proposition 3.67 Countability of the plane of naturals

open in the book · parts/02-mathematical-methods/01-logic-sets.tex:1694 · p. 36

Rests on

depends_on definition 3.65 Countable set ¶
- 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 ¶
depends_on definition 3.47 Bijective map ¶ ↺
depends_on definition 3.38 Ordered pair and Cartesian product ¶
proves proof ch:01-logic-sets@proof-13 ¶

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 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 proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶ ↺
- depends_on remark 7.94 Three numbers, two kinds ¶ ↺
depends_on lemma A.6 Lindenbaum ¶
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`	→	Countable set	declared	parts/02-mathematical-methods/01-logic-sets.tex:1697
`depends_on`	→	Bijective map	declared	parts/02-mathematical-methods/01-logic-sets.tex:1697
`depends_on`	→	Ordered pair and Cartesian product	declared	parts/02-mathematical-methods/01-logic-sets.tex:1697
`depends_on`	←	The rationals are countable	declared	parts/02-mathematical-methods/01-logic-sets.tex:1720
`depends_on`	←	A countable union of null sets is null	declared	parts/02-mathematical-methods/05-real-analysis.tex:4733
`depends_on`	←	Lindenbaum	declared	appendices/A-long-proofs.tex:1560
`depends_on`	←	Three numbers, two kinds	declared	parts/02-mathematical-methods/05-real-analysis.tex:2849
`proves`	←	ch:01-logic-sets@proof-13	declared	parts/02-mathematical-methods/01-logic-sets.tex:1700