Is infinity a concept or a word empty of meaning?

Question

I'm wondering if infinity is a concept. We know from experience that there are things for which one cannot reach the end. A long way through the space is an example. One cannot reach the end of the space because the time needed is not humanly reachable. To get the end of the time is impossible. So there are things for which, if they have an end, this end is not reachable. Intuitively, it seems that for every beginning of thinking, the lived time is the first element implied in the conceptualization of infinite.

Indeed, I ask myself if infinity is really a concept rather than a word coming out from an experience.

Therefore, talking about infinite sets give me a strange impression.

Mathematics shows that we can make a one to one correspondence natural numbers with even numbers. This is not right If there is a last number. But because of the inexhaustible number of numbers it's indeed possible to make a one to one correspondence. But it's obvious that the fact that there is also odd numbers implies that the set of all numbers is greater than the set of even numbers. We cannot reach this because we cannot end the process of counting.

If we introduce the time (I use the time as a model to show precession in causality) to reach numbers, we reach a different result. E.g. if one is walking along a road to pick up fruits. After one meter I pick up an apple and a meter further I pick up a pear and so on. If I pick up all the fruits indifferently, my basket is fuller than if I pick up pears only.

Whatever the moment I compare the basket of one who pick up all fruits and the basket of who pick up pears only, the basket of the former is always fuller.

Moreover, If there is not a temporal delay for fulling the basket there is a "step delay" which has nothing to do with time (see examples from programing in the answer of @Rushi).

So can we say that the one to one correspondence in set theory talks about nothing in real world? (Moreover there is a problem of steps).

And consequently how can we know that there are not only empty words because we are conceptually limited in a finite mind?

Answer 1

Mathematics shows that we can make a one to one correspondence [of the] natural numbers with [the] even numbers. This is not right If there is a last number.

There are infinite sets that look exactly like the set of natural numbers N apart from having some finitely more elements after all natural numbers, and there being last ones: they are precisely the 'successor ordinals' of N: N+1, N+2, etc., and their forms look like

N+3 := {0, 1, 2, ..., N, N+1, N+2}

for example, so there is in fact a last element, to wit, 'N+2', and we still can put it into bijective/one-to-one correspondence with N, simply taking

f: N+3 → N

f(N) := 0

f(N+1) := 1

f(N+2) := 2

f(0) := 3

f(1) := 4

and so on

But it's obvious that the fact that there is also odd numbers implies that the set of all numbers is greater than the set of even numbers.

Proper containment is indeed one sense under which N is 'greater' than the set of even natural numbers (let's call it) E: since containment ⊆ is a 'kind of' ordering for sets, it's reasonable to interpret E ⊊ N as such, but as you seem to know, this is not really the same thing as 'size comparison'

If we introduce the time to reach numbers, we reach a different result.

In current mainstream mathematics at least, there is no notion of 'time' related to or imposed upon the objects of inquiry: they just kinda sit there statically. Even if we speak of, say, 'a function varying with (respect to) time', it really still is a list of ordered pairs, nothing more, nothing less, so that the analogy

E.g. if one is walking along a road to pick up fruits. [...]

doesn't quite make sense

So can we say that the one to one correspondence in set theory talks about nothing in real world?

One-to-one bijective correspondences for finite sets does correspond to (in a sense, is) our ordinary use of finite numbers to speak of finite collections in the real world, but yeah, since there seems to be no actual infinite things in the real world, such a correspondence can not extend in this direction

Answer 2

Your comment seems to be the nub of your problem:

I cannot think about infinity because my finite meaning cannot grasp even conceptually non finite objects, despite that there are perhaps infinite objects

I see from your stackoverflow page you are familiar with Python. So consider this very basic Python program:

from time import sleep
i = 0
while True:
  print(i)
  sleep(1)
  i += 1 

It works as you can check out.
[You can drop the sleep if you like; it just keeps the output more manageable]

Q: How many numbers does it print?
A: All elements of ℕ
Q: When does it stop?
A: Never!
[Unless you press Control-C. Or exit Python. Or shut the machine. Or mains power goes off. Or...]

Note: This program is 6 lines/74 characters — very very finite. Like your "finite meaning" in your comment.
But more correctly the syntax is finite; the meaning (also called semantics) is infinite.

Such programs are called infinite loops. For a reason.
What is there to not understand in it?

Note: This particular program is a bit silly. But all web programs that run some service that you can point your browser towards are infinite loops!

Which means they run "forever" until some external event that is the equivalent of a control-C happens.

As it happens the above shows an infinite loop ie. the infinity is in the code. Your comment suggests you want the infinity in data. This is harder to do in Python though it can be done. But if you take a modern functional language like Haskell ie. a language that sets out to have fidelity to mathematics you can write (as in Python) lists like:

[1,2,3]
[1,4,9,16]
[0..9] (which is same as [0,1,2,3,4,5,6,7,8,9])

So far it's similar to Python, except the last which you must write in Python as range(10).

However you can also write in Haskell:

[0..]

This – quite like the Python infinite loop program – will print all natural numbers. But unlike Python it's not an infinite loop but an infinite data-structure. {You can get a sloppy equivalent with Python's count iterator}.

You could say that [0..] is the Haskell equivalent of ℕ in mathematics ie. the archetypal infinite set.
[Except its not a set but a list]

Summary

• The Haskell data structure [1...]
• The Python program above
• The math set ℕ

All the above are three sides of the same concept. The name(s) mathematician give to this concept is ℵ₀ or ω.

---

Why mathematicians use these arcane names and not simply say "infinity" is a Pandora's box I'll leave alone. If you're interested look up The Transfinite and the bunch of mathematicians who don't approve of this as good, kosher math.

As for your questions regarding the paradoxical sizes of Evens, Odds and all naturals, you could start with the allegory of Hilbert's Hotel.

Answer 3

Literally, infinity is both a concept and a word- that should be clear to you from the fact that you have been thinking about the concept and have typed the word more than once in your post.

Infinity is a valid mathematical entity. Whether it has any real-world analogues is another question. The examples you have raised, such as long distances that cannot be reached owing to practical limitations, have nothing to do with infinity per se, and nor does time. We don't know if time is infinite, and whether or not it is has no relevance to the validity of infinity in mathematics.

If you write off infinity as an empty word because it is not applicable to your limited understanding of reality, then you can write off vast areas of mathematics on the same basis; but that will not diminish the validity of mathematics one jot.

Answer 4

Infinity is a concept which has been clarified by mathematics.

• Definition 1: Two sets have the same number of elements (i.e. the same cardinality), if there is a bijective map between both sets.

  As a consequence the set of all natural numbers and the set of all even natural numbers have the same cardinality: Map each natural number to its double.

• Definition 2: A set is infinite (i.e. has an infinite cardinality), if it has a proper subset of the same cardinality.

  According to this definition the set of all natural numbers is infinite: It has the proper subset of even numbers which has the same cardinality.

Developing precise concepts to deal with infinite sets is due to Georg Cantor from the 19th century. It takes some time to understand his concepts and ideas about infinite sets like cardinals and ordinals. But he was the first, and his work was groundbreaking.

Aristotle accepted infinity as the possibility to go on counting (potential infinity). But Cantor accepted also infinity as a completed cardinality (actual infinity), which Aristotle did not. For a challenging introduction see infinity

Of course “infinity” is also a word from the dictionary.

Answer 5

I am not sure if this is a proper answer but here goes nothing.

I believe, Infinity is a concept, it is not a number but A CONCEPT, a concept of INCREMENT at that (the manipulation of increment beyond our limits). I personally think that is all it is. The application of increment. To show this I will make a slight detour into math. It is actually logically established that some infinities are bigger than other infinities. This statement actually fully encapsulates the core philosophical concept of infinity which is increment or the rate of increment usually in terms of value in numbers. George Cantor with his famous Cantor's diagonal shows that some sets of infinity are bigger than others. Basically, from my perspective, by showing the set of rational numbers (therefore the infinity in this set) is bigger than the set of integers (again the infinite number in this set) by just taking these two cases you can see that by decreasing the increment from whole jumps (from 1 to 2 in the case of integers) instead we have decreased the rate of increment from 1 to 1.(infinite zeros)1 then to 1.(infinite zeros)2 for the case of rational numbers, this is an extreme case but you see my analogy. (Similar yet slightly different method can be used for comparison between rational and real numbers, the method kind of fails when comparing complex numbers because it introduces another dimension)

Another interesting application of infinity as a concept is Calculus, where we can find the areas for different regions one of the notable ones being the circle (pi) by infinitesimally (i.e., decreasing the rate of increment just like my analogy for rational numbers) to create a sense of continuity and consecutively measure the perimeter/area of region accurately, to the tee (to the point it shows us the "objectively" accurate measurement). I have an assumption that Pi is irrational because of accuracy that arises from the application of infinity. (The actual full number of Pi is not known because, you guessed it, it continues infinitely. We still are on the quest to reach the longest digit calculated; so far we have 100 trillion digits, thanks to Google Cloud)

Another spectacular example, would be e(2.71..) which can technically be written as:

If one considered infinity to be a number and therefore its interaction with operators (arithmetic) to be like other numbers one would first calculate inside the bracket to obtain 1 and the result would be 1 but this is not the case infinity being a different concept changes the usual arithmetic/math, that one uses with finite numbers.

I personally think that infinity developed as a concept in human beings as soon as numbers and therefore counting emerged. Our ancestors might have realized (as heuristics) that one can keep counting (start from 1 increase to 2 then 3 and so on and so forth) i.e., test the limits of counting and in doing so establish that one can continue forever and therefore packaged this realization into the concept of infinity. (Pure unfounded assumption, take with a grain of salt)

Answer 6

Your comments about it being "obvious" that there are more numbers than even numbers is a bit misguided.

The fact that there is a one-two-one correspondence shows that the Cardinality of the Natural Numbers is the same as the Cardinality of the Even Natural Numbers.

It is not obvious that the Cardinality of one should be different than the other. Cardinality is not "size", or "amount". It is a statement about the existence of a bijection/surjection/injection.

Your problem is that you are confusing the word "cardinality" for the word greater. This is incorrect.

There are many notions of greater in mathematics, besides cardinality. It is unfortunate that in common media, Cardinality is expressed as expressing an amount of something.

It ends up computing amounts in the finite case, but it is a much more generalized concept than amount- and there isn't any reason to expect it to look anything like a notion of amount in general.

Of course, even if it was in general a notion of amount - there is no reason to expect that non-finite amounts behave anything like finite amounts.

Your analogy is conceiving of infinity as a never ending process, and noticing that even if two processes are never ending- one may "grow" faster than another.

This is a concept well known to mathematicians, and is covered in the case when they study infinite sequences, a faster growing sequence "dominates" a slower growing sequence. The study of limits in Calculus takes this notion, and formalizes it to great effect. In this sense of infinity, we see many mathematical models projecting behavior as things get arbitrarily large or small- which is essentially the defining notion of calculus. Infinity here, means arbitrarily large. If the universe will last forever, then it is possible this notion of infinity is realistic. As, it makes sense to say- after an arbitrarily large amount of time, x happens. etc

This is a very distinct notion than the set theoretic infinity, which is not about a never ending process. Set Theoretic Infinity, is about considering collections of objects, defined by a property- for which infinitely many distinct objects satisfy.

So, if we take the collection of "all events that will happen in the universe"- it's possible that such a set is, literally infinite- and makes sense as a claim about the real world.

Answer 7

Since your question directly refers to an issue of 'meaning', my answer begins with a reference...

Michael Dummett's excellent paper on Intuitionist logic...

https://sites.unimi.it/zucchi/NuoviFile/Dummett73.pdf

...where you will find this quote (first sentence of the second paragraph)

"Any justification for adopting one logic rather than another as the logic for mathematics must turn on questions of meaning."

As Dummett makes clear, he holds to Wittgenstein's idea that, with regards to language, meaning is use.

Dummett was also a scholar of the works of Frege...the author who most clearly expounded the 'modern' idea of translating 'concepts' into logical structures. In Dummett's paper we find this quote:

"These considerations appear, at first sight, to be reinforced by reflection upon Frege’s dictum, ‘Only in the context of a sentence does a name stand for anything’."

Therefore, the meaning of the word 'Infinity' must be understood from its use, from the context within which we find it.

Since mathematicians make extensive use of the word 'infinity' both explicitly and implicitly, one place to gain some understanding of its meaning would be in mathematics literature. What you will find there is a 'mainstream' explication of infinity via the ideas of Cantor. Set theory is assumed as foundational in much of mathematical literature. You will also find alternative ideas related to infinity that are not in agreement with the Cantor/Weierstrass/Hilbert view of a timeless Platonic realm (that most mathematicians take for granted). Many of these alternatives are related to the word 'infinitesimal'. An excellent reference for an understanding of the 'infinitesimal' is a work by J. L. Bell

https://publish.uwo.ca/~jbell/basic.pdf

In this paper Bell addresses the connection between the ideas 'infinite' and 'continuum'. He makes clear the distinction between a 'punctate' versus a 'non-punctate' view of the continuum. This distinction is significant for any understanding of the word 'infinity'.

As for my own understanding as it applies to your question I offer this:

There is no such thing as a perfectly impermeable boundary in Nature. Boundaries only appear to the bounded. The unbounded by definition 'knows' no boundary. Therein lies infinity. Therefore, 'infinity' like 'true' is essentially and permanently beyond 'definition' by us, the bounded. Any attempt to arrive at a definitive result ends in an undecidable, self-referential conundrum. The best we can offer are pragmatic, provisional proposals justified by their utility for purposes we choose based on cultural values.

For those advocates of Cantor's ideas I offer this question: Given the usual two-dimensional array of binary digits shown in the explication of Cantor's diagonal argument...one has to ask: why is it that the string of binary digits in each row is assumed to be a 'completed infinity' instead of the quite obvious: either both the rows and the columns of binary digits are 'complete' or neither is, and by the diagonal argument clearly neither can be. Infinity cannot be bounded. As far as I am concerned, the argument based on that 2D array is an affront to common sense (and to my eyes). It is a cultural artifact of late 19th century Europe.

Answer 8

The one-to-one correspondence approach in set theory is based on Cantor-Hume principle, which postulates that two sets, whose elements can be set in one-to-one correspondence are equally big. On this princeple is based the Cantorian cardinality.

But there is an alterntive to this approach, and it is called Euclid's principle, which asserts that the whole is greater than the parts. It was Galileo who first pointed out the contradiction of Euclid's principle and the principle of one-to-one correspondence in his work about perfect squares. In this context people often talk about numerocity of a set. For instance, the numerocity of even numbers is half of the numerocity of integers.

As you correctly noted, while cardinality can be defined on sets of any objects regardless of their position and order, to compare numerocities you need to introduce order or metric on the sets you are comparing (what you are calling "time").

For instance, a more densely distributed infinite subset of integers has greater numerocity than the less dense one. The numerocity of the odd positive integers has numerocity greater than the numerocity of even positive integers by 1/2 because one set is shifted towards zero, compared to the other.

Answer 9

Perhaps it's best to think of infinity as being foremost an abstraction. Yes, it's something we can't touch and feel. But there are many similar ideas that we find it useful to talk and reason about: consider, for example, a point (in space or time). Or for that matter, negative numbers.

Your question asks "I ask myself if infinity is really a concept rather than a word coming out from an experience." Well, the point about abstractions (and the words we use to identify them) is that they extrapolate from our direct experience and enable us to draw valid conclusions about the world beyond our direct experience. That's true both for mathematical abstractions like infinity and zero, and for abstractions used in other fields of intellectual endeavour, like the concept of a "legal person" or a "mental illness". It's not really meaningful to ask whether such abstractions actually "exist" (do we discover them or invent them?); the point is that they are a vital part of our intellectual apparatus and enable us to reason about the world we live in.

Answer 10

The concept of inifinity seems perplexing but we must remember that the basic definition is quite simple. It is just anything "without and end". I can construct a set S, which firstly contains the number zero and secondly if it contains the number n it also contains n+1. I put no stop condition on this set, so automatically it contains all numbers. I could add the condition that this only works up to n=10 to construct a finite set. In this case the infinite set is even easier to construct. It is hard to reason about infinities because we never see one in our daily lives, but therefore we must remember an infinite thing is just something without end.

For example, you could ask yourself is the universe infinite? This is a daunting question, but you must remember that either the universe 1) has a boundary, 2) it has no boundary and is infinite or 3) it has no boundary and wraps around to itself (like a circle). An infinite universe seems weird until you realize the alternative: a universe which somehow ends abruptly??

Finally I'd like to show why your reasoning

But it's obvious that the fact that there is also odd numbers implies that the set of all numbers is greater than the set of even numbers.

fails because of our finite intuition. Let's say we have two sets: S1 = {1,2,3,4,5,6} and S2={2,4,6}. To prove that S1 is larger than S2, we could go two ways. We could relate the size of set S1 to S2 and show that it is bigger, or we could show that it is impossible to construct a bijection between S1 and S2, i.e. we pair up every element in S2 with an element in S1, but at the end there are still elements left over.

Let's try the first method. Let's call the size of set S1 n and that of S2 m. We know that every element in S2 is also in S1, so n is larger than or equal to m. We can also find an element of S1 that is not in S2, therefore n is larger than m and we have shown that S1 is larger.

Now, for an infinite set this reasoning fail! We cannot assign a number to the size of the set, so we have to resort to our second method: either finding or not finding a bijection. This then tells us that in the infinite case, the number of integers is the same as even integers!

Answer 11

Abraham Robinson is on record in several of his writings as stating that an infinite set is "literally meaningless". In his article

Robinson, Abraham. Concerning progress in the philosophy of mathematics. Logic Colloquium '73 (Bristol, 1973), pp. 41–52, Stud. Logic Found. Math., Vol. 80, North-Holland, Amsterdam-Oxford, 1975

he clarified that what he means by this is that they "lack a reference" (i.e., possess no referent). This is consistent with Robinson's philosophy of Formalism (he was certainly no Platonist).

He usually goes on to say that we should do mathematics as usual, i.e., proceeding "as if" infinite sets existed. The mathematics we do with such entities is not meaningless.

Answer 12

All mathematical objects are purely conceptual. Math defines conceptual worlds built on sets of axioms and builds, via logic, more and more complex statements that hold in that conceptual world. Those concepts sometimes have an isomorphism to the real world, but even in simple cases, the isomorphisms break down. For example, a "perfect circle" doesn't exist. Or more simply, 1 apple minus 2 apples equals -1 apples, but negative apples don't exist in the real world. Even further a trivial statement like "1 apple = 1 apple" is not exactly a real thing, two identical apples don't exist, so the number 1 is conceptual as well.

Infinity is just more obviously conceptual, but that doesn't make it invalid as a concept or not-useful in mathematical and logical reasoning.

Answer 13

Infinite set has various rigorous definitions in mathematical theories. It’s a property of sets, but each infinite set is considered an abstract mathematical object. Classes of infinite sets, such as countably infinite sets (like the natural numbers) and continuous sets (like the real numbers) are considered equivalent (that is, the elements of any two sets in the class can be put into one-to-one correspondence with each other; they are the same size and have the same structure).

The most common framework used in mathematics, Zermelo-Fraenkel set theory, has an Axiom of Infinity that essentially starts with a set of zero elements, and adds the previous set in the sequence to it at each step. That is, the smallest element is the empty set, the next-smallest contains the empty set, the next-smallest contains the two smallest sets, the next-smallest contains the three smallest sets, and so on This gives you a set containing a set of zero elements, a set of one element, a set of two elements, and so on up. If you don’t assume that such a set exists, it’s not possible to prove from the other axioms of the theory.

Richard Dedekind instead defined an infinite set as one that has the same number of elements as a proper subset of itself; that is, a set that you can remove at least one or more elements from and still have as many elements as before. This is today called Dedekind-Infinite.

The ∞ and -∞ symbols are also used in analysis, in series (pronounced “from one to infinity”) and limit notation (pronounced “the limit as x approaches infinity”), both of which also have rigorous definitions. This is not considered a number, set or mathematical object on its own, but as a shorthand for a statement that the operation converges closer and closer to some value for all large enough bounds.

Answer 14

Aristotle distiguished between the All and the Infinite. The former is understood just what its conventional meaning states - it is everything. Whilst infinite means not finite. So no matter how much you have of something that is infinite, it is always possible to add more.

Another meaning of infinite is in Christian/Islamic theological discussion where human beings, as well as any created thing is called finite in opposition to God who is infinite.

In the early 20th Centiry two mathematical formalisations of mathematical infinity were found. These are the cardinals and the ordinal. Both generalise ordinary counting of finite sets to infinite sets.

So no, it is not empty of meaning.

Answer 15

Infinity is a concept. A concept is that which can be indicated but not necessarily observed.
To play peek-a-boo with a baby is to experience a tiny moment of absolute joy. Over and over you hide your face, and when you reveal it, both you and the child enter a world devoid of all negative thought or emotion. In that moment laughter and love rule, excluding all other forms of consciousness. It transcends language, and all differences and motivations. It is a universal human activity, impossible to experience with any human except one in the most primal state of existence. A baby. That got me to thinking…

A Finite Number of Words on Infinity/singularity (Or Peek-a-Boo With God) Infinity and singularity are mere perceptions of the same condition. This is because there can be no position relative to the edge of infinity (because the edge of infinity does not exist), so all that comprises infinity must be at the exact same point, which is of course a singularity. Infinity/singularity does not exist in space time, yet space time would not have come into existence as we know it without it.
Space time is the exception to infinity, and therefore the exception to singularity. We know this to be possible, because infinity comprises infinite possibilities, and exception is one of those possibilities. The conditions previous to the big bang were by necessity outside space time, and therefore were infinity/singularity. I submit that as cold is the absence of heat, infinity/singularity is the absence of space time, at least from the space/time perspective.
At universal singularity, space time physics does not “break down,” it is simply no longer relevant. All that can be said of infinity/singularity for certain, is that beyond concept, it does not exist in the space time universe, and the space time universe cannot directly observe infinity/singularity. Because all that exists in space time is derived from infinity/singularity, the very concepts of infinity/singularity, and space time must exist in infinity/singularity. Therein lies the Great Mystery. What would motivate infinity/singularity to bring into existence, or allow to come into existence, the obviously inferior state of space time? We cannot extrapolate that motivation from space time, so like a good movie, we must wait until the end of space time to find out for sure. Anecdotally, I suspect infinity/singularity wanted to know if it could be consciously perceived by a separate entity, which of course cannot exist in infinity/singularity. It therefore created space time to contain the separate entity, which does in fact perceive infinity/singularity, although it cannot directly observe it. You cannot play peek-a-boo by yourself, inside or outside of space /time.