In this paper, we consider, for a set $$\mathcal {A}$$A of natural numbers, the following notions of finitenessFIN1:There are a natural number l and a bijection f between $$\{ x\in \mathbb {N}:x<l\}$${x∈N:x<l} and $$\mathcal {A}$$A;FIN2:There is an upper bound for $$\mathcal {A}$$A;FIN3:There is l such that $$\forall \mathcal {B}\subseteq \mathcal {A}(|\mathcal {B}|<l)$$∀B⊆A(|B|<l);FIN4:It is not the case that $$\forall y(\exists x>y)(x\in \mathcal {A})$$∀y(∃x>y)(x∈A);FIN5:It is not the case that $$\forall l\exists \mathcal {B}\subseteq \mathcal {A}(|\mathcal {B}|=l)$$∀l∃B⊆A(|B|=l), and infinitenessINF1:There are not a natural number l and a bijection f between $$\{ x\in \mathbb {N}:x<l\}$${x∈N:x<l} and $$\mathcal {A}$$A;INF2:There is no upper bound for $$\mathcal {A}$$A;INF3:There is no l such that $$\forall \mathcal {B}\subseteq \mathcal {A}(|\mathcal {B}|<l)$$∀B⊆A(|B|<l);INF4:$$\forall y(\exists x>y)(x\in \mathcal {A})$$∀y(∃x>y)(x∈A);INF5:$$\forall l\exists \mathcal {B}\subseteq \mathcal {A}(|\mathcal {B}|=l)$$∀l∃B⊆A(|B|=l). In this paper, we systematically compare them in the method of constructive reverse mathematics. We show that the equivalence among them can be characterized by various combinations of induction axioms and non-constructive principles, including the axiom of bounded comprehension.
[1]
Yohji Akama,et al.
An arithmetical hierarchy of the law of excluded middle and related principles
,
2004,
LICS 2004.
[2]
Kai Frederick Wehmeier.
Fragments of
$HA$ based on
$\Sigma_1$-induction
,
1997,
Arch. Math. Log..
[3]
F. Richman,et al.
Varieties of Constructive Mathematics: CONSTRUCTIVE ALGEBRA
,
1987
.
[4]
Danna Zhou,et al.
d.
,
1934,
Microbial pathogenesis.
[5]
Samuel R. Buss,et al.
Intuitionistic Validity in T-Normal Kripke Structures
,
1993,
Ann. Pure Appl. Log..
[6]
Dirk van Dalen,et al.
Logic and structure
,
1980
.
[7]
R. Soare.
Recursively enumerable sets and degrees
,
1987
.
[8]
A. Troelstra.
Constructivism in mathematics
,
1988
.
[9]
Petr Hájek,et al.
Metamathematics of First-Order Arithmetic
,
1993,
Perspectives in mathematical logic.
[10]
E. Bishop.
Foundations of Constructive Analysis
,
2012
.
[11]
Hajime Ishihara,et al.
Constructive reverse mathematics: compactness properties
,
2005,
From sets and types to topology and analysis.
[12]
Wim Veldman,et al.
Two simple sets that are not positively Borel
,
2005,
Ann. Pure Appl. Log..
[13]
Kai F. Wehmeier.
Fragments of HA based on 1-induction
,
1997
.