In this article we derived an important example of the inconsistent countable set in second order ZFC (ZFC_2) with the full second-order semantics. Main results: (i) :~Con(ZFC2_); (ii) let k be an inaccessible cardinal, V is an standard model of ZFC (ZFC_2) and H_k is a set of all sets having hereditary size less then k; then : ~Con(ZFC + E(V)(V = Hk)):.

We define and prove a formal semantics divided into two complementary interacting components: the strictly linguistic semantics, we call linguistic agent, and the strictly logical and referential semantics, we call rational agent. This Linguistic \ Rational Agents’ Semantics applies to Deep Dependency trees or more generally, to discourses, i.e. sequences of DD-trees, and interprets them by functional structures we call Meaning Representation Structures, similar to the DRT, but interpreted very differently. LRA semantics incrementally interprets the discourses by minimal finite models, (...) called proto-models, in a monotonic logic of the LA and checks the proto-models with respect to the classical models of the RA. The proto-model is considered as the linguistic sense of the discourse. We define in full detail the LA which, as we believe, must be universal. On the other hand, we don’t propose a particular RA. We only define the scheme of interaction between the two agents and the stimuli of the RA used by the LA. After all, every discourse has in LRA semantics the single meaning and the single sense for every Rational Agent used to interact with the Linguistic Agent. (shrink)

Cantor proved that no set has a bijection between itself and its power set. This is widely taken to have shown that there infinitely many sizes of infinite sets. The argument depends on the princip...