[4] **viXra:1409.0174 [pdf]**
*replaced on 2016-02-27 04:59:59*

**Authors:** Misha Mikhaylov

**Comments:** 7 Pages.

It was shown here that just transitive relation may be considered as closure and its presence is necessary and sufficient to order a set linearly, and it is not possible to do this by using other relation’s property – neither reflexivity nor symmetry. By interaction, it occurs due to ambiguity of their definition – it was shown earlier (ref. [3]) they have various appearances. Among them just the only transitivity is determined uniquely. At the same time the last one doesn’t exist separately from any others. Circumstances of their joint existence are clarifying in this article.

**Category:** Set Theory and Logic

[3] **viXra:1409.0056 [pdf]**
*replaced on 2016-02-23 12:30:09*

**Authors:** Misha Mikhaylov

**Comments:** 11 Pages.

Usually itemizing relations’ properties those of them are always pointed out – some appearance of reflexivity, symmetry and transitivity. Also it is not so clear whether they are introduced artificially – i.e. axiomatically, rather for the sake of convenience or it may be done due to inartificial reasons. At the same time an origin of them is not so clear – whether they appear chaotically and independently on each other or there should be rigorous association between them. It is shown here that request of relation’s reversibility leads to these properties’ presence or absence. Often symmetry appearances are defined by using of ambiguous way. In fact, anti-symmetry is not direct negation for symmetry – there is also something that may be called as asymmetry or it may be something else. To avoid it here there was found the unified method of their definition. The same thing may be told about reflexivity and it was shown that just the only intransitivity may be represented as direct negation of transitivity.

**Category:** Set Theory and Logic

[2] **viXra:1409.0041 [pdf]**
*replaced on 2015-06-01 08:54:57*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

In this paper, we redefine the infinitesimals by using axiomatic method.

**Category:** Set Theory and Logic

[1] **viXra:1409.0006 [pdf]**
*replaced on 2014-10-20 00:20:20*

**Authors:** Felix M. Lev

**Comments:** 7 Pages. A figure added

Standard mathematics involves such notions as infinitely small/large, continuity and standard division. This mathematics is usually treated as fundamental while finite mathematics is treated as inferior. Standard mathematics has foundational problems (as follows, for example, from G\"{o}del's incompleteness theorems) but it is usually believed that this is less important than the fact that it describes many experimental data with high accuracy. We argue that the situation is the opposite: standard mathematics is only a degenerate case of finite one in the formal limit when the characteristic of the ring or field used in finite mathematics goes to infinity. Therefore foundational problems in standard mathematics are not fundamental.

**Category:** Set Theory and Logic