Alphabeta MathCountability and Uncountability
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✓ 15 results · all verified · 11 also independently AI-judged
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1 · Prerequisites
Real Analysis
2 · Logical flowchart
3 · Items
Definition
Equinumerous sets, A ≈ B and A ⪯ B
Lemma
On ℕ the order is membership: m < n ⟺ m ∈ n
Lemma
The pigeonhole principle on ℕ
Definition
Finite, countably infinite, countable, uncountable
Theorem
The Schröder-Bernstein theorem
Lemma
Every subset of an at most countable set is at most countable
Lemma
A nonempty set is at most countable iff it is a surjective image of ℕ
Lemma
The nonempty finite subsets of ℝ are exactly the listable ones
Theorem
ℕ × ℕ ≈ ℕ
Theorem
A product of two at most countable sets is at most countable
Definition
The Axiom of Countable Choice (AC_ω)
Definition
The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
Definition
Choice for pairs and countable finite choice
Theorem
AC implies DC implies countable choice
Theorem
Countable unions of at most countable sets, assuming AC_ω
Theorem
ℚ is countably infinite
Theorem
Cantor's theorem: A ≺ P(A)
Theorem
ℝ is uncountable (Cantor's nested intervals, 1874)
Corollary
Every nondegenerate interval of ℝ is uncountable
Corollary
The irrationals are uncountable
Remark
The continuum hypothesis and the later local consistency results
False statement
FALSE: every uncountable subset of ℝ contains an interval
Sources
- J. K. Hunter, An Introduction to Real Analysis
- J. Lebl, Basic Analysis: Introduction to Real Analysis, basic set theory
- Equinumerosity (Wikipedia)
- Countable set (Wikipedia)
- J. Zapletal, Set Theory Notes
- Set-theoretic definition of natural numbers (Wikipedia)
- Ordinal number (Wikipedia)
- T. Tao, Analysis I, 3rd ed., §2.2
- Pigeonhole principle (Wikipedia)
- Finite set (Wikipedia)
- Dedekind-infinite set (Wikipedia)
- T. Tao, Analysis I, 3rd ed., §3.6 (Cardinality of sets)
- T. Tao, Analysis I, 3rd ed., §3.6 and §8.1
- Schröder-Bernstein theorem (Wikipedia)
- T. Tao, Analysis I, 3rd ed., §8.1
- Maximum and minimum (Wikipedia)
- Pairing function (Wikipedia)
- D. H. Fremlin, Measure Theory, Chapter 56
- Axiom of countable choice (Wikipedia)
- Axiom of dependent choice (Wikipedia)
- Axiom of choice (Wikipedia)
- H. Herrlich, Axiom of Choice, Lecture Notes in Mathematics 1876, Springer 2006
- Jech, The Axiom of Choice, §8.2, Theorem 8.3(ii), p.123
- Morillon, Synthèse, §2.2.1, printed p.6
- Jech, The Axiom of Choice, §2.4, pp.22–23
- Rational number (Wikipedia)
- Cantor's theorem (Wikipedia)
- Cantor's diagonal argument (Wikipedia)
- J. Lebl, Basic Analysis I
- Cantor's first set theory article (Wikipedia)
- Nested intervals (Wikipedia)
- W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 2
- Irrational number (Wikipedia)
- Stanford Encyclopedia of Philosophy, The Continuum Hypothesis
- Sierpiński's theorem: GCH implies AC
- Continuum hypothesis (Wikipedia)
- Cardinality of the continuum (Wikipedia)
- Interval (mathematics) (Wikipedia)