A Course on Mathematical Logic

On the Consistency of the Arithmetic System

“…In consequence, one can obtain another arithmetic system (cf. [1], [2], [4], [6], [7], [10], [13], [14], [16], [17], [18], [21], [23], [25], [28], [30], [31], [32], [34], [36], [37], [40], [47], [48], [50], [51], [52], [53], [54], [58], [59], [62], [71], [72], [77], [78]), namely,…”

“…In consequence, one can obtain another arithmetic system (cf. [1], [2], [4], [6], [7], [10], [13], [14], [16], [17], [18], [21], [23], [25], [28], [30], [31], [32], [34], [36], [37], [40], [47], [48], [50], [51], [52], [53], [54], [58], [59], [62], [71], [72], [77], [78]), namely, 〈𝑅𝑅 𝑜𝑜+ 𝑃𝑃 , 𝐿𝐿 2 𝑟𝑟′ ∪ 𝐴𝐴 𝑟𝑟 ′ 〉 , where 𝐿𝐿 2 𝑟𝑟′ is the set of all logical axioms. 𝐿𝐿 1 2 denotes the well-known subset of the set 𝐿𝐿 2…”

“…The problem of consistency of the Arithmetic System, is the well-known problem (see [1], [3], [4], [5], [7], [8], [9], [12], [14], [15], [16], [17], [19], [20], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [37], [38], [39], [41], [42], [43], [46], [47], [48], [49], [50], [51], [52], [55], [56], [57], [58], [59], [60], [61], [63], [64],…”

Topologies and all that -- A Tutorial

“…The crucial part is the construction of a model for a consistent set of sentences. This is usually done through Henkin's approach, see, e.g., [Sho67,4.2], [CK90, Chapter 2] or [Sri08,5.1]. Rasiowa and Sikorski [RS50] followed a completely different path in their topological proof by making use of Baire's Category Theorem and using the observation that in a compact topological space the intersection of a sequence of open and dense sets is dense again.…”