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Single Idea 10110
[filed under theme 18. Thought / D. Concepts / 1. Concepts / a. Nature of concepts
]
Full Idea
Corresponding to every concept there is a class (some classes will be sets, the others proper classes).
Clarification
Sets are more precisely defined than classes
Gist of Idea
Corresponding to every concept there is a class (some of them sets)
Source
A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.4)
Book Ref
George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.90
The
41 ideas
from A.George / D.J.Velleman
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10131
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If mathematics is not about particulars, observing particulars must be irrelevant
[George/Velleman]
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10089
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Talk of 'abstract entities' is more a label for the problem than a solution to it
[George/Velleman]
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9955
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Contextual definitions replace a complete sentence containing the expression
[George/Velleman]
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10031
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Impredicative definitions quantify over the thing being defined
[George/Velleman]
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9946
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Logicists say mathematics is applicable because it is totally general
[George/Velleman]
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10106
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Rational numbers give answers to division problems with integers
[George/Velleman]
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10102
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The integers are answers to subtraction problems involving natural numbers
[George/Velleman]
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10107
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Real numbers provide answers to square root problems
[George/Velleman]
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10105
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Differences between isomorphic structures seem unimportant
[George/Velleman]
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10098
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The 'power set' of A is all the subsets of A
[George/Velleman]
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10101
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Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B
[George/Velleman]
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10099
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The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}}
[George/Velleman]
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10103
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Grouping by property is common in mathematics, usually using equivalence
[George/Velleman]
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10104
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'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words
[George/Velleman]
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10096
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Even the elements of sets in ZFC are sets, resting on the pure empty set
[George/Velleman]
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10097
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Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y
[George/Velleman]
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10100
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Axiom of Pairing: for all sets x and y, there is a set z containing just x and y
[George/Velleman]
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17900
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The Axiom of Reducibility made impredicative definitions possible
[George/Velleman]
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17902
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A successor is the union of a set with its singleton
[George/Velleman]
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10092
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In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc.
[George/Velleman]
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10094
|
The theory of types seems to rule out harmless sets as well as paradoxical ones.
[George/Velleman]
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10095
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Type theory has only finitely many items at each level, which is a problem for mathematics
[George/Velleman]
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17901
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Type theory prohibits (oddly) a set containing an individual and a set of individuals
[George/Velleman]
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10110
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Corresponding to every concept there is a class (some of them sets)
[George/Velleman]
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10109
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ZFC can prove that there is no set corresponding to the concept 'set'
[George/Velleman]
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10108
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As a reduction of arithmetic, set theory is not fully general, and so not logical
[George/Velleman]
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10111
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Asserting Excluded Middle is a hallmark of realism about the natural world
[George/Velleman]
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10120
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Soundness is a semantic property, unlike the purely syntactic property of consistency
[George/Velleman]
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10125
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The classical mathematician believes the real numbers form an actual set
[George/Velleman]
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10114
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Bounded quantification is originally finitary, as conjunctions and disjunctions
[George/Velleman]
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10123
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The intuitionists are the idealists of mathematics
[George/Velleman]
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10119
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Consistency is a purely syntactic property, unlike the semantic property of soundness
[George/Velleman]
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10128
|
The Incompleteness proofs use arithmetic to talk about formal arithmetic
[George/Velleman]
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10130
|
Set theory can prove the Peano Postulates
[George/Velleman]
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10126
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A 'consistent' theory cannot contain both a sentence and its negation
[George/Velleman]
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10127
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A 'complete' theory contains either any sentence or its negation
[George/Velleman]
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10129
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A 'model' is a meaning-assignment which makes all the axioms true
[George/Velleman]
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17899
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Second-order induction is stronger as it covers all concepts, not just first-order definable ones
[George/Velleman]
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10134
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Much infinite mathematics can still be justified finitely
[George/Velleman]
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10124
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Gödel's First Theorem suggests there are truths which are independent of proof
[George/Velleman]
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10133
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Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle
[George/Velleman]
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