean University of St. Thomas St. Paul, MN.

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Mathematical Logic

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  1. Mathematical Logic “The point of philosophy is to start with something so simple as not to seem worth stating and to end with something so paradoxical that no one will believe it.” Bertrand Russell Colleen Duffy Advisor: Dr. Jeff McLean University of St. Thomas St. Paul, MN

  2. Do Numbers Exist? • Realism in ontology: mathematical objects exist objectively, independent of the mathematician • Idealism: mathematical objects exist, but they depend on the human mind • Nominalism: mathematical objects are linguistic constructions, or mathematical objects do not exist

  3. A few philosophical views • Rationalism • Empiricism • Naturalism • Logicism • Formalism • Intuitionism • Structuralism

  4. What is a number?Frege’s definitions • Number – the Number which belongs to the concept F is the extension of the concept ‘equal to the concept F’ • Zero – the Number which belongs to the concept ‘not identical to itself’ • One – the Number which belongs to the concept ‘identical with zero’ • After every number there follows in the series of natural numbers a number • Infinity – the Number which belongs to the concept ‘finite number’ is an infinite number

  5. Questions with Infinity

  6. Zermelo-Fraenkel set theory • Most widely used • Formal system expressed in 1st order predicate logic • Formed of axioms (infinite in number) that form the basis of the theory, what is true and what can be done.

  7. Cantor Set • Georg Cantor (1845-1918) • founder of set theory • infinite sets • different levels of infinity. • Cantor set – start with the line segment [0,1] and remove the middle third. Then remove the middle third of the remaining segments. Continue forever.

  8. Continuum hypothesis (CH) • Cantor’s hypothesis • Generalized Continuum Hypothesis (GCH) – 2alepha = alepha+1 • Independent of ZF set theory.

  9. Axiom of Choice • Axiom of Choice (AC) – Let X be a collection of non-empty sets. Then we can choose a member from each set in that collection.

  10. More Axiom of Choice • Equivalents: Well-ordering principle, trichotomy of cardinals, Zorn’s Lemma • Consequences in algebra: every ring has a maximal ideal, every field has algebraic closure • Independent of ZF • If GCH is true, then AC is true.

  11. Banach-Tarski paradox • It is possible to partition a 3-D object into finitely many pieces then rearrange them with rigid motions to form two copies of the original object. =

  12. Conclusion “In mathematics you don’t understand things; you just get used to them.” Johann von Neumann

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