UNIVERSITY OF OSLO
Faculty of Mathematics and Natural Sciences
Examination in:
MAT-INF3600 — Mathematical logic.
Day of examination: Wednesday, December 3, 2014.
Examination hours:
9:00 – 13:00.
This problem set consists of 3 pages.
Appendices:
None.
Permitted aids:
None.
Please make sure that your copy of the problem set is
complete before you attempt to answer anything.
Part I
Let P and Q be unary relation symbols, and let a be a constant symbol. Let L be the
language {P, Q, a}. Furthermore, let
• Σ1 = { ∀x[P x → Qx] , P a }
• Σ2 = { ∀x[P x] → ∀x[Qx] , P a }
• Σ3 = { ¬(∀x[P x] → ∀x[Qx]) , ¬Qa }
• Σ4 = { ¬(∀x[P x] → ∀x[Qx]) , ¬P a }.
Problem 1
Answer each of the following questions by a YES or a NO. There is no need to justify your
answer.
(a) Does Qa follow tautologically from Σ1 ?
(b) Does Qa follow logically from Σ1 ?
(c) Does Qa follow tautologically from Σ2 ?
(d) Does Qa follow logically from Σ2 ?
(e) Does ∀x[P x] follow tautologically from Σ3 ?
(f) Does ∀x[P x] follow logically from Σ3 ?
(g) Does P a follow tautologically from Σ4 ?
(h) Does P a follow logically from Σ4 ?
Let Σ5 = { ∀x[P x → Qx] , ¬Qa }.
(Continued on page 2.)
Examination in MAT-INF3600, Wednesday, December 3, 2014.
Page 2
Problem 2
Give a Σ5 -deduction of ¬P a. Give a full deduction. Name all the logical axioms and
inference rules involved in the deduction.
Let Σ6 = { ∀x[P x → Qx] , Qa }.
Problem 3
Give an L-structure A such that A |= Σ6 and A 6|= P a.
Problem 4
Let Σ be a set of L-formulas and let φ an L-formula. Explain why it follows from the
Completeness Theorem that we either have
there exists a Σ-deduction of φ
(1)
there exists an L-structure A such that A |= Σ and A 6|= φ
(2)
or
Explain why it follows from the Soundness Theorem that we do not have both (1) and (2).
Part II
Let < be a binary relation symbol, let f be a binary function symbol, and let 0 and 1 be
constant symbols. Let L be the language {<, f, 0, 1}. Let T − be the L-theory where we
have the following non-logical axioms:
(T1) ∀x[¬x < x]
(T2) ∀xyz[x < y ∧ y < z → x < z]
(T3) ∀xy[x < y ∨ x = y ∨ y < x]
(T4) ∀xy[x < y → (x < f (x, y) ∧ f (x, y) < y)].
Let T be T − extended with the axiom
(T5) 0 < 1.
Problem 5
Prove that
T − ` ∀xy[x < y → ∃z[x < z ∧ z < y]]
by giving a T − -deduction. Name logical axioms and inference rules involved in the the
deduction.
Let t0 denote the L-term 0, and let tn+1 denote the L-term f (tn , 1).
(Continued on page 3.)
Examination in MAT-INF3600, Wednesday, December 3, 2014.
Page 3
Problem 6
Prove that T ` tk < 1 (for any natural number k).
Problem 7
Prove that T − 6` tk < 1 (for any natural number k).
Problem 8
Explain why any model for T is infinite.
Problem 9
Let A and B be two finite models for T − . Prove that A and B are isomorphic.
Part III
Let N be the standard LN T -structure, and let A be a set of LN T -axioms. We say that A
is ω-inconsistent if there is a formula φ(x) such that A ` ∃x[φ(x)], but for each natural
number n, A ` ¬φ(n). We say that A is ω-consistent if A is not ω-inconsistent.
Problem 10
Explain why N |= A implies that A is ω-consistent.
Problem 11
Explain why A is ω-consistent implies that A is consistent.
END