People usually say that the number greater than any assignable quantity is infinity and probably same in the case of -ve ∞.
We are dealing with infinity ∞ in our mathematical or statistical calculations, sometimes we assume, sometimes we come up with it. But whats the physical significance of infinity.
Or
Anyone with some philosophical comments?
Dear Abhy,
Infinity is not a real number. It is in fact a symbol of a large quantity without bounds.
Mohamed
The concept of infinity is a useful but merely abstract construction, and contradicts our logic if it is applied to the physical reality. In this line, I like, for example, what Wittgenstein said about this: "Let's imagine a man whose life goes back for an infinite time and who says to us: 'I'm just writing down the last digit to Pi and it's 2. Every day of his life he has written down a digit, without ever having begun; he has just finished. This seems utter nonsense, and a reductio ad absurdum of the concept of infinite totality."
Of course, the classical Zeno's paradoxes are a typical reference to the infinity concept and his application to reality.
Best regards.
ctually theres a discussion about if inifinite exist nowdays in my opinión inifinite is just a way to say " we don't know how to solve the scaling problem so this is the only thing we have" our problem is we haven't be able to work without linear functions, this has to deal with infinite becouse in linear geometry the space has curvature 1/R equal to 0, so classical maths says that theres no real such can do that, and here begins our headache.
infinity is the quality or state of endlessness or having no limits in terms of time, space, or other quantity
Infinity is undefined!
Infinity is usually used to describe an open ended number.
Infinity is symbol that tells us, try to find as much numbers as you can; since their finding is endless.
Look at Cantor's Theory of Transfinite Numbers: https://www.learner.org/courses/learningmath/number/pdfs/session2/session2.pdf
i=√(-1) = -∞
see
New Philosophical View Merging Infinity, Imaginary Number and Zero and a New Arithmetic Solving Indeterminants.
Article New Philosophical View Merging Infinity, Imaginary number an...
Mathematicians have a precise definition of infinity that can be used to prove theorems about it. A set S has infinite size if it is possible to create a 1-to-1 correspondence of the elements of S with the elements of a proper subset of S. For example, the positive integers S = {1,2,3,...} is an infinite set because there is a 1-to-1 correspondence between S and the even integers in S: 1 2, 2 4, 3 6, 4 8, ... Such a 1-to-1 correspondence is impossible for finite sets.
Dear Abhay Pratap Singh,
It is well known, to the undergraduate secondary school students that the seven indeterminate forms are
0/0 , ∞/∞, ∞ - ∞ , 0x∞ , ∞0 , 00 , 1^∞.
Infinity ∞ is NOT a real number it used to solve problems related to (limits) for some value that goes to large positive and negative unbounded values; they are familiar with L'hopital rule.
To understand ∞ we need some topology and compactification of R.
You can study the nice example of the stereographic function ( it is a diffeomorphism) between the {circle - one point } and R. This point represents +- infinity. It includes a nice geometrical explanation of infinity. You can add this point to obtain a new topological space diffeomorphic to the circle.
The same idea between sphere the stereographic function ( it is a diffeomorphism) between the {sphere - one point } and the plane, etc.
See for example the link:
https://en.wikipedia.org/wiki/Stereographic_projection
Best wishes
@ Salah Mabkhout,
I read the mentioned article. New Philosophical View Merging Infinity, Imaginary number an...
The main issue is:
What is your field where we can apply the mentioned axioms with well-defined operations on that field? As well you considered many examples of paradoxes related to different spaces with the same axioms.
One can solve a problem in some field but not in another one.
As an example, the last Fermat's theorem has solutions over complex numbers and matrices but not over Z.
Back to your arithmetic. What is i ? is it i = (0,1) in the field of complex numbers? or just notation?
I can't agree with you that i=√(-1) = - ∞ in C,
then i^2 = ∞ = -1 which is a contradiction in the complex numbers
which is a complete (Non Ordered) field.
All such ( fake calculus computations with ∞) appeared in a first course in real analysis. One can rearrange different (non convergent series to obtain different fake answers. As
0 = 0+0+0+0+0.....= ( a-a)+ ( a-a)+ ( a-a)+ ( a-a)+ ..
=a-a+a-a+a.......= a+(-a+a)+(-a+a)+(-a+a)+(-a+a)..= a+0+0+0+0....= a
In algebra:
In the language of algebra: 5 + 2 = 7 = 0 = -7 =.... this is true in Z7,
but not over Z.
In fact, many relations are true on some metric space with (algebraic structure) but not on another one.
Best wishes
@ Issam Kaddoura
" I can't agree with you that i=√(-1) = - ∞ in C"
1- Well, I didn`t assume that i= - ∞, it is not an opinion. I derived it through a rigorous mathematical steps. So, you can`t agree only if and only if you found a flaw in the mathematical steps.
2- I didn`t say (( i^2 = ∞ = -1 )), i^2 is not equal ∞.
instead -1=i^2=(-∞)(-∞).
3- The border separating reals from complex is an illusion.
@ Salah Mabkhout,
You didn't answer my question yet.
Again, what is your field?
In general, if a*b=c and a*b=d, then c=d, if not, the operation is not well defined, and the axioms are inconsistent. Consequently, it is not accepted.
Thank so much all.
Thank you @Donald for the sharing the link.
Thank a lot @Issam and @Salah for discussion.
There are many other ancient philosophical text which explains infinity form their perspective.
Best Regards,
Abhay
@ Qefsere,
The circle is bounded, where are the ends of the circle?
The whole line is included in the circle via the stereographic function. But the line is not bounded.
Is it a paradox?
Could you please define: without end.
The following is an excerpt from an article that I am working on (kindly forgive the embedded TeX.):
A very important concern in set theory is its treatment of \textit{infinity}. and to explain and tie up its relevance we begin with Cantor who is credited with the invention of set theory.\\
\textit{"Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. In fact, Cantor's method of proof of this theorem implies the existence of an 'infinity of infinities' "}\ref{ref:cantor}. Without entering a narrative on the matter of objections on Cantor's work by his peers, it suffices for our purposes to point out that at the time, his ideas were met with much objection. Before we continue further, it will serve to point out that Cantor's work began with an idea, one that he began extrapolating and formalizing at the same time. So one is on the same page, our interpretation of formalization is the use of logic and mathematically precise linguistics so as to relay an idea. There are two routes one can take at a junction where the idea is almost clear but not formal enough to be entirely challenged, and those are either to reject it as absurd simply because it does not pass the bar of perfection, or to try and understand the 'end goal' and assist in the formalization process and reveal either its truth or fallacy in a manner accepted.\\
Notably Bertrand Russle was one of the few that took on the challenge after deep contemplation on his part before arriving at the paradox that bares his name. The problems Cantor encountered however was well before this. Before Cantor only finite sets were considered and these were trivial enough to understand. Cantor began formalization with the concepts of ordinals and cardinals, further to this he established a means of comparing \textit{infinite sets}, and this is where the concept of cardinalities becomes clearer.\\ In simple terms, one is able to say that two infinite sets are 'equal in size/measure' if they can be put into one to one correspondence. A simple example of this is the sets: $s_1:=\{x\vert x\in 3n
\vert n\in \mathbb{N} \}$ and $s_2:=\{y\vert y\in n
\vert n\in \mathbb{N} \}$ have the same cardinality as the following one-to-one pairing is possible: $3n\rightarrow n \forall n\in \mathbb{N}$. This mapping takes the form: $(3,1),(6,2),...,(3k,k),..$, and we are able to know that the mapping is 'complete' as it exists for all $n\in \mathbb{N}$. From this Cantor established that the cardinalities of both $(\mathbb{N},\mathbb{Q})$ are in fact the same.\\ Take for instance the following assimilation:
\[
\left[ {\begin{array}{cccccc}
\frac{1}{1}&\frac{-1}{1}&\frac{2}{1}&\frac{-2}{1}&\frac{3}{1}&..\\
\frac{1}{2}&\frac{-1}{2}&\frac{2}{2}&\frac{-2}{2}&\frac{3}{2}&..\\
\frac{1}{3}&\frac{-1}{3}&\frac{2}{3}&\frac{-2}{3}&\frac{3}{3}&..\\
\frac{1}{4}&\frac{-1}{4}&\frac{2}{4}&\frac{-2}{4}&\frac{3}{4}&..\\
\end{array}
} \right]
\]
With little effort, one can see that in keeping the denominator same in all rows of the matrix, that all the rational numbers can be covered. Consider now the following mappings: $(1,\frac{1}{1}),(2,\frac{1}{2}),(3,\frac{-1}{1}),(4,\frac{-1}{1}),(5,\frac{1}{3}),(6,\frac{-1}{2}),(7,\frac{2}{1}),...$. Where the numbers to the left within each pair belong to $\mathbb{N}$ and the values to the right are selected in a zig-zag fashion from the matrix above. The takeaway point here is that we have found a way to list all the $\mathbb{Q}$ information and we have shown that it is pairable with $\mathbb{N}$.\\
\\
In the way of establishing cardinality of the reals is where Cantor's genius really shows. It is also from here that the intension with regards to Cantor's Set Theory becomes clear. Following Cantor's formalization, the concept of cardinality highlights limitations in our previous understanding of \textit{infinity}, and this is best seen when comparing $\kappa (\mathbb{N}):\kappa (\mathbb{R})$\footnote{Here $\kappa$ means cardinality of.}.\\
Keeping in mind that a formal technique of comparing cardinalities of sets has been established and it is via establishing a one-to-one corrospondence between sets. We will now show, using techniques of Cantor, that $\kappa (\mathbb{N})
In mathematics the concept of infinity is no longer anything weird and actually is used in different contexts.
The main of those for me is when we are talking about cardinalities of sets. A set can be finite, which means that it has finite number of elements. In other words a set is finite if its elements can be counted one by one and this process ends. I think that the concept of finite set is immediately clear to everyone. Thus, a set is infinite if it is not finite. And that is it. It is as simple as this. If it is not finite, it is infinite.
Now, you may ask whether or why infinite sets exist at all. The answer may be not very satisfactory, because in mathematics the existence of an infinite set is simply implied by one of its axioms, namely the axiom of infinity. The axiom of infinity is usually formulated in the following way: there exists the set of all natural (i.e. positive integers including 0) numbers. It is quite clear that if such a set exists it cannot be finite (thus it is infinite), because if it were finite there would exist a natural number n which equals to its cardinality, but 0,1,....,n are natural numbers and there are n+1 of them so the set of all natural numbers must have cardinality at least n+1, which is a contradiction with the assumption that it is of cardinality n. Nevertheless, we need the axiom of infinity to have infinite sets in mathematics.
Astonishingly, if you assume this axiom (which all the mainstream mathematicians do) you get the whole range of different infinities, which were discovered by Cantor in late 19th century. Namely, given two, let's say, four-element sets you can check that each of them has the same number of elements in two different ways. Either you can count the elements in each of the sets, or you can arrange the elements of both sets into pairs with each other. The second method can be applied also to infinite sets, and thus we say that two sets are of the same cardinality if the elements of those set can be arranged into pairs with each other. It is quite easy to prove that the sets of natural numbers, the set of all positive integers and the set of all rational numbers have the same cardinality, and this is the smallest possible infinity. But, already Cantor proved that this is not the case for the set of all real numbers. It can be proven that the elements of the set of all real numbers cannot be arranged into pairs with elements of the set of all natural numbers. Thus we end up with a different and bigger infinity. Actually, by Cantor Theorem the set of all subsets of a given set has always greater cardinality than the set itself. So starting with the set of all natural numbers and applying Cantor's Theorem repeatedly you can obtain infinite number of greater and greater infinities.
And that is not even the first chapter of the story... : )
The main corollary that I want you to get from this is that the infinity is a concept which can be, and actually is, studied with success by human mind. Which at least for me is a relatively comforting thought.
Cannot define something thet is per definition greater than a definition itself. Can approximatly describe it only (as a limit to infinity)
As hawking said, if the equation goes infinite something is wrong. i know its just an ancient greek trick to say it reached the upper bound, for example natural numbers always touch the upper bound , after riemann everytime a formula is not upper bounded we say something is wrong, instead of saying it goes to infinite we say we need to define it to give a number. becouse infinite is useless, what can you do with a set without an structure? nothing, instead cantor tried to make sets of infinites and then ramannujan calculated numbers asociated to the series. but for now it keeps a lot of work to give it an use
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