[3] **viXra:1505.0173 [pdf]**
*submitted on 2015-05-24 14:58:20*

**Authors:** A.D Albert

**Comments:** Pages.

Traditionally Fuzzy Logic is defined in the unit interval of [0,1]. These notes explore what will happen if this interval was larger than the unit interval (where normalization of truth values will violate the definition of a particular fuzzy set). These notes also derive suitable logical operators applicable to this extended interval utilizing first principles.

**Category:** Set Theory and Logic

[2] **viXra:1505.0122 [pdf]**
*replaced on 2016-03-02 13:16:32*

**Authors:** Misha Mikhaylov

**Comments:** 24 Pages.

It seems that statements determining features of some algebraic structures behavior are based on just intuitive assumptions or empiric observations and for sake of convenience (simplest example is the phrase: “let’s consider 0! =1”… perhaps, just because Sir Isaac Newton entrusted, so, why not choose any: e.g. 2, 5, or 7.65). So, without logical explanation these are looking a little mysterious or sometimes even magic. This article is a humble attempt to get it straight rather formally. Some troubles may appear on the way – e.g. as it was shown earlier (in the ref. [2], for example), there are at least two binary relations having properties of idempotent equivalences – algebra’s elements that may aspire to be an identity. Apparently, probable obtaining of some well-known results in the text is not an attempt of their re-discovering, but it is rather “check-points” that confirm theory validity, more by token that it was made by using of the only exceptionally formal way, while usually they are obtained rather intuitively. Usually the notion of tensor product is determined for each kind of algebraic structure – especially for modulus (in group theory it is often called direct product – but this is a matter of semantics, so, it’s rather negligible). Here it is shown that tensor product may be introduced without defining of concrete algebraic structure. Without such introduction defining of algebraic operation is strongly complicated.

**Category:** Set Theory and Logic

[1] **viXra:1505.0079 [pdf]**
*submitted on 2015-05-10 13:10:05*

**Authors:** Alexander S. Nudelman

**Comments:** 5 Pages.

In this paper we define an arithmetic theory PAM, which is an extension of Peano arithmetic PA, and prove that theory PAM has only one (up to isomorphism) model, which is the standard PA–model.

**Category:** Set Theory and Logic