Mathematical Logic

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?Freges 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

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Questions with Infinity
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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.

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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.

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Continuum hypothesis (CH)

• Cantors hypothesis
• Generalized Continuum Hypothesis (GCH) 2alepha
  alepha1
• Independent of ZF set theory.

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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.

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More Axiom of Choice

• Equivalents Well-ordering principle, trichotomy
  of cardinals, Zorns 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.

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Conclusion
In mathematics you dont understand things you
just get used to them. Johann von Neumann