I'm familiar with the famous excerpt from Principia Mathematica by Bertrand Russel and Alfred Whitehead. However, as Zermelo–Fraenkel Set Theory is today's most used foundation of mathematics (or ZF if you believe the axiom of Choice is implied), I was wondering if there is a postulation expressed in the symbolism of first-order logic for $2 + 2 = 4$ following ZFC. I'm not a logician, not even an undergraduate yet, and I was unable to find such a proof online, maybe I'm just overlooking some elementary understanding of what ZFC is.
[Math] How to prove $2+2=4$ using Zermelo–Fraenkel Set Theory
first-order-logiclogicset-theory
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In mathematics we have an idea, some concept. For example the idea of "set" was modeled after the idea that two collections are equal if they have the same members.
If I list your family it doesn't matter if I'm listing your father before you mother, or vice versa. I am only interested in the members. Similarly if I listed exactly the members of your family then this is a list of your family.
So sets were models after this idea. But we want more from sets, we want them to have certain properties. And at first we assumed these properties must always hold, but it turned out that there is an inherent contradiction in naive set theory.
So mathematicians of the early $20^{\text{th}}$ century suggested axioms, which are strict rules that the mathematical objects called "sets" must obey. Amongst those mathematicians there were Ernst Zermelo and Abraham Fraenkel, after which the theory is named.
Before we start, let me just add one definition. Given two sets, $A$ and $B$ we say that $A$ is a subset of $B$ if all the members of $A$ are members of $B$. For example, all the students in your class are also students in your high school, so the set of your classmates if a subset of the set of your schoolmates.
The axioms of the Zermelo-Fraenkel set theory describe the properties we expect sets to have, in a mathematical way. The incredible part is that our language only contains one binary relation symbol, $\in$, which is the membership symbol. Now remember that set theory only talks about sets, so all our objects are sets, including their members. This means that the most primitive notion is the question "Is this set a member of that set?.
This may sound baffling but we can encode numbers by sets and define all the mathematics within this universe, all this with only the membership relation to work with. The axioms tell us that sets have the following properties:
- Two sets are equal if and only if they have the same members.
- If we have a set, then the collection of all of its subsets is another set.
- If we have a set of sets $I$, then we have a set $U$ which is the union of those sets, namely all the members of $I$ are subsets of $U$, and $U$ is the smallest possible set with this property.
- There exists a set which has no members.
- If we have a set $x$ and a property $\varphi$, then the collection of all the members of $x$ with the property $\varphi$ is a set.
- If we can describe a function whose domain is a set, then its image is a set.
- There exists an infinite set$^*$.
- In every set $x$ which is not empty, there is a member $y$ which does not have any members shared with $x$.
These may sound a bit strange, but they serve a function. They tell us that some sets exist (empty set, infinite sets); they tell us how to create new sets from old sets; they tell us when two sets are equal; and they give us some strange condition on the membership relation (the last axiom).
It should be pointed that axioms 5 and 6 are not really axioms. They are schemata of axioms. This means that those are actually infinite lists of axioms which are easily described formally, and a computer can easily verify whether or not a certain sentence is an axiom from these lists or not.
Amongst the contradictions which these axioms solve is the one given by Russell, which is described by the collection of "all sets which are not members of themselves", this collection cannot be a set -- although we can describe this collection. With our axioms we have that if this collection is a set then we have a contradiction, so this is not a set to begin with.
But the axioms are important, because they give us a rigid framework for sets. Otherwise we just think of sets naively and so every collection we can describe must be a set, but Russell's collection cannot be a set. And that is the contradiction.
$^*$ I slightly dumb down this axiom, we require slightly more than just an infinite set, but it is easier to understand it this way. What we do require is that there is a set which "looks" like the collection of the natural numbers.
Best Answer
Zermelo–Fraenkel set theory is formulated in a language that has a binary relation $\in$, and that's it. The symbols $1,2,3,4$ and $+$ make "no sense" from a syntactic point of view.
So first one needs to explain what is $2$ in the context of set theory. This means that you need to define some sets that will behave like you would expect of the natural numbers. Normally, these are the finite von Neumann ordinals, defined recursively as $0=\varnothing$ and $n+1=n\cup\{n\}$. There are other ways to define the natural numbers, or you could be thinking about the real numbers, and then you need to define those as well first, but for now let's stick with the von Neumann ordinals and see where it takes us.
Note that even in the case of Peano arithmetic or ring theory, the symbol for $2$ and the symbol for $4$ are not part of the language. They are used as shorthand for either repeated sums of $1$ or repeated application of the successor function to $0$, or so on.
So if $2$ is defined as $1+1$ and $4$ is defined as $((1+1)+1)+1$, and you have an axiom saying that $(x+y)+z=x+(y+z)$, then you're practically done: $(1+1)+(1+1)=((1+1)+1)+1$, so $2+2=4$. But okay, let's go back to set theory.
We have the natural numbers, so $2=\{\varnothing,\{\varnothing\}\}$ and $4$, well, let's just write $4$. After this you need to ask what is $+$. Do you think about $+$ as a recursive definition of applying the successor function? Even the Peano axioms which usually form the basis for arithmetic have $+$ as a standalone object, and there is a connection between $+$ and the successor function. There are two standard ways to define what is $+$ on the natural numbers in the context of set theory:
Successive application of the successor function, so $x+0=x$ and $x+S(y)=S(x+y)$. In that case, $2+2=S(1)+S(1)=S(S(0))+S(S(0))=\dots=S(S(S(S(0))))$. And now we can also translate back into sets to get "a fully set theoretic statement". But it's awful, and I don't want to do it.
Using cardinality of sets. Note that $n$ is a set with exactly $n$ elements. $0$ has no elements and $1$ has exactly one element (which is $0$, as it turns out). So we can define $n+m=k$ if and only if the number of elements in a disjoint union of a copy of $n$ and a copy of $m$, is $k$. Or, in modern terms, there is a bijection between the two sets.
In that case, one needs to write down a function from $\{\langle 0,0\rangle,\langle 0,1\rangle\}\cup\{\langle 1,0\rangle,\langle 1,1\rangle\}$, or $\{0\}\times2\cup\{1\}\times2$, and $4$ which is $\{0,1,2,3\}$. Of course, this starts with defining what is an ordered pair from a set theoretic view, what is a function, etc., and then write this as a set theoretic expression again. Which, as before, is an awful exercise in futility.
The easiest way, at the end, is to prove this "by blocks". First prove there is a way to formalize the natural numbers, then formalize addition, then show that no matter what formalization you chose, as long as it satisfies certain properties (which you would expect from the natural numbers), it has to be the case that the object corresponding to $2+2$ is the object corresponding to $4$.
Or just be annoying and declare that you interpret $2,+$ and $4$ in a non-standard way so $2+2=5\neq 4$. Whatever floats your boat.