more on this theme
|
more from this text
Single Idea 10124
[filed under theme 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
]
Full Idea
For intuitionists, truth is not independent of proof, but this independence is precisely what seems to be suggested by Gödel's First Incompleteness Theorem.
Gist of Idea
Gödel's First Theorem suggests there are truths which are independent of proof
Source
A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.8)
Book Ref
George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.168
A Reaction
Thus Gödel was worse news for the Intuitionists than he was for Hilbert's Programme. Gödel himself responded by becoming a platonist about his unprovable truths.
The
41 ideas
from A.George / D.J.Velleman
|
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]
|
|
9946
|
Logicists say mathematics is applicable because it is totally general
[George/Velleman]
|
|
9955
|
Contextual definitions replace a complete sentence containing the expression
[George/Velleman]
|
|
10031
|
Impredicative definitions quantify over the thing being defined
[George/Velleman]
|
|
10100
|
Axiom of Pairing: for all sets x and y, there is a set z containing just x and y
[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]
|
|
10101
|
Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B
[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]
|
|
17900
|
The Axiom of Reducibility made impredicative definitions possible
[George/Velleman]
|
|
10105
|
Differences between isomorphic structures seem unimportant
[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]
|
|
17902
|
A successor is the union of a set with its singleton
[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]
|
|
17901
|
Type theory prohibits (oddly) a set containing an individual and a set of individuals
[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]
|
|
10123
|
The intuitionists are the idealists of mathematics
[George/Velleman]
|
|
10114
|
Bounded quantification is originally finitary, as conjunctions and disjunctions
[George/Velleman]
|
|
10125
|
The classical mathematician believes the real numbers form an actual set
[George/Velleman]
|
|
10130
|
Set theory can prove the Peano Postulates
[George/Velleman]
|
|
10128
|
The Incompleteness proofs use arithmetic to talk about formal arithmetic
[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]
|