green numbers give full details.
|
back to list of philosophers
|
expand these ideas
Ideas of Robert S. Wolf, by Text
[American, fl. 2005, Teaches mathematics at California Polytechnic State University.]
|
2005
|
A Tour through Mathematical Logic
|
|
Pref
|
p.-8
|
13519
|
Model theory uses sets to show that mathematical deduction fits mathematical truth
|
|
Pref
|
p.-8
|
13518
|
Modern mathematics has unified all of its objects within set theory
|
|
1.2
|
p.11
|
13520
|
A 'tautology' must include connectives
|
|
1.3
|
p.20
|
13521
|
Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance
|
|
1.3
|
p.20
|
13522
|
Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x)
|
|
1.3
|
p.20
|
13523
|
Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P
|
|
1.3
|
p.31
|
13524
|
Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof
|
|
1.7
|
p.54
|
13525
|
Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens
|
|
2.2
|
p.62
|
13526
|
Comprehension Axiom: if a collection is clearly specified, it is a set
|
|
2.3
|
p.70
|
13529
|
Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists
|
|
2.4
|
p.77
|
13530
|
An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive
|
|
5.1
|
p.165
|
13531
|
Model theory reveals the structures of mathematics
|
|
5.2
|
p.167
|
13532
|
Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants'
|
|
5.3
|
p.172
|
13533
|
First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem
|
|
5.3
|
p.172
|
13534
|
In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide
|
|
5.3
|
p.174
|
13535
|
First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation
|
|
5.4
|
p.181
|
13537
|
An 'isomorphism' is a bijection that preserves all structural components
|
|
5.5
|
p.191
|
13538
|
If a theory is complete, only a more powerful language can strengthen it
|
|
5.7
|
p.224
|
13539
|
The LST Theorem is a serious limitation of first-order logic
|