more from this thinker
|
more from this text
Single Idea 13496
[filed under theme 5. Theory of Logic / K. Features of Logics / 6. Compactness
]
Full Idea
First-order logic is 'compact', which means that any logical consequence of a set (finite or infinite) of first-order sentences is a logical consequence of a finite subset of those sentences.
Gist of Idea
First-order logic is 'compact': consequences of a set are consequences of a finite subset
Source
William D. Hart (The Evolution of Logic [2010], 3)
Book Ref
Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.80
The
17 ideas
with the same theme
[satisfaction by satisfying the finite subsets]:
9995
|
Proof in finite subsets is sufficient for proof in an infinite set
[Enderton]
|
10771
|
Compactness is important for major theories which have infinitely many axioms
[Tharp]
|
10772
|
Compactness blocks infinite expansion, and admits non-standard models
[Tharp]
|
13544
|
Inconsistency or entailment just from functors and quantifiers is finitely based, if compact
[Bostock]
|
13618
|
Compactness means an infinity of sequents on the left will add nothing new
[Bostock]
|
13841
|
Why should compactness be definitive of logic?
[Boolos, by Hacking]
|
10287
|
If a first-order theory entails a sentence, there is a finite subset of the theory which entails it
[Hodges,W]
|
13496
|
First-order logic is 'compact': consequences of a set are consequences of a finite subset
[Hart,WD]
|
17789
|
No logic which can axiomatise arithmetic can be compact or complete
[Mayberry]
|
13630
|
Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures
[Shapiro]
|
13646
|
Compactness is derived from soundness and completeness
[Shapiro]
|
13699
|
Compactness surprisingly says that no contradictions can emerge when the set goes infinite
[Sider]
|
10975
|
Compactness does not deny that an inference can have infinitely many premisses
[Read]
|
10977
|
Compactness blocks the proof of 'for every n, A(n)' (as the proof would be infinite)
[Read]
|
10976
|
Compactness makes consequence manageable, but restricts expressive power
[Read]
|
10974
|
Compactness is when any consequence of infinite propositions is the consequence of a finite subset
[Read]
|
17867
|
If a concept is not compact, it will not be presentable to finite minds
[Almog]
|