# How is 0 defined?

I know that the naturals are assumed by the axiom of infinity, but the relationship between them (eg 1+0=1), must be rule based or defined at the very least.

Basically I want to know what makes 0 or 1 abstract. For me to figure that out, I must also have their precise definitions?

I know that 0 is an abstract concept, but I want to know what or what not in its definition makes it so.

Is abstraction another name for commonality?

• In which system? Peano? Von Neumann? ZF? The axiom of infinity is usually an axiom of ZF. Apr 6, 2018 at 8:50
• You should change the title, because it attracts mathematicians who give the definition but don't help you with your actual problem of abstraction. Apr 6, 2018 at 11:15
• All numbers are "abstract". Do you have a particular reason for asking about 0 and 1 specifically?
– user6559
Apr 6, 2018 at 12:23
• This question is impeccably vague. What does "Basically I want to know what makes 0 or 1 abstract." purport to mean? If what you want is the definition of a number then you should ask for that; everything else about "abstraction" and "commonality" in this question is just confusing and vague. If you want to ask a question about the nature abstract objects you should ask that question instead, and express what the terms you're using mean to you so that people can help you understand the answer to you question better. Apr 6, 2018 at 15:02
• Zero is the additive identity. It is the element x such that, for every y, x + y = y. This is true even when + assumes a different meaning than that of ordinary addition. Apr 6, 2018 at 18:08

According to Peano's axioms zero is the number which is not the successor of a number. For each natural m, addition by n is defined by induction on n:

m+0 := m, m+(n') := (m+n)'

here the symbol ' denotes the successor.

According to von Neumann's definition of natural numbers, the number zero is the empty set, and the number 1 is the set with single element the empty set. And in general: the successor of n is the union of the set n and the set with the set n as a single element: n+1 = n ∪ {n}. See https://en.wikipedia.org/wiki/Natural_number#Von_Neumann_construction

I know that von Neumann's definition seems a bit strange. But that's only if one is not used to this construction. The von Neumann definition reduces the natural numbers - and other ordinals/cardinals - to set theory resp. class theory.

In both cases, numbers are abstract entities, i.e. general mental concepts. All mathematical objects are abstract. In particular, also the sets from von Neumann's definition are abstract.

Mathematics is a structural science:

1. Historically, mathematics started by considering only the arithmetical or geometrical aspect of objects from a certain domain of reality. This step makes the abstraction.
2. A second step in mathematics investigates the relations between the abstracted objects, e.g. the possibility to define an addition for numbers.
3. Eventually, the main content of mathematics is to study the relations which derive from the definitions, e.g., rules like commutativity which hold for the addition of numbers.
• I like the Von Neumann definition, quite intuitive and natural. Apr 6, 2018 at 6:32
• You could complete this answer by deducting from von Neumann's definition that all numbers are abstract if sets are abstract objects. (I guess sets are abstract, but I leave that to the experts.) Apr 6, 2018 at 11:18
• I see some analogy between Von Neumann's definition and the idea of the gauge boson as the particle which makes matter massive. Very elegant.
– user30980
Apr 6, 2018 at 15:44
• Von Neumann's construction shows, to answer one of novice's questions, that sets are more abstract than natural numbers. Apr 8, 2018 at 7:14
• I would call sets more general than natural numbers. Why more abstract? Apr 8, 2018 at 8:29

In broader mathematics, the defining property of 0 is that it's the additive identity — that is, adding zero to another number doesn't change that number.

This isn't inherent in the Peano axioms. The Peano axioms simply say that there is a natural number which isn't the successor of any other natural number, and that the symbol 0 represents that number. In fact, Peano originally formalized the natural numbers beginning with 1, not with 0, and the axioms were otherwise exactly the same.

But when you go on to define arithmetic and the `+` operator, you gain the axiom `x + 0 = x`. When extending mathematics to integers, rationals, complex and surreal numbers, etc. the concept of a "first" number quickly becomes useless, but the idea that there is a number which doesn't change other numbers through the operation of addition (and that this number is unique, as long as we're working in a group) remains valid and important.

• When one advances from groups to rings, one gains 1 as the multiplicative identity, and also gains the additional bonus property that multiplying anything by the additive identity yields the additive identity. Apr 6, 2018 at 22:15

In Zermelo–Fraenkel (ZF) set theory, the natural numbers are defined recursively by letting 0 = {} be the empty set and n + 1 = n ∪ {n} for each n. In this way n = {0, 1, ..., n − 1} for each natural number n.

With the above definitions and the Axiom of infinity the theory defines the set ω of all natural numbers, and then proves that it satisfies the Peano postulates.

Finally, the Recursion theorem is used to define addition and multiplication on ω.

For details see e.g.:

Gottlob Frege and Bertrand Russell each proposed defining a natural number n as the collection of all sets with n elements. More formally, a natural number is an equivalence class of finite sets under the equivalence relation of equinumerosity. [...] This definition works in naive set theory, type theory, and in set theories that grew out of type theory, such as New Foundations and related systems. But it does not work in the axiomatic set theory ZFC and related systems, because in such systems the equivalence classes under equinumerosity are proper classes rather than sets.

• Aren't in ZF natural numbers treated as a subset of cardinal numbers rather than sets themselves? So `0 = card({})` and so on? Apr 6, 2018 at 9:01
• @Ister No, in modern logic ordinals are sets. Cardinals are equivalence classes of ordinals that can be mapped into one another both directions, but they are represented by their lowest ordinal. So for ZF 0 = {}. In some sense the cardinality of {} is exactly {{}}, so in a more algebraic sense card({}) = min({{}}) = {} = 0
– user9166
Apr 6, 2018 at 17:55

One common formulation of natural numbers is the Peano axioms.

• Axiom 1 defines 0 as a natural number.
• Axiom 6 states that if n is a natural number, then S(n) is a natural number. This is what allows the natural numbers to be infinite.

The Peano axioms define zero as a base case, because it is the only natural number that is not of the form S(n), so it prevents expressions from being infinite.

Here's another way to do it, using set theory:

• The natural number n is represented as the set of all natural numbers less than n, expressed in the same representation.

This way, zero is `Ø`, one is `{Ø}`, two is `{Ø,{Ø}}`, three is `{Ø,{Ø},{Ø,{Ø}}}`, etc.

This doesn't explicitly have zero as a base case, but it is the only set that is not required to contain other sets, so it prevents expressions from being infinite.

This representation has the advantage that m < n is represented as m ∈ n (since n contains all the naturals less than it). We can define the successor of n to be n ∪ {n}, which defines a new set with the elements of n, plus n itself.

In both cases, if we have some representation of 0 and some predecessor operation for nonzero naturals, we can define addition as follows:

Given two numbers n and m:

• If n = 0, then n + m = m.
• If n = S(k), then n + m = S(k + m).
• Could you please also tell me what S() is defined as? Apr 6, 2018 at 6:44
• @novice S is the successor function in whatever axiom system we're using. In the case of the Peano axioms, it's integral to them and thus can't be "defined" any more than 0 is "defined." We're just saying that it exists and is a member of the set. In the set-theoretic viewpoint, it's just the set-theoretic operation that I described. Apr 6, 2018 at 6:46
• What does S(n) mean? the successor of n? Some operation or axiom must give the successor right? Apr 6, 2018 at 6:47
• @novice One of Peanos axioms states that every number n has a successor S(n). Apr 6, 2018 at 15:00
• The existence of zero does not prevent expressions from being infinite. Only induction and the fact that \$0\$ is not a successor prevents that (e.g. you can prove by induction that \$n \not= S(n)\$ for any \$n\$). Apr 7, 2018 at 12:18

You got some nice descriptions of the Peano or the ZF/von Neumann number systems already.

I know that the naturals are assumed by the axiom of infinity,

That is a weird way to put it, maybe I am just not understanding it correctly. The AoI closely resembles the purely mechanical way in which numbers are constructed in the von Neumann system. It puts forth that there is an infinite set containing all sets which represent individual natural number; it is necessary because no natural number is itself infinitely large.

Maybe you are thinking about the Axiom of Induction, which constructively gives you all naturals?

but the relationship between them (eg 1+0=1), must be rule based or defined at the very least.

That is true. Numbers in general are rule based, and well defined. There is very little "philosophical" in the number constuctions of Peano or the ZF/von Neumann variation.

Basically I want to know what makes 0 or 1 abstract. For me to figure that out, I must also have their precise definitions?

Peano Axioms, which define or construct the natural numbers, are what you are looking for.

• The very first axiom reads `0 is a natural number.`.
• The 8th says `[...] there is no natural number whose successor is 0.`

Quite literally, 0 is defined to be the first natural number. As mentioned elsewhere, there were earlier variants where Peano started with 1, and it did not change much. I guess it was changed to 0 later because of practical reasons, mostly.

I know that 0 is an abstract concept, but I want to know what or what not in its definition makes it so.

0 is not more abstract than any other mathematical term. There is nothing special about it - it is as abstract as 1, -1, infinity or whatever else you could think of.

That said, in both Peano as well as ZFC constructions, 0 or {} are actually less abstract than any other natural number in the sense that they are the only natural numbers that are defined explicitely as such (without resolving to a successor construction).

Is abstraction another name for commonality?

Wikipedia to the rescue: Abstraction `in its main sense is a conceptual process where general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or "concrete") signifiers, first principles, or other methods.`

Abstraction is not a deep philosophical term, but simply and exactly what the definition of the word says. You go from concrete examples (a bag with 3 eggs, another bag with 5 eggs) and figure out a concept of natural numbers which tell you how you can systematically work with bags of any number of eggs, even 1000000 eggs, which you would never see in the real world.

I won't tackle the question whether abstractions are real. ;) Philosophers of all ages (which I am not one of) seem to have fought long and hard about it.

Commonality is not the same: that just means that you have a bunch of objects which have some identical features. The moon is yellow, cheese is yellow. Yellow is a commonality between the moon and cheese, but there probably is no rule, concept or theorem linking the two.

• But there is the commonality which lies underneath. For example, I say that "3 apples", and "3". 3 apples + 1 apple = 4 apples, and 3 + 1 = 4. So the term 3 apples, has some features in common with the number 3, and has an extra feature that it applies to "number of apples", in the bag. Apr 7, 2018 at 23:28
• So in Wiki's definition of abstraction, the general rules are what are common between specific examples. Apr 7, 2018 at 23:35

There are different degrees of abstraction: something is abstract to the extent that it omits details. The user interface for a technological device is abstract: it hides the inner workings of that device.

Basically I want to know what makes 0 or 1 abstract. For me to figure that out, I must also have their precise definitions?

Mathematical definitions of quantity are irrelevant here, because even a layman's concept of quantity is abstract. “Three” is abstract, it omits saying what is counted. “Three oranges” is abstract, because it omits saying exactly which oranges are spoken of. “The sun” is abstract, because it omits a lot of details about the nature of the sun.

According to this view, though, zero is not abstract, because it means “an empty quantity”, and there is no lacking information here, no need to wonder “an empty quantity of what?”. Similarly, the empty set is not abstract, because there is no need to answer the question “an empty set of what?”. That the empty set is not abstract may be the reason why it can be used as the “starting point” of mathematics.

If you want zero or the empty set to be abstractions, you need to use another sense of the word “abstraction”, like “imaginary concept”.

Is abstraction another name for commonality?

Abstraction is not the same as commonality, because there is only one sun, yet “the sun” is an abstraction (as explained above). Commonality may inspire the creation of abstractions, because it can make clearer which details we should omit. By seeing a lot of stars, we can come up with the concept of star, by trying to articulate what all stars have in common. “Star” omits information about physical location, in constrast with “the sun” which is “the closest star to the Earth”.

Abstraction makes us more powerful by simplifying things and focusing exclusively on the most relevant details. Clear, simple thought, at an abstract level, can be more effective than confused, complicated thought, lost in details. Abstraction harnesses the power of focus. It allows us to step back and better see the big picture. It makes easier the wielding of complicated tools (conceptual tools in mathematics, physical tools in the physical world). The more complex something is, the more details we need to hide, and the more abstract the resulting concept. Mathematics involves objects more abstract than numbers. For example, sets of numbers can be manipulated without explicit references to the content of these sets. Hiding the least relevant details is especially helpful when dealing with complicated things like sets of sets of sets. Powerful abstractions allow saying vertiginous things with just a few symbols, by hiding all the work that has been done before to create these powerful abstractions in the first place. Mathematicians stand on the shoulders of giants. (And frequently also on their own shoulders! I guess mathematicians are contortionists.)

Conceptual tools are not just used in mathematics, they are also used in natural languages. For example, the concept of “car” is a helpful abstraction: thankfully, we do not always need to mention brand, model and color when mentioning a car. This is especially useful when speaking of a lot of different cars.

An important difference between abstractions for physical things and abstractions for mathematical things is that physical abstractions are usually simplistic (because the world is so complex), whereas mathematicians strive to make their abstractions “rigorous.” For example, the abstraction “planet Earth” assumes that there is a clear spatial boundary between the Earth and the rest of the cosmos. However, the real situation seems more complicated: does the Earth include its own atmosphere? If yes, where does its atmosphere precisely stop? Is the Earth an absolutely precise zone? I am no physicist, but the uncertainty principle seems to make that doubtful. In contrast, mathematical objects seem more docile when it comes to making abstractions based on them. Mathematicians try to make their abstractions simple, but not simplistic.

Abstraction allows generality. By omitting to mention which natural number a mathematician is studying, whatever she finds out about it is true of all natural numbers. For example, let n be a natural number; then the sum of n and its successor is odd, because n+(n+1) = (n+n)+1 = 2n+1. There is no need to take all natural numbers one by one, we can speak of them all simultaneously. This is very expressive.

Abstraction allows uncoupling and flexibility. As others have mentioned, natural numbers can be defined in different ways. However, no matter the chosen definition, the behavior of the resulting numbers should be basically the same. This allows someone to study natural numbers without the need to choose a definition. If mathematicians discover a much better definition of the natural numbers, all the work that has already been done before on natural numbers may still be valid with that new definition.

Someone who wants to build a computer screen does not know which images the user will want to display. The builder only has an abstract knowledge of which images the screen should be capable of displaying. The builder knows the amount of pixels, and which colors each pixel should be able to display. By keeping in mind the image as an abstraction (“set of pixels”) instead of a concrete picture, she manages to make up an extremely versatile device, able to display nearly any image she wants.

Hopefully, this has helped you achieve a better grasp of the nature of abstraction.

0 is number which has no any importance if it used before any number,but is meaningful when it is added after any number. ex 01----no importance 10-----here 0 makes values larger than earlier value.

• I think you are talking about the digit 0 rather than the number 0. If we use the notation S() for 0 this doesn't make sense, but the properties of numbers themselves doesn't depend on notation. Apr 7, 2018 at 10:06
• I think the author is looking for a precise definition of 0, not its use. The author is also asking a question about "abstraction" and "commonality". You might want to add more to this answer. Apr 7, 2018 at 13:27
• Strictly speaking there is a case where a number prefixed with 0 is important - namely to say that it it's not in base 10. I realise that's not really relevant to the question as such but I wanted to point out that if you're talking about the digit '0' then it's not as simple as to say that it is only meaningful when it occurs before any other digits. But if this seems out of touch forgive the computer programmer in me - I just felt the need to point it out... Apr 8, 2018 at 0:17