I am taking an introduction class on set theory. We have formally constructed the natural numbers, integers, rationals and reals. I am now trying to think of how to define the complex numbers inside of set theory. My idea is to follow the idea of how the reals were constructed from Dedekind cuts. The complex numbers would be Dedekind cuts of the real numbers that allow for the imaginary numbers to be defined. I am not sure exactly how to formally explain this or if this would be the correct way of going about defining the complex numbers. Any answers and comments would be greatly appreciated. Thank you in advance.
[Math] Construction of Complex Numbers Inside of Set Theory
complex numberselementary-set-theory
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There are a few possibilities, but here is the one approach. Even the starting point—the set of natural numbers $\mathbb{N}$—can be defined in several ways, but the standard definition takes $\mathbb{N}$ to be the set of finite von Neumann ordinals. Let us assume that we do have a set $\mathbb{N}$, a constant $0$, a unary operation $s$, and binary operations $+$ and $\cdot$ satisfying the axioms of second-order Peano arithmetic.
First, we need to construct the set of integers $\mathbb{Z}$. This we can do canonically as follows: we define $\mathbb{Z}$ to be the quotient of $\mathbb{N} \times \mathbb{N}$ by the equivalence relation $$\langle a, b \rangle \sim \langle c, d \rangle \text{ if and only if } a + d = b + c$$ The intended interpretation is that the equivalence class of $\langle a, b \rangle$ represents the integer $a - b$. Arithmetic operations can be defined on $\mathbb{Z}$ in the obvious fashion: $$\langle a, b \rangle + \langle c, d \rangle = \langle a + c, b + d \rangle$$ $$\langle a, b \rangle \cdot \langle c, d \rangle = \langle a c + b d, a d + b c \rangle$$ (Check that these respect the equivalence relation.) Again, this is not the only way to construct $\mathbb{Z}$; we can give a second-order axiomatisation of the integers which is categorical (i.e. any two models are isomorphic). For example, we may replace the set $\mathbb{Z}$ by $\mathbb{N}$, since the two sets are in bijection; the only thing we have to be careful about is to distinguish between the arithmetic operations for $\mathbb{Z}$ and for $\mathbb{N}$. (In other words, $\mathbb{Z}$ is more than just the set of its elements; it is also equipped with operations making it into a ring.)
Next, we need to construct the set of rational numbers $\mathbb{Q}$. This we may do using equivalence relations as well: we can define $\mathbb{Q}$ to be the quotient of $\mathbb{Z} \times (\mathbb{Z} \setminus \{ 0 \})$ by the equivalence relation $$\langle a, b \rangle \sim \langle c, d \rangle \text{ if and only if } a d = b c$$ The intended interpretation is that the equivalence class of $\langle a, b \rangle$ represents the fraction $a / b$. Arithmetic operations are defined by $$\langle a, b \rangle + \langle c, d \rangle = \langle a d + b c, b d \rangle$$ $$\langle a, b \rangle \cdot \langle c, d \rangle = \langle a c, b d \rangle$$ And as before, we can give an axiomatisation of the rational numbers which is categorical.
Now we can construct the set of real numbers $\mathbb{R}$. I describe the construction of Dedekind cuts, which is probably the simplest. A Dedekind cut is a pair of sets of rational numbers $\langle L, R \rangle$, satisfying the following axioms:
- If $x < y$, and $y \in L$, then $x \in L$. ($L$ is a lower set.)
- If $x < y$, and $x \in R$, then $y \in R$. ($R$ is an upper set.)
- If $x \in L$, then there is a $y$ in $L$ greater than $x$. ($L$ is open above.)
- If $y \in R$, then there is an $x$ in $R$ less than $y$. ($R$ is open below.)
- If $x < y$, then either $x \in L$ or $y \in R$. (The pair $\langle L, R \rangle$ is located.)
- For all $x$, we do not have both $x \in L$ and $x \in R$. ($L$ and $R$ are disjoint.)
- Neither $L$ nor $R$ are empty. (So $L$ is bounded above by everything in $R$ and $R$ is bounded below by everything in $L$.)
The intended interpretation is that $\langle L, R \rangle$ is the real number $z$ such that $L = \{ x \in \mathbb{Q} : x < z \}$ and $R = \{ y \in \mathbb{Q} : z < y \}$. The set of real numbers is defined to be the set of all Dedekind cuts. (No quotients by equivalence relations!) Arithmetic operations are defined as follows:
- If $\langle L, R \rangle$ and $\langle L', R' \rangle$ are Dedekind cuts, their sum is defined to be $\langle L + L', R + R' \rangle$, where $L + L' = \{ x + x' : x \in L, x' \in L' \}$ and similarly for $R + R'$.
- The negative of $\langle L, R \rangle$ is defined to be $\langle -R, -L \rangle$, where $-L = \{ -x : x \in L \}$ and similarly for $-R$.
- If $\langle L, R \rangle$ and $\langle L', R' \rangle$ are Dedekind cuts, and $0 \notin R$ and $0 \notin R'$ (i.e. they both represent positive numbers), then their product is $\langle L \cdot L' , R \cdot R' \rangle$, where $L \cdot L' = \{ x \cdot x' : x \in L, x' \in L', x \ge 0, x' \ge 0 \} \cup \{ x \in \mathbb{Q} : x < 0 \}$ and $R \cdot R' = \{ y \cdot y' : y \in R, y \in R' \}$. We extend this to negative numbers by the usual laws: $(-z) \cdot z' = -(z \cdot z') = z \cdot -z'$ and $z \cdot z' = (-z) \cdot -z'$.
John Conway gives an alternative approach generalising the Dedekind cuts described above in his book On Numbers and Games. This eventually yields Conway's surreal numbers.
Since we start with an axiom C: $(\exists i)i^2=-1$ which does not explicitly contradict any other axiom in $\mathbb R$ (since $i$ does not appear in other axioms), there is no way to directly disprove it. But in a slight modification to the aforementioned, there are some axioms which are actually disproven by (C), like trichotomy:
$$(\forall x)[x<0\vee x=0\vee x>0]$$
(read "every number is either negative, positive, or zero".) Because $i^2=-1$ and $0^2=0$ implies $i\neq0$, $i>0$ implies $i^2=-1>0$ implies $1<0$ implies $1^2>0$ which is a contradiction, and $i<0$ implies $i^2=-1>0$ similarly. What we do, then, is we just drop the offending axioms. Importantly, $\mathbb C$ does not contradict the "important" axioms, like commutativity and associativity of addition and multiplication, and the existence of inverses to non-zero elements. The important part is that although we can't keep everything, we can keep some things, and we can work usefully with what remains.
The problem with your example is that you have not just defined a new element $x$, but you have also defined how it multiplies, and your definition contradicts another one which has already been defined. In $\mathbb C$, we just define the part that can not otherwise be derived, and let the other axioms "figure out the rest", so that we don't have any danger of redefining things incorrectly.
By the way, you asked on the previous question what the axioms of the reals are, so I thought I'd list them here, courtesy of Spivak's Calculus.
- (addition is associative) $(\forall a,b,c)\ a+(b+c)=(a+b)+c$
- (additive identity) $(\exists0)(\forall a)\ a+0=a$
- (additive inverse) $(\forall a)(\exists b)\ a+b=0$
- (addition is commutative) $(\forall a,b)\ a+b=b+a$
- (multiplication is associative) $(\forall a,b,c)\ a(bc)=(ab)c$
- (multiplicative identity) $(\exists1)(\forall a)\ a\cdot1=a$
- (multiplicative inverse) $(\forall a\ne0)(\exists b)\ ab=1$
- (multiplication is commutative) $(\forall a,b)\ ab=ba$
- (distributive law) $(\forall a,b,c)\ a(b+c)=ab+ac$
- (trichotomy) $(\forall x)[x<0\vee x=0\vee x>0]$
- (positives closed under addition) $(\forall a>0,b>0)[a+b>0]$
- (positives closed under multiplication) $(\forall a>0,b>0)[ab>0]$
- (least upper bound) $(\forall S\subseteq\mathbb R)[\mbox{if }S\mbox{ has an upper bound, then }S\mbox{ has a least upper bound}]$
The first four just define all the "nice" properties of addition, the next four do the same for multiplication, number 9 covers the distributive property, 10-12 cover the semantics of an order relation, and number 13 "fills in the gaps" in the rational numbers. That last one is a bit complicated to write in symbols, but it allows you to say that all sorts of numbers like $\sqrt2$ and $\pi$ are actually numbers. The relationship with $\mathbb C$ is that numbers 10-12 are thrown away, but the first 9, which define a field, are still okay. Number 13 makes explicit reference to $\mathbb R$, and it needs an ordering to work properly, so we just leave that one as-is. That way you can still have numbers like $i\pi$ and identify them as elements of $\mathbb C$.
I know you didn't want so much notation all at once, but nothing in here is too out there to get from Intro Calc. (By the way, $\forall$ means "for all", $\exists$ means "there exists", and $\vee$ means "or", in case you haven't seen those before.)
Best Answer
The simplest thing is just to define a complex number to be an ordered pair $(x,y)$ of reals. Define the sum in the obvious way, define the product $$(x,y)(a,b)=(ax-by,ay+bx),$$and show you have a field.
Then $x\mapsto(x,0)$ is an embedding of $\Bbb R$ into $\Bbb C$. If you define $i=(0,1)$ you get $i^2=(-1,0)$. Complex numbers.