It does not say anything about whether, for a particular theory T satisfying the conditions of the theorem, the statement "T is consistent" can be proved in the sense of being shown to be true by a conclusive argument, or by a proof generally acceptable for mathematicians.
Theorem 17 The recurrence equation (5) has a unique solution f T in C c. Proof The completeness of ( C c, p C c ) and Lemma 15 provide the conditions in the statement of Theorem 9.
In other words, T satisfies the conditions in the statement of Theorem 15 and hence has a fixed point.
Since Theorem 3.2 is independent of axioms and conditions of Section 2, the statement of Theorem 3.3 holds whenever Theorem 2.12 holds.
In [40 42] the reader can find many systems that satisfy the conditions considered in the statement of Corollary 3.2.
Let ({y_{n}}subset Tx) be a sequence such that each (y_{n}) satisfies the conditions as in the statement of Lemma 2.4.
Positivists have often tended to conflate the truth conditions of a statement with its confirming evidence, a procedure which has led to certain absurdities committed by phenomenalists and operationalists, such as the pronouncement that the meanings of statements about past events consist in their (forthcoming future) evidence.
In this paper, we notice that the techniques used in [37, 38] are valid if the contraction condition in the statement of the theorem can be expressed in two variables.
However, Karapinar and Agarwal [55] noticed that the techniques used in [54, 63] are valid if the contraction condition in the statement of the theorem can be expressed in two variables and they proved some theorems on the existence and uniqueness of a common fixed point for which the techniques of the papers [54, 63] are not applicable.
