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