# Set Theory and Logic

## 1608 Submissions

[3] **viXra:1608.0395 [pdf]**
*replaced on 2016-11-03 03:06:10*

### The Topology on a Complete Semilattice

**Authors:** Max Null, Sergey Belov

**Comments:** 23 Pages.

We define the topology atop(χ) on a complete upper semilattice χ = (M, ≤).
The limit points are determined by the formula
lim (X) = sup{a ∈ M | {x ∈ X| a ≤ x} ∈ D},
D
where X ⊆ M is an arbitrary set, D is an arbitrary non-principal ultrafilter
on X. We investigate lim (X) and topology atop(χ) properties. In particular,
D
we prove the compactness of the topology atop(χ).

**Category:** Set Theory and Logic

[2] **viXra:1608.0358 [pdf]**
*replaced on 2017-10-27 07:50:32*

### Natural Non-Godel Definitions of Incompleteness

**Authors:** Vatolin Dm.

**Comments:** 7 Pages. Russian

Here are definitions of «completeness» and «incompleteness» for mathematical theories. These definitions are different from those that gave Godel. Сontradictions of the Godel's arguments have been eliminated. Found are theo-rems that put everything in its place.

**Category:** Set Theory and Logic

[1] **viXra:1608.0057 [pdf]**
*replaced on 2016-09-07 19:32:34*

### Curry's Non-Paradox and Its False Definition

**Authors:** Adrian Chira

**Comments:** 7 Pages.

Curry's paradox is generally considered to be one of the hardest paradoxes to solve. It is shown here that the paradox can be arrived in fewer steps and also for a different term of the original biconditional. Further, using different approaches, it is also shown that the conclusion of the paradox must always be false and this is not paradoxical but it is expected to be so. One of the approaches points out that the starting biconditional of the paradox amounts to a false definition or assertion which consequently leads to a false conclusion. Therefore, the solution is trivial and the paradox turns out to be no paradox at all. Despite that fact that verifying the truth value of the first biconditional of the paradox is trivial, mathematicians and logicians have failed to do so and merely assumed that it is true. Taking this into consideration that it is false, the paradox is however dismissed. This conclusion puts to rest an important paradox that preoccupies logicians and points out the importance of verifying one's assumptions.

**Category:** Set Theory and Logic