Ideas of Keith Hossack, by Theme
[British, fl. 2007, Lecturer at Birkbeck College, London.]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
10676

The Axiom of Choice is a nonlogical principle of settheory

10686

The Axiom of Choice guarantees a oneone correspondence from sets to ordinals

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Naïve logical sets
23623

Predicativism says only predicated sets exist

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
23624

The iterative conception has to appropriate Replacement, to justify the ordinals

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
23625

Limitation of Size justifies Replacement, but then has to appropriate Power Set

4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
10687

Maybe we reduce sets to ordinals, rather than the other way round

4. Formal Logic / G. Formal Mereology / 3. Axioms of Mereology
10677

Extensional mereology needs two definitions and two axioms

5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / d. and
23628

The connective 'and' can have an ordersensitive meaning, as 'and then'

5. Theory of Logic / E. Structures of Logic / 6. Relations in Logic
23627

'Before' and 'after' are not two relations, but one relation with two orders

5. Theory of Logic / F. Referring in Logic / 2. Descriptions / b. Definite descriptions
10671

Plural definite descriptions pick out the largest class of things that fit the description

5. Theory of Logic / G. Quantification / 6. Plural Quantification
10666

Plural reference will refer to complex facts without postulating complex things

10675

A plural comprehension principle says there are some things one of which meets some condition

10669

Plural reference is just an abbreviation when properties are distributive, but not otherwise

5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / d. Russell's paradox
10673

Plural language can discuss without inconsistency things that are not members of themselves

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
10680

The theory of the transfinite needs the ordinal numbers

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
10684

I take the real numbers to be just lengths

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
23626

Transfinite ordinals are needed in proof theory, and for recursive functions and computability

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2ndorder
10674

A plural language gives a single comprehensive induction axiom for arithmetic

6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
10681

In arithmetic singularists need sets as the instantiator of numeric properties

10685

Set theory is the science of infinity

6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
23621

Numbers are properties, not sets (because numbers are magnitudes)

6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
23622

We can only mentally construct potential infinities, but maths needs actual infinities

7. Existence / D. Theories of Reality / 10. Ontological Commitment / a. Ontological commitment
10668

We are committed to a 'group' of children, if they are sitting in a circle

9. Objects / C. Structure of Objects / 5. Composition of an Object
10664

Complex particulars are either masses, or composites, or sets

10678

The relation of composition is indispensable to the partwhole relation for individuals

9. Objects / C. Structure of Objects / 8. Parts of Objects / c. Wholes from parts
10665

Leibniz's Law argues against atomism  water is wet, unlike water molecules

10682

The fusion of five rectangles can decompose into more than five parts that are rectangles

18. Thought / A. Modes of Thought / 1. Thought
10663

A thought can refer to many things, but only predicate a universal and affirm a state of affairs

27. Natural Reality / C. Space / 2. Space
10683

We could ignore space, and just talk of the shape of matter
