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All the ideas for 'fragments/reports', 'A Mathematical Introduction to Logic (2nd)' and 'What Required for Foundation for Maths?'

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62 ideas

2. Reason / D. Definition / 2. Aims of Definition
Definitions make our intuitions mathematically useful [Mayberry]
2. Reason / E. Argument / 6. Conclusive Proof
Proof shows that it is true, but also why it must be true [Mayberry]
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
Until the 1960s the only semantics was truth-tables [Enderton]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / a. Symbols of ST
'F(x)' is the unique value which F assumes for a value of x [Enderton]
'fld R' indicates the 'field' of all objects in the relation [Enderton]
'ran R' indicates the 'range' of objects being related to [Enderton]
'dom R' indicates the 'domain' of objects having a relation [Enderton]
We write F:A→B to indicate that A maps into B (the output of F on A is in B) [Enderton]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
The 'powerset' of a set is all the subsets of a given set [Enderton]
Two sets are 'disjoint' iff their intersection is empty [Enderton]
A relation is 'symmetric' on a set if every ordered pair has the relation in both directions [Enderton]
A relation is 'transitive' if it can be carried over from two ordered pairs to a third [Enderton]
A 'relation' is a set of ordered pairs [Enderton]
A 'domain' of a relation is the set of members of ordered pairs in the relation [Enderton]
A function 'maps A into B' if the relating things are set A, and the things related to are all in B [Enderton]
A function 'maps A onto B' if the relating things are set A, and the things related to are set B [Enderton]
A relation is 'reflexive' on a set if every member bears the relation to itself [Enderton]
A 'function' is a relation in which each object is related to just one other object [Enderton]
A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects [Enderton]
A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second [Enderton]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / e. Equivalence classes
We 'partition' a set into distinct subsets, according to each relation on its objects [Enderton]
An 'equivalence relation' is a reflexive, symmetric and transitive binary relation [Enderton]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
There is a semi-categorical axiomatisation of set-theory [Mayberry]
Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
The misnamed Axiom of Infinity says the natural numbers are finite in size [Mayberry]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
The set hierarchy doesn't rely on the dubious notion of 'generating' them [Mayberry]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
Limitation of size is part of the very conception of a set [Mayberry]
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
Inference not from content, but from the fact that it was said, is 'conversational implicature' [Enderton]
5. Theory of Logic / A. Overview of Logic / 2. History of Logic
The mainstream of modern logic sees it as a branch of mathematics [Mayberry]
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
First-order logic only has its main theorems because it is so weak [Mayberry]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Only second-order logic can capture mathematical structure up to isomorphism [Mayberry]
5. Theory of Logic / B. Logical Consequence / 2. Types of Consequence
Validity is either semantic (what preserves truth), or proof-theoretic (following procedures) [Enderton]
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
Big logic has one fixed domain, but standard logic has a domain for each interpretation [Mayberry]
5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
A logical truth or tautology is a logical consequence of the empty set [Enderton]
5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
A truth assignment to the components of a wff 'satisfy' it if the wff is then True [Enderton]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
No Löwenheim-Skolem logic can axiomatise real analysis [Mayberry]
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
'Classificatory' axioms aim at revealing similarity in morphology of structures [Mayberry]
Axiomatiation relies on isomorphic structures being essentially the same [Mayberry]
'Eliminatory' axioms get rid of traditional ideal and abstract objects [Mayberry]
5. Theory of Logic / K. Features of Logics / 3. Soundness
A proof theory is 'sound' if its valid inferences entail semantic validity [Enderton]
5. Theory of Logic / K. Features of Logics / 4. Completeness
A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity [Enderton]
5. Theory of Logic / K. Features of Logics / 6. Compactness
Proof in finite subsets is sufficient for proof in an infinite set [Enderton]
No logic which can axiomatise arithmetic can be compact or complete [Mayberry]
5. Theory of Logic / K. Features of Logics / 7. Decidability
Expressions are 'decidable' if inclusion in them (or not) can be proved [Enderton]
5. Theory of Logic / K. Features of Logics / 8. Enumerability
For a reasonable language, the set of valid wff's can always be enumerated [Enderton]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / b. Quantity
Greek quantities were concrete, and ratio and proportion were their science [Mayberry]
Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry]
Cantor extended the finite (rather than 'taming the infinite') [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
If proof and definition are central, then mathematics needs and possesses foundations [Mayberry]
The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry]
Foundations need concepts, definition rules, premises, and proof rules [Mayberry]
Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry]
We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry]
Set theory is not just another axiomatised part of mathematics [Mayberry]
9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry]
10. Modality / B. Possibility / 8. Conditionals / f. Pragmatics of conditionals
Sentences with 'if' are only conditionals if they can read as A-implies-B [Enderton]
23. Ethics / C. Virtue Theory / 1. Virtue Theory / a. Nature of virtue
A virtue is a combination of intelligence, strength and luck [Ion]