PL 3014 - Metalogic
Important topics in Chapter 7
Church's Thesis. A partial function is computable by algorithm iff it is a recursive partial function.
Def. A set is effectively enumerable if there is an effective procedure by which its members may be li

PL 3014 - Metalogic
Prop. (Gdel's Second Theorem)
Let S be a consistent recursively axiomatizable extension of N. There is no proof in S of the consistency
of S.
Proof Outline
1. Construct a wf C of LN that asserts that N is consistent.
2. Demonstrate N C

PL 3014 - Metalogic
Important topics in Chapter 6 (Gdel's Incompleteness Theorem)
N is the first order system of arithmetic in the language LN.
N is the (intended) interpretation of LN in which DN = N.
Notation:
0(n) is an abbreviation for the closed term

PL 3014 - Metalogic
Important topics in Chapter 5 (Mathematical Systems)
Let S be a first order system.
1. Logical axioms of S are axioms of S that are logically valid (i.e., (K1)-(K6).
2. Proper axioms of S are axioms of S that are not logically valid.
D

PL 3014 - Metalogic
Proposition 4.40
1
Preliminaries
Suppose we enlarge L by adding new constants b0, b1, . to form L+ . Let S be an extension of K. Now
construct an extension S+ of S by including as axioms all axioms of S and all instances of S-axioms th

PL 3014 - Metalogic
Important Definitions in Chapter 3 (Predicate Calculus)
Def. 3.14. An interpretation I of L consists of:
1. A non-emtpy set DI (the domain of I).
2.
A (possibly empty) collection of distiguished elements ai of DI.
3.
A (possibly empty)

PL 3014 - Metalogic
Important Definitions and Propositions for Statement Calculus
Def. 2.12. A valuation of L is a function v : cfw_wfs of L cfw_T, F such that,
for any wfs A, B of L,
(i) v(A) v(A)
(ii) v(A B) = F iff v(A) = T and v(B) = F.
Def. 2.13. A w

PL 3014 - Metalogic
The formal language of statement calculus:
1. Symbol alphabet:
statement variables: p1, p2, .
punctuation:
(, ), ,
connectives:
,
2. Grammar: A well-formed formula (wf) of statement calculus:
(i) pi is a wf, for i 1.
(ii) If A and B a