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Class 19

Class 19

Reviewing Infinities

Mohammad Mahmoody

October 31, 2017
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  1. Reminder • Problem set 8 due this Friday • 2nd

    exam next Thursday. (We will post last years’ exam this week.) 1
  2. Infinite Sets 3 The cardinality of the set ℕ =

    ∈ ℕ ∧ < } is . If there is a bijection between two sets, they have the same cardinality. A set is infinite, if there is no bijection between and any ℕ .
  3. Equal Cardinalities 4 If there exists a bijection between sets

    and , they have the same cardinality: || = ||
  4. Comparing Cardinalities 5 If there exists a surjective function from

    sets to , then we say ≤ || Equivalent: total injective function from to .
  5. 6 1st : Definition. A set is Dedekind-infinite if and

    only if it has the same cardinality as some strict subset of itself. 2nd : Standard Definition. A set is infinite, if there is no bijection between and any ℕ . 3rd Definition. A set is 3rd def-infinite, if ℕ ≤ ||. Namely, there is a surjective function from to ℕ 4th Definition. A set is 4th def-infinite, if there is a total injective function from ℕ to
  6. Countable 8 Definition. A set is countable if and only

    if ≤ |ℕ| Namely, there is a surjective function from ℕ to . Equivalently:
  7. Countably Infinite 11 A set is countable if and only

    if there exists a surjective function from ℕ to . Namely ≤ |ℕ| A set is infinite, if and only if there is a surjective function from to ℕ. Namely ℕ ≤ || A set is countably infinite iff it is countable and it is infinite. Equivalent: a set is countably infinite iff there exists a bijection between and ℕ
  8. Charge Infinities are Spooky, Strange, and Surprising, We will mostly

    deal with countable infinities PS8 due this Friday Exam on Thurs next week 15