Dear Professor Juan-Esteban

I came across this article By Professor John D. Barrow titled "does Infinity exist?". In it the author distinguishes between three types of infinity, Mathematical, Physical and Cosmological. It is a very interesting reference.

http://plus.maths.org/content/does-infinity-exist

'Unboundedness of any Set, Space or any thing is termed as Infinite, if some one succeeds in finding in its bounds, then it is not infinite and it becomes finite, so infinity means beyond our reach hypothetically, which could be attemped to reach, without success and definitely it is an endless process.

Dear Ajaib Banyal,

I also think that infinity denotes an endless process and it has a potential or virtual reality.

Nevertheless, this claim requires a multivalued logic, or at least, a three valued one, in which the truth-values are True, False and Undefined. If you prefer numerical values, True = 1.0, False = 0.0 and Undefined = 0.5.

To see this fact, consider an infinite set E with an infinite subset X = {a_1, a_2, a_3…} ⊆ E. Assume that the members of X are chosen at random, hence there is no predictable pattern in order to discern whether or not some member x of E belongs to X.

Obviously, if for some finite index n, a member x of E is equal to a_n, then the statement x ∈ X is true.

Now, consider any algorithm A, that in successive steps compares x and a_n in order to decide whether or not x belongs to X. The algorithm stops when, for some n, x = a_n. If the algorithm stops at some n, then the statement x ∈ X. is true. By contrast, if the algorithm becomes an endless process, then the expression x ∈ X is undecidable, unless we assume that the endless nature of the process is a human limitation and actual infinity exists; that is to say, we assume that infinity is reachable.

At the center of a black hole space-time has infinite curvature and matter is crushed to infinite density under the pull of infinite gravity. At a singularity, space and time cease to exist as we know them. The laws of physics as we know them break down at a singularity, so it's not really possible to envision something with infinite density and zero volume.

There is general consensus that supermassive black holes exist in the centers of most galaxies.

Dear Edwin,

The infinity I am talking about, is a mathematical concept, closely related with the cardinal notion. I think that it is very interesting to analyze the infinity in the scope of cosmology, but it is a topic which falls far from my skills.

In any case, thank you for your invitation.

Best regards-

Dear contributors,

Please find the attached paper by A. Zenkin, to illustrate this topic. I think that it is very interesting and very unknown.

Dear contributors,

Please find the attached paper by A. Zenkin, to illustrate this topic. I think that it is very interesting and very unknown.

Probably the most convincing answer was given by P. Vopenka in his Alternative Set Theory - axiomatically developed theory where he replaces actual infinity by the natural one - a phenomenon that raises when dealing with very large sets. According to him, actual infinity is only an illusion and the whole classical set theory can be embedded into AST where actual infinity does not exist.

Unfortunately, the only book of him written in English is below, other books of him are only in Czech.

@book{vop:ast,

author = {Vop\v{e}nka, P.},

year = {1979},

title = {Mathematics in the Alternative Set Theory},

address = {Leipzig},

publisher = {Teubner}

}

You may also learn about AST from my book (otherwise devoted to other topic):

@book{nov:aml,

author = {Nov\'ak, V.},

year = {1992},

title = {The Alternative Mathematical Model of Linguistic Semantics and Pragmatics},

address = {New York},

publisher = {Plenum}

}

Let me remark that the problem is related to the vagueness phenomenon whose best working mathematical model is provided using mathematical fuzzy logic

@article{nov:FSS2012Reason,

author = {Nov\'ak, V.},

title = {Reasoning about Mathematical Fuzzy Logic and its Future},

year = {2012},

volume = {192},

journal = {Fuzzy Sets and Systems},

pages = {25-44}

}

cf. also

@incollection{nov:Fdays06,

author = {Nov\'ak, V.},

year = {2006},

title = {Fuzzy Sets as a Special Mathematical Model of Vagueness Phenomenon},

editor = {Reusch, B.},

booktitle = {Computational Intelligence, Theory and Applications},

publisher = {Springer},

address = {Heidelberg},

pages = {683-690}

}

@article{nov:evalexpress,

author = {Nov\'ak, V.},

year = {2008},

title = {A Comprehensive Theory of Trichotomous Evaluative Linguistic Expressions},

volume = {159},

number = {22},

pages = {2939–-2969},

journal = {Fuzzy Sets and Systems}

}

Dear Vilem Novak

Thank you for the bibliography you have recommended. It is very useful for me.

Regards.

My soul is an entangled knot,

Upon a liquid vortex wrought

By Intellect in the Unseen residing,

And thine doth like a convict sit,

With marlinespike untwisting it,

Only to find its knottiness abiding;

Since all the tools for its untying

In four-dimensional space are lying,

Wherein they fancy intersperses

Long avenues of universes,

While Klein and Clifford fill the void

With one finite, unbouded homaloid,

And think the Infinite is now at last destroyed.

James Clerk Maxwell (1831 - 1879)

infinity actually does not exist. Even cantor did not used actual infinity. he only created theory of cARDINALITY, which really distingushes into discrete and continum. a great theory . it is really poor of the material world where we value contributuons of Gauss mostly applied but understimate Cantor, Cauchy, euler who were as or probably grater mathematicians than Gauss. Euler was in particular a much greater mathematical genius.

but one argument iin favor of infinity is that even natural numbers give rise to this notion. for infinitesmal one needs atleast a set of rational numbers.

there w2as apaper in ACm transactonson nlogic where a consistent algebra was created using 0 , 1 and infinity and it wwas shown that alll usual mathematical systems derive from it.

Dear Professor Juan-Esteban

I came across this article By Professor John D. Barrow titled "does Infinity exist?". In it the author distinguishes between three types of infinity, Mathematical, Physical and Cosmological. It is a very interesting reference.

http://plus.maths.org/content/does-infinity-exist

Reply to: Hélio Barnabé Caramuru

You have raised quite a very important question about infinity a reality per mathematically or physically possible, although not experimentally provable.

My publication with Journal Article: ABSOLUTE GENESIS FIRE FIFTH DIMENSION MATHEMATICAL PHYSICS

PRESIDENT RAJAN IYER, PH.D.,

per your request is highly appreciated. Although at present I am not able to upload that, definitely hope to present with you during conference either invited or sponsored physics global event.

Please continue associations through RESEARCHGATE especially and the forum to discuss matters of great importance to science & humanity specifically.

Thank you

Rajan Iyer Engineeringinc International Scientist

This may be naive physics but doesn't the answer depend on whether empty space exists independently of matter? If, as I think most physicists tend to believe, space is impossible without matter then the states of the universe are limited to the possible permutations of the atoms in the universe and their internal structure. This appears to be finite and it is difficult to state that a number that could not be represented in the universe is real. (I am assuming that the current perception that the universe has a finite dimension is correct so we could assume a mathematical limit on the number of particles in it)

Not clear how the 'atual' and 'infinity' comes together. For any Mathematical, Physical or Cosmological concept of infinity perhaps no one can add the word actual.

The problem is in the word "exist". As a pure mathematician I Immediately think, "Duh! The integers." This set "exists" in every sense that is mathematically interesting.

If you mean "as a physical object" then I've got trouble with other things you routinely except as "actually existing". Like your upcoming doctor's appointment. If you say, well, that actually exists -- I'll say "where?" If you say "in his doctor's office" I'll point out that that is not where it is right now! Where is it now? On a calendar? You mean if you burn the calendar the appointment doesn't exist? Or do you mean later on the space-time-continuum? Well what if you miss that appointment? Does it exist then? If not, then why are you billed for it anyway?

A black hole is an "actual singularity", shown to "exist" by Penrose, and studied by Hawking his whole life. In a sense a singularity is an "actual infinity". But just as sensibly we can think of it as an "actual zero" -- depending on which ball our eye is on.

Here's another actual infinity: The number of possible strings, in principle, that can be made by concatenating English letters A, B, C, ... This would include all English words and also an awful lot of nonsense. Since we do not bound the length the number of possibilities is infinite. Do you object that they do not exist because they aren't actually put down on paper? I insist that they exist in precisely the same sense as your doctor's appointment exists.

You are right that the problem is with the word 'exist'. If we take it to mean 'have some counterpart in reality' your final example is not infinite because you run into the same situation I describe. You will eventually use all the matter in the universe writing your strings. You will then try to make the lettering smaller until you are making maximal use of the different configurations of atoms in the universe. Then you will stop unable to go further.

IDear Juan,

For me, nfinity is an encephalic concept, not measurable in any meaningful dimensional form in space.

Regards from Athens,

Panagiotis Stefanides

http://www.stefanides.gr

I query was how the 'atual' and 'infinity' comes together. Then I said perhaps no one can add the word actual for any Mathematical, Physical or Cosmological concept of infinity. Am I right ?

Just because we have a hard time understanding infinity does not mean it does not exist. The question should be are all systems finite. If the answer to this question is yes then infinite does not exist if the answer is there are an infinite set of finite systems then it does. Yes to the question. Infinite is the universe. Not finite but curved to seem to be infinite. There is always something more.

Dear George,

Perhaps denoting the question in the following terms becomes easier.

I consider potential infinity as a synonymous of "endless", that is to say, a mathematical construction is potentially infinite provided that it cannot be completed. By contrast, actual infinity must be considered as a mathematical object the construction of which can be completed. Both concepts are incompatible; therefore if by infinity we mean "endless", then cannot be actual.

Dear Juan-Esteban,

Math seems to be the problem here. I understand infinity, math does not. Any function that cannot be resolved exactly such as divide by zero or goes to zero but never real gets there are functions that really are undefined. This means that math has the problem. Just by changing the name of a duck does not make it something else. We must update our math to accommodate nature. It has always seemed strange to me that nature has numbers that can not be defined exactly in math. Pi is one of them, infinite is another one but there are many more called transcendental numbers. Even the definition of them sends us to a spiritual or nonphysical realm. I am not saying that math is wrong, just that it is so very limiting in its current form.

The mind is infinite in a finite way. Our ability to process and then create new things gives us the ability to have infinite possibilities with a finite physical space that is the mind.

Endless is just another way to say infinite. To solve infinite problems we need a math that is infinite and so far we do not have that. Reality is different. It has to do with the existence of a finite number of solutions that do occur in space time even though there is an infinite number possible. Again math is the problem.

Quantum Mechanics is a great example of this. We know that there is a chance that something will happen and that it will not happen. QM tells us all about that but it does not explain why only one solution really happens each time. A limit of the math not the mind.

I really enjoy the talk but have to do some work. Talk to you soon.

It seems to me that by fiat we posit as True both "ability for endless construction" and "infinity as an object" based on extrapolation-abstraction from experience-logic. Presumably no one has ever directly experienced either. Essentially by fiat, we have declared existence of two objects 1) potential infinity which has certain properties, and 2) actual infinity that has certain properties.

The extent to which these two ideas conflict mathematically seems an indication that we've got something wrong (mathematically speaking). Or we have two (unrecognized?) axiomatic systems.

Perhaps we face a situation analogous to Cantor's "continuum hypothesis [which] has no solution within our system of mathematics" (Aczel, 2000, p 155)? Can the problem be recast as a form of the hypothesis?

In any case actual-inifinity and potential one are not two different names of the same object because the have different definitions.

Actual or completed or finished infinity is in contradiction with fundamental principles of mathematics.

Proof.   It is a theorem of ZFC (the most common version of transfinite set theory) that there are aleph_0 (A0) natural numbers. More precisely: The cardinality of the set of natural numbers {1, 2, 3, ...} is A0. This A0 is, according to Cantor, "a fixed quantity" in trichotomy with all cardinal numbers including all natural numbers. That means the statement

         For all natural numbers n: n < A0

makes sense. See any text book of set theory.

According to set theory the arithmogeometrical figure

1

2, 1

3, 2, 1

…

contains an infinite number of elements, namely all A0 natural numbers.

However, no row of the figure contains A0 elements.

According to mathematics, the figure is an inclusion-monotonic sequence of finite rows. That means: Every union of finite rows is contained in one of the unioned rows. (The principle of construction shows, that this property is independent of the number of finite rows. Otherwise there would be a first finite row that does not contain all elements of its predecessors.) If the whole figure contains a fixed quantity of A0 elements, then this fixed quantity must be in one of its rows. Contradiction. Therefore the idea of the set of natural numbers and its cardinality being a fixed quantity A0 can be excluded.

Another way to see this is shifting all rows into the first row, maintaining the column of each number. No row has A0 elements, but finally, after shifting, the first row has A0 elements. That is in contradiction with the principle of translation invariance of letters and symbols.

An introduction into transfinite set theory, a lot of statements about infinity and several contradictions can be found in https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf

Regards, WM

Actual infinity does not exist as anumber so that the system follows usual number lawas of Fields , or in case of natural numbers uniqueness of identity for addition is lost 

 but toplogically or when viewed as ordered sets the infinity actually exits . one can assume real line or rational doted line is completed by additing two points at infinity. since Mathematics is a ligical system of symbolism infinity exists in thes econtexts. in integration theory functions can take infinite values and infinity into 0 is 0 is astandard property.

in applid mathematics one can extend measure theoratical by compromising on this property to have integral of dirac delta function which is infinity at only 0

There is no complete real line. The line that you draw on paper consists of a very small number of molecules. The number of thought points that you can identify by co-ordinates is limited by the number of different words that are possible in mathematical discourse (including monologue). Discourse requires physical media and co-ordinates require memory space in brains or other storage media. The whole accessible universe contains about 1080 atoms. It is hardly possible to use all of them, but even then at least one atom is required to store a co-ordinate. That means: No line contains more than 1080 thought points. Sorry, but mathematics cannot exist independently of reality and physics.

Regards, WM

Dear Wolfgang,

I appreciate your manuscript. It is very valuable for me. Congratulations and best regards,

Juan-Esteban.

Dear Juan-Esteban,

I intend to add this statement:

Juan-Esteban Palomar Tarancon    I also think that infinity denotes an endless process and it has a potential or virtual reality. Nevertheless, this claim requires a multivalued logic, or at least, a three valued one, in which the truth-values are True, False and Undefined. [Juan-Esteban Palomar Tarancon in "Does actual infinity exist?", ResearchGate (26 Aug 2012)]

to chapter V. It is absolutely correct since an infinite sequence of digits cannot define a number. (Every definition has to have an end signal. Actual infinity would not change that situation because even there the last digit is missing (and infinitely many others, since every digit, even there, has only a finite number of predecessors but an infinite number of successors.)) My question: Should I order your statement under P or T?

Regards, WM

Dear Wolfgang,

I agree with you. In addition, your previous statement:

Sorry, but mathematics cannot exist independently of reality and physics.

leads me to the following question.

Let x = 0.c₁c₂c₃.... and y = d₁d₂d₃... be the decimal expressions of two real number in (0.1). If we know that, for some m ∈ ℕ, the following statement is true

∀ n ≤ m: cₙ = dₙ

we can deduce that |x-y| ≤ 1/10m

In such a comparision, can m be arbitary large? I think that, in real world, the Heisenberg's uncertainty principle negates this possibility and for m large enough we need introduce a fuzzy logic to assign truth-values to both statements x = y and x ≠ y.

Dear Juan-Esteban,

m cannot be arbitrary large because by means of less than 10100 particles no natural number with a greater digit-complexity can be represented. (Under digit-complexity I understand a simplified version of Kolmogorov-complexity. To explain it by a short example: You can produce numbers like 1099 on the display of a common pocket calculator, but you cannot display a number like 256808734108 which requires more than 10 digits there.)  Of course, the uncertainty principle would also prohibit an infinitely exact reading of a pointer on a scale.

By the way Jim Slattery (I don't know how to answer him directly) raised the question of the finity or infinity of the universe. This question however is irrelevant in this context because that part of the universe that is accessible to us will be finite forever.

Regards, WM

Dear Juan-Esteban Palomar Tarancon ,

It's an interesting question. For "infinity" to exist, "exist", it would relate to a physical construct, and not necessarily mathematical, for instance in regard to space or time. The idea of "infinity" indeed was pioneered by Cantor and then taken up by Hilbert, the aim there of Hilbert was to understand how space could be infinite in addressing the possible expansion of space in the context of the redshift effect of light. I'm able to explain this a little better in a recent paper:

Preprint Space, and the Propagation of Light

In that paper, which is really a neat history of the mathematics of space together with observed ideas of physical space, the idea of how infinity mathematically relates with space is discussed, why it was employed, and the success of that employment as an idea to solve ideas in special and general relativity, and of course the redshift of light.