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Single Idea 13519
[filed under theme 5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
]
Full Idea
Model theory uses set theory to show that the theorem-proving power of the usual methods of deduction in mathematics corresponds perfectly to what must be true in actual mathematical structures.
Gist of Idea
Model theory uses sets to show that mathematical deduction fits mathematical truth
Source
Robert S. Wolf (A Tour through Mathematical Logic [2005], Pref)
Book Ref
Wolf,Robert S.: 'A Tour Through Mathematical Logic' [Carus Maths Monographs 2005], p.-8
A Reaction
That more or less says that model theory demonstrates the 'soundness' of mathematics (though normal arithmetic is famously not 'complete'). Of course, he says they 'correspond' to the truths, rather than entailing them.
The
19 ideas
from Robert S. Wolf
13518
|
Modern mathematics has unified all of its objects within set theory
[Wolf,RS]
|
13519
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Model theory uses sets to show that mathematical deduction fits mathematical truth
[Wolf,RS]
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13520
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A 'tautology' must include connectives
[Wolf,RS]
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13522
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Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x)
[Wolf,RS]
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13521
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Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance
[Wolf,RS]
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13523
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Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P
[Wolf,RS]
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13524
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Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof
[Wolf,RS]
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13525
|
Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens
[Wolf,RS]
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13526
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Comprehension Axiom: if a collection is clearly specified, it is a set
[Wolf,RS]
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13529
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Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists
[Wolf,RS]
|
13530
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An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive
[Wolf,RS]
|
13531
|
Model theory reveals the structures of mathematics
[Wolf,RS]
|
13532
|
Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants'
[Wolf,RS]
|
13533
|
First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem
[Wolf,RS]
|
13534
|
In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide
[Wolf,RS]
|
13535
|
First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation
[Wolf,RS]
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13537
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An 'isomorphism' is a bijection that preserves all structural components
[Wolf,RS]
|
13538
|
If a theory is complete, only a more powerful language can strengthen it
[Wolf,RS]
|
13539
|
The LST Theorem is a serious limitation of first-order logic
[Wolf,RS]
|