What is the most useful non-existing object of your field?

Question

When many proofs by contradiction end with "we have built an object with such, such and such properties, which does not exist", it seems relevant to give this object a name, even though (in fact because) it does not exist. The most striking example in my field of research is the following.

Definition : A random variable $X$ is said to be uniform in $\mathbb{Z}$ if it is $\mathbb{Z}$-valued and has the same distribution as $X+1$.

Theorem : No random variable is uniform in $\mathbb{Z}$.

What are the non-existing objects you have come across?

Answer 1

The elliptic curve attached to a nontrivial solution of $x^n+y^n=z^n,\quad n>2$.

Answer 2

An infinite strictly decreasing sequence of positive integers.

Silly as it might seem, it is the clue for a technique called Fermat descent and which is still nowadays of crucial importance in arithmetic/diophantine geometry. Fermat's idea was to take an integer solution in positive integers $(a_1,b_1,\dots,c_1)$ to some diophantine equation he had in mind, and then to massage it in order to create another solution $(a_2,b_2,\dots,c_2)$ still in positive integers but with $a_2<a_1,b_2<b_1,\dots c_2<c_1$: due to the non-existence quoted above, this was a contradiction and he won in proving the diophantine equation in his hands had no solution at all (in positive integers, it might sometime had "the trivial one", namely $(0,\dots,0)$). Although only vaguely reminiscent of this technique, the much more sophisticated "étale descent" owes its name to Fermat's.

Answer 3

A very well-known example, whose importance in set theory and the foundations of mathematics can't be easily overstated, would be Russell's set of the sets which don't contain themselves.

Answer 4

A polynomial-time algorithm for SAT (satisfiability), the problem of whether a boolean logical formula has a setting of its variables that makes it true.

(It's not quite in the letter of the question because we do not know that it does not exist.)

Primarily, we show that problems are NP-hard by reducing SAT (or another NP-hard problem) to those problems in polynomial time. The argument is thus that, if we have a polytime algorithm for those problems, then this constructs a polytime algorithm for SAT. Since we do not believe this mythical creature exists, we do not think those problems can be solved efficiently. (not sure if all mathematicians are already aware of this or whether the summary is useful.)

If we had this polynomial-time algorithm for SAT, then we could prove theorems quickly and automatically, we could break cryptosystems, we could improve massively in all sorts of scheduling, routing, resource allocation, and other optimization problems -- in short, "useful" would be an understatement.

(Let me add -- what's really "useful" is the converse: if this object does not exist, then we know that these sorts of tasks cannot be accomplished.)

Answer 5

The path integral!

I realize that it's not in the spirit of the question, but it's too good not to mention.

Answer 6

A number which is less than 1 and greater than 1.

EDIT: Since my attempt at provocation was understandably taken as mere frivolity, or rudeness for its own sake, let me risk self-advertising by pointing to some examples: the argument just before the statement of Corollary 4.9 in arxiv.org/abs/0801.3415; or Lemma 3.6 in arxiv.org/abs/0811.4432; or Lemma 3.2 in arxiv.org/abs/0906.2253

Answer 7

Three examples that come to my mind (not from my field)

• The free complete lattice on three generators. On a first sight, it seems harmful to construct this structure by transfinite induction, that is "from below", as an increasing union of sets each labeled by some ordinal, starting from $\{a,b,c\}$. The problem is that one would need all ordinals: in other words, the free complete lattice on three generators is a proper class (for a proof, see e.g. P.T.Johnstone's Stone Spaces, ch I ).

• Non-commutative finite fields. These have a lot of useful and interesting properties, the most relevant of which, after Wedderburn's theorem, is possibly non-existence.

Answer 8

Generic filters, in forcing. As long as $\mathbb{P}$ is not trivial, no truly generic filters through $\mathbb{P}$ exist, yet we use "them" all the time.

Okay, one's mileage might vary with this answer depending on philosphy: for example, if we're working in something like a set-theoretic multiverse, then we can say that generic filters always exist, in an appropriate sense; in the opposite direction, one could argue that - insofar as what we are using forcing for is producing independence results - we only really use filters which are generic with respect to some countable model, which certainly exist. And a multiverse-type approach can subsume this perspective, if we view every universe as potentially countable; when I'm thinking seriously about the philosophy of set theory, this is certainly the point of view I adopt. But when I'm actually doing set theory, I naively assume that (1) there is a "real" set-theoretic universe $V$, and (2) generic filters over $V$ "exist," so in that sense I'm using nonexistent objects.

Answer 9

Not quite in my field, but: Reinhardt cardinals.

Answer 10

In intuitionistic mathematics, a non-constant function from $ \mathbb R $ to $ \{ 0 , 1 \} $.

Many classical theorems can be proved to fail intuitionistically by showing that they imply this or something much like it. (Probably the most common thing is to show that the classical theorem implies the theorem $$ \forall \, x , y \in \mathbb R , \; x = y \; \vee \; x \ne y \text , $$ which doesn't look like the existence of a thing; but this is equivalent to the existence, for each real number $ x $, of a function $ f $ from $ \mathbb R $ to $ \{ 0 , 1 \} $ such that $ f ( y ) = 1 $ iff $ x = y $.)

More generally, in constructive mathematics, we don't usually assume that such functions don't exist, but we also understand that we can't prove that they do. So this still demonstrates that classical theorems can't be proved constructively (at least, not without being modified).

In a more neutral framework, we might speak of a non-constant continuous function from $ \mathbb R $ to $ \{ 0 , 1 \} $, or of a non-constant computable function from $ \mathbb R $ to $ \{ 0 , 1 \} $.

Answer 11

A field $F$ with algebraic closure of degree $3$ over $F$.

Useful because: It is the first restriction on the structure of absolute Galois groups of fields: they have no torsion except for involutions. This result, due to Artin and Schreier was the starting point of much of modern Galois theory.

Answer 12

Another "not quite my field" example, plus it is not known yet if this is an answer to your question: Siegel zeros.

Answer 13

The program $H$ which computes the function $$h(P,x)=\begin{cases} 1 & \text{If program $P$ will terminate on input $x$}\\ 0 & \text{otherwise} \end{cases}$$

This function (and the program that computes it) forms the basis of the most common proof of the impossibility of a solution solving the Halting Problem..

Thus it forms the basis of many proofs of in-computability, by showing that if some function $g$ (computed by a Program $G$), then $g$ would have the properties of $h$ and thus the would not be computable (and thus $G$ does not exist)

These impossible programs are known as "halting oracles"; in fact, there's a whole hierarchy of them! h above only solves the program halting-problem. Since the oracle can't be a program, it can't solve its own halting problem. We can define an oracle h2 to solve the program-halting-oracle-halting-problem, but then we need another oracle h3 to solve the program-halting-oracle-halting-oracle-halting-problem, and so on.

Answer 14

The complex number $i$, which does not exist in the field of real numbers. (Please note the pun)

Answer 15

It is not my field, but I would like to mention this example anyway since when I learned it some time ago I was very impressed. In quantum field theory, in particular in quantum electrodynamics, one assumes existence of the whole theory, namely operator valued functions on $\mathbb{R}^{3+1}$ which should satisfy various properties, e.g. equivariance under the Poincare group, equal time commutation relations, existence of in and out states. However existence of such objects is not proven in physically interesting situations, e.g. for quantum electrodynamics in 4d. For me, as a mathematician, it was quite shocking and took a long time to realize that such advanced and non-trivial objects are only believed to exist, and were not constructed even in any non-rigorous sense. Moreover as far as I heard, now it is believed that some of these theories even should not exist (!), but they worked well so far since they are expected to be good approximations to more sophisticated (probably) existing theories.

Answer 16

How about an infinite (strictly) descending sequence in a well-founded relation? It does not have a special name because it does not exist for trivial reasons. But it fits the description, there are tons of proofs where people construct infinite descending sequences of natural numbers, ordinals etc.

Answer 17

Not my field per se, but...

A scale-invariant, rich, and consistent clustering scheme does not exist.

That observation led to the discovery that there is essentially a unique functorial hierarchical clustering scheme.

Answer 18

Non-trivial approximate subrings of ${\bf R}$ or of ${\bf F}_p$.

The existence of such objects is ruled out by a number of "sum-product theorems", a typical one of which asserts that given a subset $A$ of ${\bf F}_p$ that is not extremely large or extremely small, either the sum set $A+A$ or the product set $A \cdot A$ has to be significantly larger than $A$.

On the other hand, one can improve upon the "trivial bound" in many arguments in arithmetic combinatorics or combinatorial geometry by analysing a putative configuration that attains this trivial bound and showing that it ultimately must arise from an approximate subring. Some early examples of this are in

Bourgain, Jean; Katz, N.; Tao, Terence C., A sum-product estimate in finite fields, and applications, Geom. Funct. Anal. 14, No. 1, 27-57 (2004). ZBL1145.11306.

There are now dozens of other places where this sort of argument appears. (Analogous arguments also appear in other fields, e.g. using the "group configuration theorem" from model theory, or the "group chunk theorem" in algebraic geometry.)

Answer 19

The core model. To a large extent, inner model theory (my area of set theory) is about building the core model K which then under any reasonable hypothesis is shown to not exist.

Answer 20

Not my field per se either, and maybe this is more pedestrian than some of the other examples offered thus far, but it seems that a lot of arguments involving Riemann surfaces rely on the fact that if $M$ is a compact Riemann surface of genus $g>0$, there is no meromorphic function on $M$ with a single, simple pole.

Answer 21

Approximation-preserving reductions between optimization problems in the same complexity class.

This may require a bit of explanation. My trade is developing polynomial approximation algorithms for various computational problems that are known to be NP-hard. Most of these problems polynomially reduce to each other; if the reductions could be extended to the corresponding approximation algorithms I (as well as numerous other mathematicians and software engineers) would be out of business.

However, one of the consequences of PCP theorem is that, provided that $NP\neq P$, the existence of a polynomial reduction between problems does not imply the existence of a polynomial reduction between the corresponding approximation problems.

Answer 22

A variety of algebras of a fixed type (e.g. groups, rings, Lie algebras, semigroups) that contains only finitely based subvarieties is called a Specht variety. It is known, as early as the 1980s, that maximal Specht varieties of semigroups do not exist. But some maximal Specht varieties of monoids were recently discovered; this is quite surprising given how close semigroups and monoids are.

Answer 23

In the study of NF(U) set theory, the (graphs of the) singleton function restricted to the universe or the ordinals. The absence of these sets is how NF avoids Cantor's paradox and Burali-Forti, and often the simplest disproof of some property comes from showing that the property would entail the existence of these functions.

Answer 24

Proper unramified extensions of $\mathbf{Q}$ (Hermite-Minkowski).

Analogously: Non-trivial Abelian schemes over $\mathrm{Spec}\,\mathbf{Z}$ and other small rings of integers of number fields (Fontaine).

Infinitely many rational points on curves of genus $>1$ over number fields (Faltings, Vojta, Bombieri).

Answer 25

The Fundamental Theorem of Hopf Modules: Every Hopf module over a Hopf algebra is trivial.

Several of the classical proofs of major results in Hopf algebras involve constructing/defining a Hopf module such that the theorem desired holds precisely when this Hopf module is trivial. So the non-existence of a non-trivial such object is quite important.

The concept of Hopf module has of course been generalized (in many different ways), though their triviality is no longer guaranteed in such broader contexts.

Answer 26

Modular cusp forms of weight 2 for $\mathrm{SL}_2(\mathbb Z)$ or $\Gamma_0(2)$. Their non existence is a key ingredient for the non existence of the elliptic curve $y^2=x(x-a^p)(x+b^p)$, where $a^p+b^p=c^p$ is a counterexample to FLT and other similar diophantine equations.

Answer 27

Great question! Gromov's proof of the nonexistence of compact exact Lagrangian submanifolds $L \subset \mathbb{R}^{2n}$ (as well as few other non-existence results proved there). Gromov'es work in general could be mentioned as the starting of modern symplectic geometry - but the result itself showed that Lagrangian submanifolds exhibit special intersection \ non-existence properties (toghether with the Lagrangian Arnold conjectures, and other results from that time) paving the way to ideas such as Lagrangian Floer homology, Fukaya categories, etc...

Answer 28

For algebraic topology:

1) there is no extension of $\mathrm{id}: S^1\to S^1$ to $D^2\to S^1$

2) there are no maps $S^{2n-1}\to S^n$ of Hopf invariant $1$ for $n\not\in \{ 1, 2, 4, 8\}$.

Answer 29

This isn't a precise 'proof by contradiction' answer (although there are potentially ways to make it so, using descriptive set theory techniques), but more a matter of guiding philosophy.

In the class of second-countable locally compact groups, there is a general dearth of 'universal' objects (e.g. a group in the class such that all groups in the class appear as closed subgroups) and embedding theorems (analogous statements to things like 'every countable group embeds in a 2-generator group'). This is in contrast both to the more special classes of countable groups and second-countable compact groups, and also to the more general class of Polish groups.

By itself, this is inconvenient: we can't hope to prove many general results about locally compact groups by first passing to our favourite universal object and then analysing its structure in detail. But what is being hinted at here is that any given (second-countable) locally compact group is in some sense 'much smaller' than the class as a whole. This gives the possibility to prove some surprisingly strong finiteness properties, once one puts the appropriate caveats around compact groups and discrete groups. This sense that any individual group in the class is small has long been known for connected locally compact groups (modulo a compact normal subgroup, such a group is a finite-dimensional Lie group; especially once you pass to the associated Lie algebra, having finite dimension is obviously a very powerful finiteness property); we are now beginning to understand it also for totally disconnected locally compact groups.

Answer 30

$\newcommand{\R}{\mathbb{R}}$ Related to the original example of the OP: In Bayesian statistics, "non-informative prior distributions", there is even a paper with that in the title! http://www.uv.es/~bernardo/Dialogue.pdf

For example, in the spirit of the OQ, a noninformative prior on $\R$ can be defined as the distribution of a random variable $X$ such that $X+c$ has the same distribution as $X$. No such random variable exists, but if we calculate formally, the constant density function $f(x)=1$ fits the bill, even if it is not a probability density function, its integral being $\infty$.