Can $1=0$ ever make sense?

Question

Can $1=0$ ever make sense?

In more detail, under which interpretation of $0$ and $1$ is $1=0$ possible? what are the consequences of such a result? Under which interpretations would $1=0$ never be possible? In particular, if $0$ and $1$ are interpreted as numbers, is $1=0$ possible?

Answer 1

Yes, define the quotient EDIT:from the comments, additive group $\mathbb R/\mathbb Z $. Then $x$~$y$ iff $x-y \in \mathbb Z$. Then, to the effects of this quotient ring, we have $0=1 $ (and $...=-1=1=2=3=...$.), as someone mentioned in the comments. You may also do this topologically, by creating a topological quotient on $[0,1]$, where we identify $0$~$1$ and we identify every other point with itself.This last space is homeomorphic to the circle $S^1$.

Answer 2

A ring is a set together with two operations, called $+$ and $\cdot$, satisfying some familiar properties from algebra. In this context, $0$ is defined as the additive neutral element (i.e., $x+0=x$ for all $x$ in the ring) and $1$ is the multiplicative neutral element (i.e., $1\cdot x=x$ for all $x$ in the ring). It is then easily shown that $0=1$ holds if, and only if, the entire ring collapses to just a single element: $R=\{0=1\}$. So, algebraically, the effect of $0=1$ is the collapse of the entire system to something quite trivial.

In the usual context of number systems, the equality $0=1$ never holds. If by number system we agree to a reasonable system containing the natural numbers, then all we have to do is look at the natural numbers and establish that $0\ne 1$. This leads to the foundational issue of how to define the natural numbers. A typical way is to define the natural as certain sets. In that case, $0=\emptyset$ and $1=\{\emptyset\}$. The fact that these two sets are different follows easily from the axioms of set theory (of course, you need to specify which axioms you choose), and thus $0\ne 1$. A way the circumvents definitions is the axiomatic approach, namely Peano's axioms. One of those axioms states practically explicitly that $0\ne 1$, so that settles it.

Periodic phenomena naturally lead to a system where $0=1$, namely the reals modulo $1$. Put simply, suppose that you are studying a system with periodic behaviour occurring at a unit length of time. That means that at times $t$ and $t+1$ the system looks the same. Intrinsically for the system then, you want to identify $t$ and $t+1$. In particular, $0=1$. Topologically, this amounts to considering a circle $S$ (obtained by taking the interval $[0,1]$ and gluing its two endpoints) and the function $\mathbb R \to S$ mappings $x$ to $x (\mod 1)$. This is the universal covering spaces of the circle. So, topologically, the effect of $0=1$ leads naturally to one of the most fundamental examples of a covering space.

Answer 3

As well as the circle group $\mathbb R/\mathbb Z$, we also have $0=1$ in the following interesting settings:

• The torsion subgroup of $\mathbb R/\mathbb Z$, which is $\mathbb Q/\mathbb Z$, isomorphic to the multiplicative group of complex roots of unity, and the simplest example of an infinite Archimedean cyclically ordered group.
• The Sylow subgroups of $\mathbb R/\mathbb Z$ and $\mathbb Q/\mathbb Z$, which are the Prüfer groups $\mathbb Z(p^\infty)=\mathbb Z[p^{-1}]/\mathbb Z=\mathbb Q_p/\mathbb Z_p$, where the last is a quotient of $p$-adic numbers and equal to the Pontryagin dual of $\mathbb Z_p$.
• The rational vector space $\mathbb R/\mathbb Q$. This is the sort of thing you need to construct Dehn invariants (sort of) and Vitali sets.

Sometimes a pre-rigorous topic seems to imply $0=1$ in a setting where it would be false, but it later turns out that the equation is avoided. I'm thinking here of Grandi's series, the divergent infinite series $1-1+1-1+\cdots$. Operating on this series as if it were convergent would imply $0=1$, and it was once ridiculed on those grounds. But we now have a rigorous theory of divergent series and an understanding of which operations apply to which summability methods, and virtually all such methods assign it a generalized sum of $\frac12$.

Finally, in folklore: The equation $0=1$ is sometimes used as the ultimate inconsistent statement, as in Akihiro Kanamori's monograph The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings, where it represents the strongest possible large cardinal axiom.