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Single Idea 13518
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
]
Full Idea
One of the great achievements of modern mathematics has been the unification of its many types of objects. It began with showing geometric objects numerically or algebraically, and culminated with set theory representing all the normal objects.
Gist of Idea
Modern mathematics has unified all of its objects within set theory
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
His use of the word 'object' begs all sorts of questions, if you are arriving from the street, where an object is something which can cause a bruise - but get used to it, because the word 'object' has been borrowed for new uses.
The
19 ideas
from 'A Tour through Mathematical Logic'
13519
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Model theory uses sets to show that mathematical deduction fits mathematical truth
[Wolf,RS]
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13518
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Modern mathematics has unified all of its objects within set theory
[Wolf,RS]
|
13520
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A 'tautology' must include connectives
[Wolf,RS]
|
13524
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Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof
[Wolf,RS]
|
13522
|
Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x)
[Wolf,RS]
|
13521
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Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance
[Wolf,RS]
|
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]
|
13525
|
Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens
[Wolf,RS]
|
13526
|
Comprehension Axiom: if a collection is clearly specified, it is a set
[Wolf,RS]
|
13529
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Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists
[Wolf,RS]
|
13530
|
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]
|
13537
|
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]
|