definition 3.65 Countable set

open in the book · parts/02-mathematical-methods/01-logic-sets.tex:1660 · p. 36

ground object -- no derivation owed

Rests on

depends_on axiom 3.33 Peano axioms ¶
depends_on definition 3.62 Equipotence ¶
- depends_on definition 3.47 Bijective map ¶
- depends_on definition 3.40 Cardinal ¶

Supports

depends_on corollary 3.68 The rationals are countable ¶
- depends_on corollary A.3 The continuum is the power set of the naturals ¶
- depends_on corollary 3.70 cor:inf-irrationals-uncountable ¶
- depends_on remark 7.94 Three numbers, two kinds ¶
depends_on definition 7.138 Set of measure zero ¶
- depends_on definition 7.140 Hausdorff measure ¶
  - depends_on definition 7.142 Box-counting dimension ¶
    - depends_on definition 32.79 Box-counting dimension, restated from Part II ¶
      - depends_on definition 32.81 Correlation dimension ¶
      - depends_on example 32.80 The middle-thirds Cantor set ¶
    - depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶
    - depends_on remark 7.144 What is developed here, and what is not ¶
  - depends_on proposition 7.141 The critical exponent, and the Hausdorff dimension ¶
    - depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶ ↺
  - depends_on remark 7.144 What is developed here, and what is not ¶ ↺
- depends_on lemma 7.139 A countable union of null sets is null ¶
  - depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶ ↺
  - depends_on proposition 7.141 The critical exponent, and the Hausdorff dimension ¶ ↺
  - depends_on remark 7.94 Three numbers, two kinds ¶ ↺
- depends_on proposition 7.141 The critical exponent, and the Hausdorff dimension ¶ ↺
- depends_on remark 7.144 What is developed here, and what is not ¶ ↺
depends_on proposition 3.67 Countability of the plane of naturals ¶
- depends_on corollary 3.68 The rationals are countable ¶ ↺
- depends_on lemma 7.139 A countable union of null sets is null ¶ ↺
- depends_on lemma A.6 Lindenbaum ¶
- depends_on remark 7.94 Three numbers, two kinds ¶ ↺
depends_on theorem 3.69 Uncountability of the continuum ¶
- depends_on corollary 3.70 cor:inf-irrationals-uncountable ¶ ↺
- depends_on remark 7.94 Three numbers, two kinds ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Peano axioms	declared	parts/02-mathematical-methods/01-logic-sets.tex:1665
`depends_on`	→	Equipotence	declared	parts/02-mathematical-methods/01-logic-sets.tex:1665
`depends_on`	←	The rationals are countable	declared	parts/02-mathematical-methods/01-logic-sets.tex:1720
`depends_on`	←	Set of measure zero	declared	parts/02-mathematical-methods/05-real-analysis.tex:4725
`depends_on`	←	Countability of the plane of naturals	declared	parts/02-mathematical-methods/01-logic-sets.tex:1697
`depends_on`	←	Uncountability of the continuum	declared	parts/02-mathematical-methods/01-logic-sets.tex:1745