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Single Idea 10097
[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
]
Full Idea
The Axiom of Extensionality says that for all sets x and y, if x and y have the same elements then x = y.
Gist of Idea
Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y
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.50
A Reaction
This seems fine in pure set theory, but hits the problem of renates and cordates in the real world. The elements coincide, but the axiom can't tell you why they coincide.
The
41 ideas
from 'Philosophies of Mathematics'
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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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9946
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Logicists say mathematics is applicable because it is totally general
[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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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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10107
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Real numbers provide answers to square root problems
[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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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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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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10094
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The theory of types seems to rule out harmless sets as well as paradoxical ones.
[George/Velleman]
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10105
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Differences between isomorphic structures seem unimportant
[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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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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10125
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The classical mathematician believes the real numbers form an actual set
[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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10119
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Consistency is a purely syntactic property, unlike the semantic property of soundness
[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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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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10130
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Set theory can prove the Peano Postulates
[George/Velleman]
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10128
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The Incompleteness proofs use arithmetic to talk about formal arithmetic
[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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10124
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Gödel's First Theorem suggests there are truths which are independent of proof
[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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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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