Results 1 
3 of
3
On Equivalents of Wellfoundedness  An experiment in Mizar
, 1998
"... Four statements equivalent to wellfoundedness (wellfounded induction, existence of recursively defined functions, uniqueness of recursively defined functions, and absence of descending omegachains) have been proved in Mizar and the proofs mechanically checked for correctness. It seems not to be w ..."
Abstract

Cited by 13 (3 self)
 Add to MetaCart
Four statements equivalent to wellfoundedness (wellfounded induction, existence of recursively defined functions, uniqueness of recursively defined functions, and absence of descending omegachains) have been proved in Mizar and the proofs mechanically checked for correctness. It seems not to be widely known that the existence (without the uniqueness assumption) of recursively defined functions implies wellfoundedness. In the proof we used regular cardinals, a fairly advanced notion of set theory. The theory of cardinals in Mizar was developed earlier by G. Bancerek. With the current state of the Mizar system, the proofs turned out to be an exercise with only minor additions at the fundamental level. We would like to stress the importance of a systematic development of a mechanized data base for mathematics in the spirit of the QED Project.
MML Identifier: TBSP_1. Totally Bounded Metric Spaces Alicia de la Cruz
"... and [14] provide the notation and terminology for this paper. For simplicity, we follow the rules: M denotes a non empty metric space, c denotes an element of M, N denotes a non empty metric structure, w denotes an element of N, G denotes a family of subsets of N, C denotes a subset of N, R denotes ..."
Abstract
 Add to MetaCart
and [14] provide the notation and terminology for this paper. For simplicity, we follow the rules: M denotes a non empty metric space, c denotes an element of M, N denotes a non empty metric structure, w denotes an element of N, G denotes a family of subsets of N, C denotes a subset of N, R denotes a Reflexive non empty metric structure, T denotes a Reflexive symmetric triangle non empty metric structure, t1 denotes an element of T, Y denotes a family of subsets of T, f denotes a function, n, m, p, k denote natural numbers, r, s, L denote real numbers, and x denotes a set. Next we state three propositions: (1) For every L such that 0 < L and L < 1 and for all n, m such that n ≤ m holds L m ≤ L n. (2) For every L such that 0 < L and L < 1 and for every k holds L k ≤ 1 and 0 < L k. (3) For every L such that 0 < L and L < 1 and for every s such that 0 < s there exists n such that L n < s. Let us consider N. We say that N is totally bounded if and only if the condition (Def. 1) is satisfied. (Def. 1) Let given r. Suppose r> 0. Then there exists G such that G is finite and the carrier of N = � G and for every C such that C ∈ G there exists w such that C = Ball(w,r). Let us consider N. We see that the sequence of N is a function and it can be characterized by the following (equivalent) condition: (Def. 2) domit = N and rngit ⊆ the carrier of N. In the sequel S1 is a sequence of M and S2 is a sequence of N. We now state the proposition (5) 1 f is a sequence of N iff dom f = N and for every n holds f (n) is an element of N. Let us consider N, S2. We say that S2 is convergent if and only if: (Def. 3) There exists an element x of N such that for every r such that r> 0 there exists n such that for every m such that n ≤ m holds ρ(S2(m),x) < r. Let us consider M, S1. Let us assume that S1 is convergent. The functor limS1 yields an element of M and is defined by: 1 The proposition (4) has been removed.
MML Identifier:TBSP_1. Totally Bounded Metric Spaces Alicia de la Cruz
"... and [14] provide the notation and terminology for this paper. For simplicity, we follow the rules: M denotes a non empty metric space, c denotes an element of M, N denotes a non empty metric structure, w denotes an element of N, G denotes a family of subsets of N, C denotes a subset of N, R denotes ..."
Abstract
 Add to MetaCart
and [14] provide the notation and terminology for this paper. For simplicity, we follow the rules: M denotes a non empty metric space, c denotes an element of M, N denotes a non empty metric structure, w denotes an element of N, G denotes a family of subsets of N, C denotes a subset of N, R denotes a Reflexive non empty metric structure, T denotes a Reflexive symmetric triangle non empty metric structure, t1 denotes an element of T, Y denotes a family of subsets of T, f denotes a function, n, m, p, k denote natural numbers, r, s, L denote real numbers, and x denotes a set. Next we state three propositions: (1) For every L such that 0 < L and L < 1 and for all n, m such that n ≤ m holds L m ≤ L n. (2) For every L such that 0 < L and L < 1 and for every k holds L k ≤ 1 and 0 < L k. (3) For every L such that 0 < L and L < 1 and for every s such that 0 < s there exists n such that L n < s. Let us consider N. We say that N is totally bounded if and only if the condition (Def. 1) is satisfied. (Def. 1) Let given r. Suppose r> 0. Then there exists G such that G is finite and the carrier of N = � G and for every C such that C ∈ G there exists w such that C = Ball(w,r). Let us consider N. We see that the sequence of N is a function and it can be characterized by the following (equivalent) condition: (Def. 2) domit = N and rngit ⊆ the carrier of N. In the sequel S1 is a sequence of M and S2 is a sequence of N. We now state the proposition (5) 1 f is a sequence of N iff dom f = N and for every n holds f(n) is an element of N. Let us consider N, S2. We say that S2 is convergent if and only if: (Def. 3) There exists an element x of N such that for every r such that r> 0 there exists n such that for every m such that n ≤ m holds ρ(S2(m),x) < r. Let us consider M, S1. Let us assume that S1 is convergent. The functor lim S1 yields an element of M and is defined by: 1 The proposition (4) has been removed.