## Is mathematics consistent? (2003)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Kitada03ismathematics,

author = {Hitoshi Kitada},

title = {Is mathematics consistent?},

year = {2003}

}

### OpenURL

### Abstract

Abstract: A question is proposed whether or not set theory is consistent. We consider a formal set theory S, where we can develop a number theory. As no generality is lost, in the following we consider a number theory that can be regarded as a subsystem of S, and will call it S (0). Definition 1. 1) We assume that a Gödel numbering of the system S (0) is given, and denote a formula with the Gödel number n by An. 2) A (0) (a, b) is a predicate meaning that “a is the Gödel number of a formula A with just one free variable (which we denote by A(a)), and b is the Gödel number of a proof of the formula A(a) in S (0), ” and B (0) (a, c) is a predicate meaning that “a is the Gödel number of a formula A(a), and c is the Gödel number of a proof of the formula ¬A(a) in S (0). ” Here a denotes the formal natural number corresponding to an intuitive natural number a of the meta level. Definition 2. Let P(x1, · · ·.xn) be an intuitive-theoretic predicate. We say that P(x1, · · ·,xn) is numeralwise expressible in the formal system S (0), if there is a formula P(x1, · · ·,xn) with no free variables other than the distinct variables x1, · · ·,xn such that, for each particular n-tuple of natural numbers x1, · · ·,xn, the following holds: i) if P(x1, · · ·,xn) is true, then ⊢ P(x1, · · ·,xn). and ii) if P(x1, · · ·,xn) is false, then ⊢ ¬P(x1, · · ·,xn).

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(Show Context)
Citation Context ...nly possibility remaining is to conclude that set theory is inconsistent. Case ii) There is a nonrecursive ordinal, thus there is a least nonrecursive ordinal ω1 usually called Church-Kleene ordinal (=-=[1]-=-, [6]). In this case the above extension of S (α) is possible if and only if α < ω1. We note that ω1 is a limit ordinal. For if it is a successor of an ordinal δ, then δ < ω1 is recursive, hence so is... |

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(Show Context)
Citation Context ...uch that the predicates A (0) (a, b) and B (0) (a, c) defined above are primitive recursive and hence numeralwise expressible in S (0) with the associated formulas A (0) (a, b) and B (0) (a, c). (See =-=[4]-=-.) Definition 3. Let q (0) be the Gödel number of a formula: Namely In particular ∀b[¬A (0) (a, b) ∨ ∃c(c ≤ b &B (0) (a, c))]. A q (0)(a) = ∀b[¬A (0) (a, b) ∨ ∃c(c ≤ b &B (0) (a, c))] A q (0)(q (0) ) ... |

19 |
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(Show Context)
Citation Context ...ossibility remaining is to conclude that set theory is inconsistent. Case ii) There is a nonrecursive ordinal, thus there is a least nonrecursive ordinal ω1 usually called Church-Kleene ordinal ([1], =-=[6]-=-). In this case the above extension of S (α) is possible if and only if α < ω1. We note that ω1 is a limit ordinal. For if it is a successor of an ordinal δ, then δ < ω1 is recursive, hence so is ω1 =... |

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(Show Context)
Citation Context ... structure that it contains the domain x and by using that domain only it defines x itself, it is not so unreasonable to think that there is no nonrecursive ordinal. There is, however, a possibility (=-=[3]-=-) that the condition whether or not a nonrecursive ordinal exists in ZFC is independent of the axioms of ZFC. In that case we have two alternatives. Case i) There is no nonrecursive ordinal, and hence... |