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Single Idea 10073
[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
]
Full Idea
By Gödel's First Incompleteness Theorem, there cannot be a negation-complete set theory.
Gist of Idea
There cannot be a set theory which is complete
Source
Peter Smith (Intro to Gödel's Theorems [2007], 01.3)
Book Ref
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.5
A Reaction
This means that we can never prove all the truths of a system of set theory.
The
33 ideas
with the same theme
[general points about the basics of set theory]:
17831
|
Cantor gives informal versions of ZF axioms as ways of getting from one set to another
[Cantor, by Lake]
|
17879
|
Axiomatising set theory makes it all relative
[Skolem]
|
17832
|
Zermelo showed that the ZF axioms in 1930 were non-categorical
[Zermelo, by Hallett,M]
|
10870
|
ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice
[Zermelo, by Clegg]
|
13012
|
Zermelo published his axioms in 1908, to secure a controversial proof
[Zermelo, by Maddy]
|
17609
|
Set theory can be reduced to a few definitions and seven independent axioms
[Zermelo]
|
9565
|
Zermelo made 'set' and 'member' undefined axioms
[Zermelo, by Chihara]
|
3339
|
For Zermelo's set theory the empty set is zero and the successor of each number is its unit set
[Zermelo, by Blackburn]
|
8679
|
We perceive the objects of set theory, just as we perceive with our senses
[Gödel]
|
17835
|
Gödel show that the incompleteness of set theory was a necessity
[Gödel, by Hallett,M]
|
3340
|
Von Neumann defines each number as the set of all smaller numbers
[Neumann, by Blackburn]
|
18189
|
ZFC could contain a contradiction, and it can never prove its own consistency
[MacLane]
|
9879
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NF has no models, but just blocks the comprehension axiom, to avoid contradictions
[Quine, by Dummett]
|
9193
|
ZF set theory has variables which range over sets, 'equals' and 'member', and extensionality
[Dummett]
|
9194
|
The main alternative to ZF is one which includes looser classes as well as sets
[Dummett]
|
10492
|
A few axioms of set theory 'force themselves on us', but most of them don't
[Boolos]
|
18115
|
We could add axioms to make sets either as small or as large as possible
[Bostock]
|
10191
|
Set theory reduces to a mereological theory with singletons as the only atoms
[Lewis, by MacBride]
|
15507
|
Set theory has some unofficial axioms, generalisations about how to understand it
[Lewis]
|
10073
|
There cannot be a set theory which is complete
[Smith,P]
|
3335
|
The standard Z-F Intuition version of set theory has about ten agreed axioms
[Benardete,JA, by PG]
|
17795
|
Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation
[Mayberry]
|
17796
|
There is a semi-categorical axiomatisation of set-theory
[Mayberry]
|
17833
|
The first-order ZF axiomatisation is highly non-categorical
[Hallett,M]
|
17834
|
Non-categoricity reveals a sort of incompleteness, with sets existing that the axioms don't reveal
[Hallett,M]
|
13011
|
New axioms are being sought, to determine the size of the continuum
[Maddy]
|
18195
|
A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy
[Maddy]
|
10886
|
Determinacy: an object is either in a set, or it isn't
[Zalabardo]
|
10166
|
ZFC set theory has only 'pure' sets, without 'urelements'
[Reck/Price]
|
10096
|
Even the elements of sets in ZFC are sets, resting on the pure empty set
[George/Velleman]
|
10653
|
Maybe set theory need not be well-founded
[Varzi]
|
8682
|
Major set theories differ in their axioms, and also over the additional axioms of choice and infinity
[Friend]
|
18843
|
The iterated conception of set requires continual increase in axiom strength
[Rumfitt]
|