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All the ideas for 'fragments/reports', 'First-order Logic, 2nd-order, Completeness' and 'Investigations in the Foundations of Set Theory I'

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27 ideas

2. Reason / C. Styles of Reason / 1. Dialectic
20768 Like spiderswebs, dialectical arguments are clever but useless [Ariston, by Diog. Laertius]
2. Reason / D. Definition / 8. Impredicative Definition
15924 Predicative definitions are acceptable in mathematics if they distinguish objects, rather than creating them? [Zermelo, by Lavine]
4. Formal Logic / F. Set Theory ST / 1. Set Theory
17608 We take set theory as given, and retain everything valuable, while avoiding contradictions [Zermelo]
17607 Set theory investigates number, order and function, showing logical foundations for mathematics [Zermelo]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
10870 ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice [Zermelo, by Clegg]
13012 Zermelo published his axioms in 1908, to secure a controversial proof [Zermelo, by Maddy]
17609 Set theory can be reduced to a few definitions and seven independent axioms [Zermelo]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
13017 Zermelo introduced Pairing in 1930, and it seems fairly obvious [Zermelo, by Maddy]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
13015 Zermelo used Foundation to block paradox, but then decided that only Separation was needed [Zermelo, by Maddy]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / m. Axiom of Separation
13486 Not every predicate has an extension, but Separation picks the members that satisfy a predicate [Zermelo, by Hart,WD]
13020 The Axiom of Separation requires set generation up to one step back from contradiction [Zermelo, by Maddy]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
10757 Henkin semantics has a second domain of predicates and relations (in upper case) [Rossberg]
10751 Second-order logic needs the sets, and its consequence has epistemological problems [Rossberg]
10759 There are at least seven possible systems of semantics for second-order logic [Rossberg]
5. Theory of Logic / B. Logical Consequence / 2. Types of Consequence
10753 Logical consequence is intuitively semantic, and captured by model theory [Rossberg]
5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-
10752 Γ |- S says S can be deduced from Γ; Γ |= S says a good model for Γ makes S true [Rossberg]
5. Theory of Logic / E. Structures of Logic / 1. Logical Form
10754 In proof-theory, logical form is shown by the logical constants [Rossberg]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
10756 A model is a domain, and an interpretation assigning objects, predicates, relations etc. [Rossberg]
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
10758 If models of a mathematical theory are all isomorphic, it is 'categorical', with essentially one model [Rossberg]
5. Theory of Logic / K. Features of Logics / 4. Completeness
10761 Completeness can always be achieved by cunning model-design [Rossberg]
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
10755 A deductive system is only incomplete with respect to a formal semantics [Rossberg]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
13487 In ZF, the Burali-Forti Paradox proves that there is no set of all ordinals [Zermelo, by Hart,WD]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers
18178 For Zermelo the successor of n is {n} (rather than n U {n}) [Zermelo, by Maddy]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
13027 Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets [Zermelo, by Maddy]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
9627 Different versions of set theory result in different underlying structures for numbers [Zermelo, by Brown,JR]
23. Ethics / C. Virtue Theory / 2. Elements of Virtue Theory / f. The Mean
3049 The chief good is indifference to what lies midway between virtue and vice [Ariston, by Diog. Laertius]
23. Ethics / D. Deontological Ethics / 1. Deontology
3549 Ariston says rules are useless for the virtuous and the non-virtuous [Ariston, by Annas]