Thursday, 19 November 2009

False infinity in mathematics


“False infinity in mathematics”

Sergey Sitnikov

Dear readers offering you the work “False infinity in mathematics” I would like to mention that I introduced a new notion “ideal infinity” and gave the definition thereof: zero relation and zero interaction between objects.
I also gave a definition to the thought and to the image. Image is an ideal reflexion of the world in our consciousness, the altered state of consciousness is not considered. We think using images, thus a thought is a constantly changing combination of images interconnected by movement, direction, logic, emotions, desire, etc.
A set is not quite successful as example of an ideal infinity, as some people may take it for a direct proof of a set being the ideal infinity and start proving the opposite. Please mention that this is just an example whereby I would like to demonstrate the essence of the ideal infinity.
The chapter “Images and words in mathematics” is also quite disputable (although it does not influence the basis of the article “False infinity in mathematics”)
I have a question to the readers. Is it possible to think out an image, which does not exist in our world?
No one has answered it yet. And here are words in mathematics possessing no image, but everyone understands them at once.
For example (-1) (minus one), try to think out an image corresponding to these two words. You shall not succeed.
But how can I be against these designations, I am only for the rational use of words free of any images of our world in mathematics. And the inability of our consciousness to perceive an image in detail, which is perfectly reflected in it, is just an issue of time and consciousness development, an issue of the infinite self-improvement of mind.



Three types of infinity:
1 Inconceivable (actual) infinity
2 Ideal infinity
3 False (potential) infinity

1 Inconceivable infinity is the infinity of mystics, it cannot be comprised by the consciousness

2 Ideal infinity is the infinity of the science, meaning zero relation and zero interaction between objects. For example, a set in mathematics, elements without interactions. Set is a population of elements indefinite in amount and indefinite in features, i.e. elements with zero interaction, without features apart from one, of course, the set consists of elements. By the way, this definition of a set generates all the paradoxes. As soon as we define features for the elements, they still are indefinite in amount, but the infinity vanishes. The elements with features are objects of our world, and those cannot be infinite. Oh no. And this is what I try to show in my article.

3 False infinity. Infinity of a process.

First I would like to quote the Internet edition V MIRE NAUKI (“In the world of science” - www.sciam.ru). Mathematics. Theory of inconsistency of the existence. 75 years of Gödel's theorem


“Where shall we search for reliability and truth, if even the mathematical thought itself misfires?” grieved Hilbert in his lecture at the congress of mathematicians in 1925.

Simplified to the maximum it can be stated as follows: mathematics can be presented as a system of derivations from a set of axioms and it can be proved that:

1. Mathematics is complete, i.e. any mathematical statement can be either proved or disproved based on the rules of the discipline.
2. Mathematics is consistent, i.e. it is impossible both to prove and to disprove a statement without violation of the the conventional rules of reasoning.
3. Mathematics is decidable, i.e. based on the rules it is possible to find out whether any mathematical statement is satisfiable or refutable.

In fact the goal of Hilbert's program was to draw up a general algorithm of answering all mathematical questions or at least to prove the existence of such an algorithm. The scientist himself was confident that all three statements were true. In his opinion mathematics was really complete, consistent and decidable. It only remained to prove this.”


“However the “Universal Axiomatization” did not take place. The whole grand and super-ambitious program, which has been elaborated by the major mathematicians of the world for several decades, was refuted by a single theorem. Its author was Kurt Gödel who barely turned 25 at that time.

In 1930 at the conference in Königsberg organized by Vienna Circle he delivered his lecture “On the completeness of a logical calculus” and at the beginning of the next year he published his article “On formally undecidable propositions of Principia
Mathematica and related systems”. The focal point of his work was to formulate and to prove the theorem, which played a fundamental role in the further development of mathematics and other sciences. I refer to the famous Gödel's incompleteness theorem. Its most widespread, although not strict definition states that “for each consistent system of axioms there is  a sentence which is neither provable nor refutable within this axiomatic system”. Thus Gödel has refuted the first statement of Hilbert's program”.
“The insidious “circumstance” was “Russel's paradox”, which later received wide publicity, representing the question: Shall the set of all sets  that are not members of themselves be a member of itself?”
Unquote

“Russel's paradox”
This paradox is based on the notion of the set of all sets including (as sub-sets) each and all sets and in the same time being a set itself. This means that along with all other sets it contains itself as a sub-set. And this is the fact made play in the Russel's paradox.
Shall the set be a member of itself?
Let us simplify the problem. Is it possible to be contained in something, which has no limits? For example, is it possible to be contained in the infinity?
It is possible and impossible to be contained in something, which has no limits.
This issue cannot be solved from outside. An element (being a set itself) is contained in the set – yes, it is; the element is not contained in the set – no, it is not. We as external observers can take either side in this dispute, as both answers are correct. Is the element contained in the infinity? - Yes, it is. Is the element not contained in the infinity? - No, it is not.
But if one proceeds from the fact that we live in the world of objects and think using images, which represent the ideal reflexion of the world in our consciousness, then an element can be contained in the set only if the set has limits.
It is possible to be contained in something, which has limits. Otherwise uncertainty. Thus it can be assumed that set and infinity are one and the same notion.
Here we speak about the inconceivable infinity or the actual infinity according to the conventional designation.
In case of a potential infinity or a false infinity, infinity of a process, an element can be contained in such an infinity, if the element itself is strictly defined. If the element is not defined, then it is for you to decide whether it is contained in the infinity or not. Both answers are correct.


IMAGES AND WORDS IN MATHEMATICS
We think using images, not necessarily visual ones. An image represents the ideal reflexion of the world in our consciousness, the altered state of consciousness is not considered.
What is our world? Our world is the world of objects. It is objects and interactions between the objects. Mathematics has started with analogies between numbers and world's objects. Interactions between the objects and between the numbers did not differ (addition, subtraction). As mathematics developed it became more and more abstract, the connection between mathematics and world must be already proved. And as a pinnacle of abstraction mathematicians started to speculate using words. Mathematicians speculate using words, hence contradictions, misunderstanding, differences, et cetera. Hence formal mathematics. This is mathematics where mathematical definitions (notions, terms) do not have corresponding images from our world behind them. For example, (-1) minus one, 0 – zero, - infinity, et cetera. The mathematics where each mathematical term has a corresponding image from our world behind them is descriptive mathematics.
A word in any language is just a designation of an object according to some its features and a designation of an interaction. When the analogy between the world's objects and the mathematics was easily traced, no major problems arose. But when mathematics abstracted away so that a new notion (word or words), e.g. set, is introduced first and then they endeavor to explain this notion again with words, the time of paradoxes arrives. Let us consider such notions as zero and infinity without digging into them. There are no objects in the world analogous to infinity or to zero, nevertheless, they exist in mathematics confusing if used unthinking.
Set is a population of elements indefinite in amount and indefinite in features, i.e. elements with zero interaction, without features apart from one, of course, the set consists of elements. By the way, this definition of a set generates all the paradoxes. As soon as we define features for the elements, they still are indefinite in amount, but the infinity vanishes. The elements with features are objects of our world, and those cannot be infinite. Oh no. And this is what I try to show in my article.
Set is a population of elements indefinite in amount and indefinite in features. Thus it can be assumed that ideal infinity and set are one and the same – zero relation and zero interaction between elements. But only until the elements are not defined in features.
Shall the set of all sets  that are not members of themselves be a member of itself? Shall the infinity of all infinities that are not members of themselves be a member of itself? If one departs a little bit from the formal mathematics there immediately appear images, objects, solutions, proofs, finally truth. In my work “False infinity in mathematics” I think it is shown, but it is not proved at a real example yet.
False infinity. Infinity of a process.
The features of the elements in any process are defined, they interact and without features it is difficult. It is assumed that if elements are not repeated, the process is not infinite, there cannot be an infinite number of elements. Our world is a closed system. And if the elements in the process are repeated? On the basis of multiple experiments the physical principle of momentum conservation in any isolated system was exactly determined. In our case the isolated system is the space. The consequence is the independence of the position of a center of mass of such a system from any processes taking place in it. Question: Can a process last infinitely in an isolated system? But first one has to answer the question, whether the system can exist for ever? The most likely, it is no.
But let us go back to the formal mathematics. How is it possible to express a definition having no image through other definitions having images behind them. There are ways confirming that our mind is a machine. The machine unable to think out an image, which does not exist in our world. For example, the definition of the ideal infinity as zero relation and zero interaction between objects does not involve the subject-matter of the infinity.
Thus I do not attempt to explain the infinity. The infinity has no image in our world. I enclosed infinity among the objects having images. And so I defined the infinity without involving its subject-matter. In this way it is possible to define zero or God. Although mathematics and belief are quite different subjects, even they have some crosspoints.

I do not argue against formal mathematics, but the use of it must be thought over in respect of the compliance with our world, our thinking using images, not words.

Sergey Sitnikov

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