Hi people,

I’m sure, the most people feel the intuition and beauty of infinitesimals but a lot of them have problem with the rigour introduction via logic by A. Robinson and E. Nelson. Several of authors try to simplify the approach of infinitesimals, among of them, I think, J. Keisler gives the most energy to make nonstandard analysis popular.  But all the approaches include logic. Can we understand infinitesimals without logic? We say YES.

The view of the mainstream in the mathematical world from K. Weierstrass up to bevor A. Robinson to the quantities of infinity (large or small) is that they are only a process, denoted and etablished in form of limit. Infinite quantities were not allowed to exist as free, independent variables. As $x\rightarrow 0$, $latex x$ is always a variable of a function $f(x)$, or as $latex n\rightarrow\infty$, $n$ must be an index of a sequence $latex (a_n)$.  Paul Du Bois-Reymond has an effort to treat infinite quantities by giving them a form of functions having the same limit of one unique variable. This approach has natural constrictions and it’s acceptance by the other mathamaticians is only very limitted. He has a theorem, which is true and according G. H. Hardy fundamental, but for us, it leads us to misconception of the substance of infinity. Since A. Robinson, the existence of infinity quantities is assured, but in form of new numbers, the so called hyperreal numbers. The infinity quantities are now free and independent numbers, and it works very well, as it must be. But by his influence and that of E. Nelson, we must see the world as two parts: the standard one and the extended non-standard one. One fact is not changed at every aproach, that the infinity quantity is a problem of the conception. We will say more, infinity quantity can be used to study matters of perception.
We now introduce not new numbers to the real numbers, but a new order to the basic order $latex <$ in the body of the real numbers.

The new axioms

The new relation is a binary relation, called as infinity order or infinity relation, denoted as $latex \sqsubset$. $latex a \sqsubset b$ means $latex a$ is infinitely small to $latex b$. This order is definited by 6 axioms.
(U1)     $latex \sqcup(a\sqsubset b,a\sim b, a\sqsupset b)$
(U2)     $latex a\sqsubset b\wedge c\neq 0\;\rightarrow\; ac\sqsubset bc$
(U3)     $latex \sqsubset$ is arbitrarily transitive.
(U4)     $latex \sim$ is finitely transitive.
(U5)     $latex a_1,a_2\sim b\;\rightarrow \;|a_1|+|a_2|\sim b$
(U6)     $latex a\sqsubset b\;\rightarrow \;a<|b|$

Notes and explanations to the axioms:

– (U1) is the trichotomy law of the infinity like the analogous of the order $latex <$ :
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I.e. it is true exactly either $latex a<b$, or $latex a=b$, or $latex a>b$.
The relations $latex =,\ge,\le$ are only secondary relations of the basic relation $latex <$. The equality = is logically definited by this trichotomy law.
In this spirit we can define some secondary relations of the infinity order as follow.

$latex a$ ist infinitely large to $latex b$ :
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$latex a$ is equivalent or comparable with $latex b$ :
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$latex a$ is not infinitely small to $latex b$ :
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$latex a$ is not infinitely large to $latex b$ :
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$latex a$ is approximate or infinitely close to $latex b$ :
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If $latex a\approx 0$ i.e. $latex a\sqsubset 1$, we say, $latex a$ is infinitely small, or infinitesimal.
If $latex a \sqsupset 1$, we say, $a$ is infinitely large.
If $latex a\sim 1$, we say, $a$ is finite.
The order $latex \sqsubset$ is not really new. In physics we know it as the relation $latex \ll$, and on the other hand we know the notions of Landau
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(Landau-notation is often defined as relation between 2 functions. Only the notation of the physicists $a\ll b$ means really a relation of two free variables.)

The half-order $latex \sqsubseteq$ is not antisymmetric. The body of the real numbers within the meaning of the standard analysis is
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where $latex \mathbb{R}$ is the set of the real numbers.
The body of the real numbers within the new meaning is
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- (U2) means, infinity order is invariant regarding the  Multiplication.
- (U3) means
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where $latex n$ can be arbitrarily infinite or finite.
- (U4) means
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- (U5) is equivalent with
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- (U6) implies $latex 1\sim 1$ and (U5) secures us, that 2,3,... are finite natural numbers.
It is $latex n\sim 1\rightarrow n+1\sim 1$, and further $latex n+1\sim 1\rightarrow n+2\sim 1$ and so on, but it can not go on forever just because of (U4).

Next time we show you a correction of the 4 axioms of the oeder $<$, and of the 11 axioms of the body of the real numbers. The axiom of the completeness of the body of the real numbers will be treated extra in the part of sequences.