# Can infinity be defined?

I know that one can look in a dictionary and find definitions for "infinity".

Similarly, texts in mathematics will give varying accounts of how "infinity" is treated.

So, I am not asking for these definitions. What I am asking about any potential or actual definition of infinity is whether it is not a contradiction in terms and a self-defeating exercise to define what is indefinite?

Formally, infinity is a negation of finity. But negation is not defining. And to refer infinity to an unlimited iteration of n=n+1 is simply to introduce circularity into the discussion; for what is unlimited but a negation of limited?

It may well be that the notions of God and infinity are very closely connected, and perhaps have a common history. This would then raise questions about the definition of God. Is it not self-defeating to attempt to define the transcendental, for the defined transcendental renders it immanent? In the same way, an instantiated infinity, an infinity one can refer to, is not infinite.

• Also useful : A.W.Moore, The Infinite (2012). – Mauro ALLEGRANZA May 27 '18 at 11:06
• "But negation is not defining" In what sense ??? We define even a number that... And then we define as odd a number that is not even. What is wrong in this ? – Mauro ALLEGRANZA May 28 '18 at 6:21
• What is your definition of a definition? It's kind of a mean question, I know, but if you are refusing to accept the accepted formal definitions for a word on the grounds that you don't think they're valid, you are indeed using a definition of definition that most do not use. Understanding what you think it means would be helpful. – Cort Ammon May 28 '18 at 15:36
• I will say, my personal favorite definition to go down the line I think you're going is the Dao. Famously, "The Dao that can be written is not the eternal Dao." And yet there is a symbol, 道, which is defined to be precisely that. – Cort Ammon May 28 '18 at 15:38

If you reject all definitions of what we would use the word "infinity" to mean, then it's tautological that "infinity" cannot be defined. Similarly for the word "infinite".

• I don't know if you wrote this before or after my edit, but I understand infinity to be the negation of finity.. As a definition, that is fairly useless, like having a concept of unbrown. – user33399 May 28 '18 at 22:26
• @HermanHofman "Not brown" is a perfectly reasonable way to define a concept. Whether that concept is useful is a different story, but that's not what you asked. The utility of the concept given by a definition is unrelated to the meaningfulness of the definition, and a definition via negation is perfectly meaningful. – Noah Schweber May 30 '18 at 20:11

Mathematics has a long and colourful history of dealing with infinity. If you know some real analysis, then you know that Cauchy (b. 1789) was the first to make rigorous our account of sequences and series, and even functions and calculus, and he did this by being the first to understand a very subtle and elegant way of expressing the infinitesimally large and small.

With a_n as a sequence and a being its limit, we take the deceptively simple expression to mean the following While initially impenetrable, this is the perfect definition for something dealing with the extremely small and large. It means that no matter how small a distance you choose, a truly converging sequence will always remain with that distance from its limit for the rest of infinity, provided you start looking at the sequence at the right point.

Infinity has always been shorthand for the unbounded: it was an important discovery to note that an unbounded function could be defined by always eventually surpassing any finite limit you might set, for example. Being human, we can more naturally work with the finite*. This is how Cauchy was able to nail down infinities: by taking finite quantities and showing that the eventual surpassing of a limit, or eventual containment within a limit, was enough for us mere mortals to harness the concept to our advantage.

*Prior to Cantor.

• I don't think the OP is looking for a mathematical definition. – Frank Hubeny Dec 23 '18 at 11:47

Here is the question:

So, I am not asking for these definitions. What I am asking about any potential or actual definition of infinity is whether it is not a contradiction in terms and a self-defeating exercise to define what is indefinite?

The OP is not looking for a definition of infinity, but whether infinity can be defined. Also the OP is not interested in a mathematical definition or one which defines the infinite as the negation of finitude. Here is the OP's concern:

It may well be that the notions of God and infinity are very closely connected, and perhaps have a common history. This would then raise questions about the definition of God. Is it not self-defeating to attempt to define the transcendental, for the defined transcendental renders it immanent? In the same way, an instantiated infinity, an infinity one can refer to, is not infinite.

Dominic J. O'Meara provides a view from Plotinus of what is involved in returning to the One, Plotinus's notion of God. Our knowledge including our definitions are at best derivative. O'Meara cites the Enneads IV. 8. 1. 1-10: (page 104)

The arguments [that allowed us to return to our true self as soul] lead us even further. They not only show soul what it is; they also lead it to see that the knowledge it has is derivative, that it derives from a higher form of thinking, the divine intellect which, unlike it, does not need to work through long logical processes, but possesses knowledge in a different and superior way.

Whatever we have for definitions are derivative from this higher form of thinking, Furthermore these arguments must be left behind: (page 105)

It should be noted that if the transformation of our lives that comes with becoming intellect is facilitated by philosophical arguments, these arguments must also be left behind: they make use of logical processes and are not required in the perfect knowledge of intellect.

This suggests, as the OP suspects, that any definition of God (or infinity) may need to be left behind at least given the neo-Platonism presented by Plotinus.

O'Meara, D. J. (1995). Plotinus: an introduction to the Enneads. Oxford University Press.

Assuming the question is regarding the very concept or property of Infinity rather than a particular instance or object. Consider what @Sputnik, above, has mentioned and try to apply Cauchy: What would the finite quantity to be surpassed be? An enumeration all possible infinities? But then the definition would contain the term, or more likely simply fail to be a well formed formula. So Infinity cannot be defined without resorting to recursion. Mostly because "by definition" we cannot know all types of infinities.

Yes, similar arguments can be applied to God: I.e. arguments cannot be applied to God

Formally, "infinite" is not just the negation of "finite". An infinite set can be defined as one that can be placed into one-to-one correspondence with a proper subset of itself. That's a definition that doesn't even refer to finite sets, but defines a particular property that infinite sets have and finite sets don't. This leads to a branch of set theory that is, as far as we can tell, not self-contradictory (there's always the chance that someone will come along and disprove most of math, but I'm not worried that it will happen). They can be counter-intuitive, but that's true of lots of math and physics.

Moreover, "definition" and "indefinite" aren't antonyms. You can look in a dictionary and get a reasonably precise definition of "vague". It's perfectly reasonable to have dictionary definitions of "indefinite" and "infinite".

• To avoid the unnecessary complexity of subsets, the infinity in the Axiom of Infinity of ZF is often even simpler. There is a set such Z such that if x is in Z then there is some y in Z with x in y. A set is infinite if it can be mapped onto some set that has this property. Things are finite if they aren't infinite. – user9166 Dec 23 '18 at 7:46