Combining Texts

Ideas for 'Unconscious Cerebral Initiative', 'Introduction to Mathematical Logic' and 'Logical Pluralism'

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2 ideas

6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry

17762	In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki]
	Full Idea: Since non-Euclidean geometry preserves all Euclid's postulates except the fifth one, all the theorems derived without the use of the fifth postulate remain valid.
	From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1)

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction

17754	Inductive proof depends on the choice of the ordering [Walicki]
	Full Idea: Inductive proof is not guaranteed to work in all cases and, particularly, it depends heavily on the choice of the ordering.
	From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.1.1)
	A reaction: There has to be an well-founded ordering for inductive proofs to be possible.