JAF26/Weak Arithmetics Days
Seville, June 11–13, 2007
VERSIONS OF THE PIGEONHOLE PRINCIPLE FOR
INCREASING FUNCTIONS
CH. CORNAROS
Abstract. We study the strength of forms of the pigeonhole principle
(PHP) for increasing functions in subsystems of P A. We prove that
almost all models of I∆0 (resp. BΣ1 ) satisfy these forms of PHP for ∆0
(resp. ∆1 ) formulas.
Definition 1. B(Σn , a) is I∆0 plus the collection schema for Σn formulae,
up to a, i.e.,
∀x<a∃yϕ(x, y) → ∃z∀x<a∃y < zϕ(x, y),
for all ϕ ∈ Σn .
Definition 2. P HP (Σn , a) is the (strong or full) pigeon hole principle axiom schema for Σn formulae ϕ up to a, i.e.,
ϕ does not define a bijective map : a → a − 1.
∗
P HP (Σn , a) will denote P HP (Σn , a), but with ϕ strictly increasing and
P HP ∗ (Σn ) denotes ∀aP HP ∗ (Σn , a).
Definition 3. We define P HP (∆n , a) to be the axiom schema for ∆n formulae, i.e., the schema
∀x<a∀y<a(ϕ(x, y, v̄) ↔ ψ(x, y, v̄)) →
ϕ does not define a bijective map from a into a − 1,
where ϕ(x, y, v̄) ∈ Σn and ψ(x, y, v̄) ∈ Πn .
Similarly, we define P HP ∗ (∆n , a).
Theorem 1. i) For n ≥ 1,
IΣn ` B(Σn+1 , a) ↔ P HP (Σn+1 , a) ↔ P HP (∆n+1 , a).
ii) I∆0 + exp ` P HP (Σ1 ) ↔ P HP ∗ (Σ1 ) ↔ P HP ∗ (∆1 ) ↔ I∆1 ↔ BΣ1 .
iii )Let K |= I∆0 not of the form aN , for any a ∈ K. Then K |= P HP ∗ (∆0 ).
iv) Let K |= BΣ1 not of the form aN , for any a ∈ K. Then K |= P HP ∗ (∆1 ).
Problem. Does P − ` P HP ∗ (∆1 ) → BΣ1 ?
A Partial Answer.
Proposition 2. P − ` P HP ∗ (∆1 , a) → I(∆1 , a),
where I(∆1 , a) is the induction schema for ∆1 formulae up to a, i.e.,
∀x<a + 1(ϕ(x, p̄) ↔ ψ(x, p̄)) →
[ϕ(0, p̄) ∧ ∀x<a(ϕ(x, p̄) → ϕ(x + 1, p̄)) → ϕ(a, p̄)],
for all ϕ ∈ Σ1 and ψ ∈ Π1 .
1
2
CH. CORNAROS
(Note that ∀aI(∆n , a) is equivalent to the usual I∆n ). So the relationships
among the above schemas are
∗
P − ` P HP (∆1 ) → BΣ1 → P HP ∗ (∆1 ) → I∆1 ,
where ∗ means “in most cases”. If we add exp to I∆1 , then the converse
implications hold. If we do have not exp, it seems that we go from I∆1 back
to BΣ1 in most cases (see [6] or [8]). So we can ask another question:
Problem. Does P − ` BΣ1 → P HP (∆1 )?
This problem seems to be related to the problem I∆0 ` P HP (∆0 )? (see [1])
References
[1] C. Dimitracopoulos and J. Paris: The pigeonhole principle and fragments of arithmetic, Z. Math. Logik Grundlag. Math. 32 (1986), 73–80.
[2] P. Hájek and P. Pudlák: Metamathematics of first-order arithmetic, Springer-Verlag,
Berlin, 1993.
[3] J. Paris and C. Dimitracopoulos: Truth definitions for ∆0 formulae, Logic and algorithmic, Zurich 1980, Univ. Genève, Geneva, 1982, 317–329.
[4] J. Paris and L. Kirby: Σn -collection schemas in arithmetic, Springer Lecture Notes
in Mathematics, 834(1980), 312-337.
[5] J. B. Paris, A. J. Wilkie and A. R. Woods: Provability of the pigeonhole principle
and the existence of infinitely many primes, J. Symbolic Logic 53 (1988), 1235–1244.
[6] A. Sirokofskich and C. Dimitracopoulos: On a problem of J. Paris, to appear in the
J. Logic Comput.
[7] Th. Slaman: Σn -bounding and ∆n -induction, Proc. Amer. Math. Soc. 132 (2004),
2449–2456.
[8] N. Thapen: A note on ∆1 induction and Σ1 collection, Fund. Math. 186 (2005), no.
1, 79–84.
[9] A. R. Woods: Some problems in logic and number theory and their connections, Ph.D.
thesis, University of Manchester, 1981.
Department of Mathematics, University of Aegean, GR-832 00 Karlovassi,
Greece
E-mail address: [email protected]