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
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10676
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The Axiom of Choice is a non-logical principle of set-theory
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10686
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The Axiom of Choice guarantees a one-one correspondence from sets to ordinals
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Naïve logical sets
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23623
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Predicativism says only predicated sets exist
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
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23624
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The iterative conception has to appropriate Replacement, to justify the ordinals
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
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23625
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Limitation of Size justifies Replacement, but then has to appropriate Power Set
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4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
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10687
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Maybe we reduce sets to ordinals, rather than the other way round
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4. Formal Logic / G. Formal Mereology / 3. Axioms of Mereology
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10677
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Extensional mereology needs two definitions and two axioms
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5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / d. and
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23628
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The connective 'and' can have an order-sensitive meaning, as 'and then'
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5. Theory of Logic / E. Structures of Logic / 6. Relations in Logic
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23627
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'Before' and 'after' are not two relations, but one relation with two orders
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5. Theory of Logic / F. Referring in Logic / 2. Descriptions / b. Definite descriptions
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10671
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Plural definite descriptions pick out the largest class of things that fit the description
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5. Theory of Logic / G. Quantification / 6. Plural Quantification
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10666
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Plural reference will refer to complex facts without postulating complex things
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10669
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Plural reference is just an abbreviation when properties are distributive, but not otherwise
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10675
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A plural comprehension principle says there are some things one of which meets some condition
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5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / d. Russell's paradox
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10673
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Plural language can discuss without inconsistency things that are not members of themselves
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
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10680
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The theory of the transfinite needs the ordinal numbers
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
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10684
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I take the real numbers to be just lengths
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
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23626
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Transfinite ordinals are needed in proof theory, and for recursive functions and computability
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
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10674
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A plural language gives a single comprehensive induction axiom for arithmetic
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
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10681
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In arithmetic singularists need sets as the instantiator of numeric properties
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10685
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Set theory is the science of infinity
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
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23621
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Numbers are properties, not sets (because numbers are magnitudes)
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6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
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23622
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We can only mentally construct potential infinities, but maths needs actual infinities
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7. Existence / D. Theories of Reality / 11. Ontological Commitment / a. Ontological commitment
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10668
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We are committed to a 'group' of children, if they are sitting in a circle
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9. Objects / C. Structure of Objects / 5. Composition of an Object
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10664
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Complex particulars are either masses, or composites, or sets
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10678
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The relation of composition is indispensable to the part-whole relation for individuals
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9. Objects / C. Structure of Objects / 8. Parts of Objects / c. Wholes from parts
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10682
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The fusion of five rectangles can decompose into more than five parts that are rectangles
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10665
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Leibniz's Law argues against atomism - water is wet, unlike water molecules
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18. Thought / A. Modes of Thought / 1. Thought
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10663
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A thought can refer to many things, but only predicate a universal and affirm a state of affairs
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27. Natural Reality / C. Space / 2. Space
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10683
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We could ignore space, and just talk of the shape of matter
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