Short Answer
The statement is made true almost directly from an assumption when we create the integers and is thus not the result of a theorem one would (normally) prove. For example, the result might follow from the assumption that a number we create is not the same as one previously created, or it is an assumption that can be easily derived as the specific case of an axiom.
Long Answer
The first thing we have to do is determine how we wish to view the integers. Possibilities include viewing them
as thoughts that developed in the human mind,
as concepts that developed historically,
as concepts that we can construct from a definition of number and
certain operations, such as a successor function, addition, or
subtraction,
in terms of a model for the integers, or
in terms of a generalization that includes the integers, such as a
field.
Let’s look at each of these approaches in more detail and the implications for our central question.
Thoughts within an individual mind will be a concept either surmised, taught, or arrived at by some other means. Although an interesting topic, this suggests that the answer we are looking for is most likely to be found within one or more of the other views.
Because the positive integers were invented so long ago, we would probably have difficulty verifying any theories we might have about their development. We know more about zero and the negative integers, but I’m not sure that necessarily helps. However, we can certainly surmise about the ways it might have developed, especially if we are exploring theories about thought itself. Whatever we come up with, we can apply to our next view.
There are a number of approaches we might take to defining number and then create the integers. Let me outline some of the critical steps in one approach that provides the result we need. Let’s assume we derived the positive integers in some way from the process of counting objects. We can use a successor process directly to create new numbers and, if we like, the development can also involve addition as a way of counting the elements in two sets and arriving at the count of the union of those two sets. As we count or add 1 to a previously “created” positive integer, we have choices with respect to the result:
- the “object” does not exist,
- the “object” is something we have already created, or
- the “object” is new.
The second choice can be used to create various mathematical systems. If, for example, we want to create only a finite number of numbers, one possible structure is the finite field (also called a Galois field) such as modular arithmetic with a prime number of elements. For example, if 2 is our prime, our two elements are 0 and 1. Then $0\equiv 2(mod2)$ and $1\equiv -1(mod2)$. Roughly stated, in such a system, we have $0=1+1=2$ and $1=-1$. In fact, all the even numbers are “equal” to 0 and all the odd numbers “equal” to 1 in such a system.
The last choice in our bulleted list above, where we decide that the “object” is new, produces the positive integers. Once we have the positive integers, we can either count backwards or reverse the process of addition to give us subtraction. Subtracting one from the integers greater than one gives us a number we have already created using addition. If we take 1 away from 1 we are confronted with the same 3 choices about a possible result of the operation. If we declare that the “object” is new, we get zero, and if we continue the process using the same assumption, we obtain the negative integers. The fact that we choose to create a system where we make the assumption that the operation leads to a new number is why $1\neq -1$.
Once we have our numbers and operations such as addition and subtraction, we expect to see this choice reflected in other ways. For example, you would expect that we might be able to then create a proof of some sort that leads to the same result. Here is an example:
Suppose that, contrary to our assumption, we have
\begin{equation*}
1=-1.
\end{equation*}
Then adding 1 to these two equal things gives
\begin{equation*}
1+1=1-1.
\end{equation*}
The right-hand side is how we constructed the number zero, so substituting this number we obtain
\begin{equation*}
1+1=0,
\end{equation*}
which we can write as
\begin{equation*}
2=0.
\end{equation*}
Now, one way to interpret this is simply that we have obtained a contradiction, so our original assumption that $1=-1$ must be false, leading to $1\neq -1$. Perhaps this is more convincing to use as a contradiction because negative numbers are not involved. If it is not more convincing to us, then we have to ask, “Why don’t we have $2=0$?”
The answer is that when we were adding 1 to 1 in order to create the positive integers, we had our 3 choices of what to make of this potential number. We saw previously that one choice is to develop 2-element modular arithmetic, and then we did have essentially $2=0$. If our choice is to consider 2 to be a new number, then that decision is what is making $1\neq -1$.
How does this decision that $2\neq 0$, which is about non-negative integers, lead to an inequality that included a negative integer? Well, if instead we decided
\begin{equation*}
2=0,
\end{equation*}
then rewriting 2, we have
\begin{equation*}
1+1=0.
\end{equation*}
At this point, we make our central hypothesis that subtracting 1 from a number creates a new number. This means that
\begin{equation*}
1+1-1=0-1
\end{equation*}
creates one new number on each side, and by our definition of 0, we have
\begin{equation*}
1+0=0-1,
\end{equation*}
and by a property of 0,
$1=-1$ .
So, we better have $2\neq 0$ rather than $2=0$ to avoid $1=-1$ .
What makes all this possible is our central assumption about the creation of new numbers as we add or subtract $1$.
- The previous section shows how we fairly directly make the assumption that $1=-1$ when our operations of addition or subtraction create new numbers. That result can also be seen in relatively simple and straightforward proofs that follow from this assumption.
Although the kind of development of the integers we have described is straightforward, intuitively understandable, and may well closely follow the historical invention of the integers, there are many other mathematical systems that claim to define the integers. Since historically, humans created integers before any axiomatic definition, perhaps we should view these creations as being axiomatic based models for the integers that have already been brought into existence rather than \textit{the} definition of the integers. Of course, people can define whatever they like, so how you wish to view these mathematical structures that create, in some form or another, either the natural numbers or structures that mirror the natural numbers is up to you. In any case, it is fair to look at such systems and see where the assumption that leads to $1\neq -1$ can be found. We look at a few representative examples.
Peano Axioms
There are many forms of Peano axioms, and the \href{https://en.wikipedia.org/wiki/Peano_axioms}{Wikipedia article} on them contains a great deal of information about the topic. These various forms are “essentially” the same in the sense that they are isomorphic to each other. However, the original Peano axioms started with 1, so they essentially created the positive integers. Thus, we are immediately confronted with the issue of what axioms are we even talking about. We have to choose some set of axioms, so, without justification, I’ll just go with the axioms presented in the Wikipedia article.
Now, these axioms allow us to create the non-negative integers, but our original question involved $-1$. We can use the fact shown in the previous section that given our usual development of the integers and the assumptions we made there, that having $2\neq 0$ is required, because otherwise it leads, again based on the way we constructed elementary arithmetic, to $1=-1$ .
Under the Peano Axioms, we are calling 2 the successor of 1, symbolically represented by $S(1)=2, $ and if we had $2=0,$ we could conclude that $S(1)=0$ .
But this conclusion violates Peano Axiom 8:
For every natural number n, S(n) = 0 is false. That is, there is no natural number whose successor is 0.
So, we see that this is where the Peano Axiom assumption can be found that prevents $2=0$. However, to show that $1\neq -1,$ we used elementary arithmetic to show that $2\neq 0$ implies $1\neq -1$. We can conclude the following: Yes, $1\neq -1$ is consistent with the Peano Axioms, and we can show that Axiom 8 is the critical assumption needed to create a proof using the axioms. However, the Peano Axioms don’t create negative numbers, and the statement needed to allow us to use the Peano Axioms ($2\neq 0$) would require us to have already made the assumption within elementary arithmetic that $1\neq -1$. Thus, nothing is gained by using the Peano Axioms.
We could, of course, extend the Peano Axioms to create negative numbers. Then using something like Axiom 8 would then assure that the process of making numbers always created new numbers. You can think of this as “no loops allowed!” But even in this case, we see that fundamentally, it is an assumption we are making about creating new numbers that leads us to $1\neq -1$.
Order Relation Axiom
Another approach is to add the order relation axiom:
If x and y are two numbers, then exactly one of the following holds: $x=y$, $x<y$, or $y<x$.
We also need to define the symbol “<”. Since we have defined the positive integers, we can use the following:
Definition: Given integers x and y, $x<y$ if and only if there exists positive integer k such that $x+k=y$.
Then $-1$ is $<1$ by definition since for $k=2$ : $-1+2=1$,
which by our assumed axiom, means we cannot have $1=-1$ .
Intuitively, if we start creating numbers starting at $-1$ by adding $1$, we successively get greater number $0$ and then greater number $1$. By the previous discussion, we need the successor of $1$ to be a new integer, and not the number we started with, which was $-1$. Thus, the order relation axiom and required definition appear to be a complicated way of expressing a very simple idea: at every creation step, we assume we have a new number.
Integers as Equivalence Classes
Rather than introduce equivalence classes here, if the reader is unfamiliar with the concept, I will refer to internet sources such as \href{https://www.math.wustl.edu/~freiwald/310integers.pdf}{https://www.math.wustl.edu/\textasciitilde{}freiwald/310integers.pdf}. Although I think “defining” numbers this way is problematic, we can use that definition to show that it won’t help us even if we accepted it.
Definition : For $\left(a,b\right)$ and $\left(c,d\right)$ members of the set of ordered pairs $\omega \times \omega $, we define a relation,
\begin{equation*}
\left(a,b\right)\simeq \left(c,d\right) \text{if and only if }\mathrm{a}+\mathrm{d}=\mathrm{b}+\mathrm{c}.
\end{equation*}
It looks mysterious, but if we rearrange terms in the defining equation, we obtain
$\left(a,b\right)\simeq \left(c,d\right) \text{ if and only if }\mathrm{a}-\mathrm{b}=\mathrm{c}-\mathrm{d}, $
so essentially all we are doing is defining numbers in terms of integer differences. We do so using only the addition operation to avoid having to define subtraction. Then each number is an infinite set of integer pairs with the same difference. Not only is the set infinite, but one has to justify calling a set a number. Thankfully, we need not pursue the problematic details here. Using the definition and applying it to the two numbers in question,
if $-1=\left(0,1\right)$ and $1=\left(1,0\right),$ then according to the definition,
$0+0=1+1$ ,
which not only requires that we assume or prove $0+0=0$ , but the application of the definition only leads to $2=0$ . We previously used elementary arithmetic to show that $2\neq 0$ implies $1=-1$ and thus we can obtain the result directly using elementary arithmetic. Therefore, we have gained nothing by defining integers as equivalence classes.
- Finally, we look into the issue of whether or not we can use generalizations of the integers in some way to answer the question of why $1\neq -1$. The problem we face here is similar to that encountered in the attempts to answer the question that we explored in the previous section. A generalization, like a model of the integers, is based on a mathematical system that we can use rather directly to assure that $1\neq -1$ . Then if the generalization has the same result, it is not some deep property of the generalization, but rather that the system simply reflected something that was built in using an assumption.
Consider the generalization known as a field. Both elementary arithmetic and modular arithmetic with two elements are fields, so the definition of a field does not capture the characteristic we are seeking. Thus, fields are going to be of no use, even if all we are interested in is where the critical assumption was made.
