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Single Idea 10092
[filed under theme 6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
]
Full Idea
In the unramified theory of types, all objects are classified into a hierarchy of types. The lowest level has individual objects that are not sets. Next come sets whose elements are individuals, then sets of sets, etc. Variables are confined to types.
Clarification
'Ramified' means having complex extensions
Gist of Idea
In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc.
Source
A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3)
Book Ref
George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.46
A Reaction
The objects are Type 0, the basic sets Type 1, etc.
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
|
'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
|
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
|
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
|
Asserting Excluded Middle is a hallmark of realism about the natural world
[George/Velleman]
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10120
|
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
|
Bounded quantification is originally finitary, as conjunctions and disjunctions
[George/Velleman]
|
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10123
|
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]
|
|
10130
|
Set theory can prove the Peano Postulates
[George/Velleman]
|
|
10126
|
A 'consistent' theory cannot contain both a sentence and its negation
[George/Velleman]
|
|
10127
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A 'complete' theory contains either any sentence or its negation
[George/Velleman]
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|
10129
|
A 'model' is a meaning-assignment which makes all the axioms true
[George/Velleman]
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|
17899
|
Second-order induction is stronger as it covers all concepts, not just first-order definable ones
[George/Velleman]
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10134
|
Much infinite mathematics can still be justified finitely
[George/Velleman]
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10124
|
Gödel's First Theorem suggests there are truths which are independent of proof
[George/Velleman]
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10133
|
Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle
[George/Velleman]
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