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Single Idea 9995
[filed under theme 5. Theory of Logic / K. Features of Logics / 6. Compactness
]
Full Idea
If a wff is tautologically implied by a set of wff's, it is implied by a finite subset of them; and if every finite subset is satisfiable, then so is the whole set of wff's.
Clarification
a 'wff' is a well-formed formula
Gist of Idea
Proof in finite subsets is sufficient for proof in an infinite set
Source
Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 2.5)
Book Ref
Enderton,Herbert B.: 'A Mathematical Introduction to Logic' [Academic Press 2001], p.142
A Reaction
[Enderton's account is more symbolic] He adds that this also applies to models. It is a 'theorem' because it can be proved. It is a major theorem in logic, because it brings the infinite under control, and who doesn't want that?
The
30 ideas
from 'A Mathematical Introduction to Logic (2nd)'
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9718
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Validity is either semantic (what preserves truth), or proof-theoretic (following procedures)
[Enderton]
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9719
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A proof theory is 'sound' if its valid inferences entail semantic validity
[Enderton]
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9720
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A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity
[Enderton]
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9724
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Until the 1960s the only semantics was truth-tables
[Enderton]
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9994
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A truth assignment to the components of a wff 'satisfy' it if the wff is then True
[Enderton]
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9721
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A logical truth or tautology is a logical consequence of the empty set
[Enderton]
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9723
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Sentences with 'if' are only conditionals if they can read as A-implies-B
[Enderton]
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9996
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Expressions are 'decidable' if inclusion in them (or not) can be proved
[Enderton]
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9722
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Inference not from content, but from the fact that it was said, is 'conversational implicature'
[Enderton]
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9997
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For a reasonable language, the set of valid wff's can always be enumerated
[Enderton]
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9995
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Proof in finite subsets is sufficient for proof in an infinite set
[Enderton]
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9707
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'F(x)' is the unique value which F assumes for a value of x
[Enderton]
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9705
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'fld R' indicates the 'field' of all objects in the relation
[Enderton]
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9704
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'ran R' indicates the 'range' of objects being related to
[Enderton]
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9703
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'dom R' indicates the 'domain' of objects having a relation
[Enderton]
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9710
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We write F:A→B to indicate that A maps into B (the output of F on A is in B)
[Enderton]
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9699
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The 'powerset' of a set is all the subsets of a given set
[Enderton]
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9700
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Two sets are 'disjoint' iff their intersection is empty
[Enderton]
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9712
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A relation is 'symmetric' on a set if every ordered pair has the relation in both directions
[Enderton]
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9713
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A relation is 'transitive' if it can be carried over from two ordered pairs to a third
[Enderton]
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9701
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A 'relation' is a set of ordered pairs
[Enderton]
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9702
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A 'domain' of a relation is the set of members of ordered pairs in the relation
[Enderton]
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9708
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A function 'maps A into B' if the relating things are set A, and the things related to are all in B
[Enderton]
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9709
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A function 'maps A onto B' if the relating things are set A, and the things related to are set B
[Enderton]
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9711
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A relation is 'reflexive' on a set if every member bears the relation to itself
[Enderton]
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9706
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A 'function' is a relation in which each object is related to just one other object
[Enderton]
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9714
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A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects
[Enderton]
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9717
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A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second
[Enderton]
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9716
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We 'partition' a set into distinct subsets, according to each relation on its objects
[Enderton]
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9715
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An 'equivalence relation' is a reflexive, symmetric and transitive binary relation
[Enderton]
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