# Set Theory and Logic

## 1212 Submissions

[2] **viXra:1212.0127 [pdf]**
*submitted on 2012-12-20 14:29:35*

### Ultra-Universal Models

**Authors:** Colin Naturman, Henry Rose

**Comments:** 10 Pages.

The concept of ultra-universal algebras in varieties is generalized to models of first order theories. Characterizations of theories which have ulta-universal models are found and general examples of ultra-universal models are investigated. In particular we show that a theory has an ultra-universal model iff it is consistent and its class of models satisfies the joint embedding property.

**Category:** Set Theory and Logic

[1] **viXra:1212.0088 [pdf]**
*replaced on 2013-06-04 05:22:15*

### Difficulties of the Set of Natural Numbers

**Authors:** Qiu Kui Zhang

**Comments:** 10 Pages.

In this article some difficulties are deduced from the set of natural numbers. The demonstrated difficulties suggest that if the set of natural numbers exists it would conflict with the axiom of regularity. As a result, we have the conclusion that the class of natural numbers is not a set but a proper class.

**Category:** Set Theory and Logic