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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?

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    $\begingroup$ Related: mathoverflow.net/questions/tagged/f-1 $\endgroup$
    – Emil Jeřábek
    Commented Sep 28, 2014 at 22:05
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    $\begingroup$ I'm not sure why this question is being so heavily downvoted; it's certainly on the really soft end, but I think it can lead to some interesting answers. +1 (if only to counteract said downvotes). $\endgroup$
    – Noah Schweber
    Commented Sep 28, 2014 at 22:21
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    $\begingroup$ Since the phrase "proof by contradiction" was invoked, I'll recall the distinction between "proof of negation" and "proof by contradiction", as explained by Andrej Bauer here: math.andrej.com/2010/03/29/…. For proofs of negation, one supposes a proposition $\phi$ is true and derives a contradiction; therefore $\phi$ is false. A proof by contradiction supposes $\phi$ is false and derives a contradiction; therefore $\phi$ is true. Intuitionists accept proofs of negation, but not proof by contradiction! $\endgroup$
    – Todd Trimble
    Commented Sep 28, 2014 at 23:59
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    $\begingroup$ There is a related question on matheducators.se. $\endgroup$
    – Ben Barber
    Commented Sep 29, 2014 at 6:36
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    $\begingroup$ @RickyDemer: A similar phenomenon occurs when one comes up with a list of nice properties for an object to have, then proves that no object can simultaneously have all of them. It is then often very interesting to search for ways to weaken the conditions to get an object that does exist. As a familiar example, there does not exist a countably additive, translation invariant, set function $\mu : \mathcal{P}(\mathbb{R}) \to [0,\infty]$ with $\mu([0,1]) = 1$. Lebesgue had the brilliant idea to fix this by defining $\mu$ only on an appropriate subset of $\mathcal{P}(\mathbb{R})$. $\endgroup$
    – Nate Eldredge
    Commented Sep 29, 2014 at 7:08

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An onto endomorphism $\phi$ of a finitely generated hopfian semigroup $S$ for which there would exist a cofinite proper subsemigroup $T$ with $\phi(T)=S$.

The mere statement and the idea of proof reminisces from far away of Poincare Recurrence Theorem, though this story is purely combinatorial and uses very delicate rewriting procedure!

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I'm a little surprised that no one answered with the following:

An incompact set of first order sentences.

That would be, a set of sentences whose finite subsets have models, but there is no model of the whole set. As a particular example, one could say that an incompact sentence is one having models of arbitrarily high finite cardinalities but no infinite models.

Vaguely related to the previous example, in my specific field I'm interested in proving that certain formulas (do not) characterize directly indecomposable structures. Most of the time, you might take two very similar finite structures $\mathbf{A}$ and $\mathbf{B}$, such that $\mathbf{A}$ has direct product decomposition, and $\mathbf{B}$ results of a minor tweak of $\mathbf{A}$ (such as adding or dropping some elements), in such a way $|\mathbf{B}|$ is prime, hence indecomposable. The relevant entity here is

A nontrivial factorization of a prime number.

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A convenient category of spectra.

In category theory, we like to work with categories satisfying some nice properties such as being Cartesian closed. If the naturally occurring category does not satifies these properties, we could try to modify it slightly to get a convenient category to work in. An example of such a category is a convenient category of topological spaces. It was proved by L.G. Lewis Jr that there is no symmetric monoidal category of spectra satisfying some natural properties that we might expect from such a category.

[This answer does not really fit the premise of the question, but still is an interesting example of a useful object that does not exist]

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  • $\begingroup$ I guess this forces users to work with concrete models, which sometimes come with "extra handles" to get a grip on. $\endgroup$
    – David Roberts ♦
    Commented Aug 14, 2017 at 8:49
  • $\begingroup$ @DavidRoberts There are several models which are "convenient enough" for many practical purposes as described here: ncatlab.org/nlab/show/symmetric+smash+product+of+spectra. Currently, many authors use concrete model not only for spectra, but also for spaces (simplicial sets, topological spaces), $\infty$-categories (quasicategories, complete Segal spaces), and other higher structures. $\endgroup$
    – Valery Isaev
    Commented Aug 14, 2017 at 9:25
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    $\begingroup$ I think it might be possible to work in an abstract setting in the future. For example, HoTT is an appropriate setting to work abstractly with spaces. There is also some work towards an abstract setting for doing $\infty$-category theory (ncatlab.org/nlab/show/infinity-cosmos, math.jhu.edu/~eriehl/synthetic.pdf). I think it should be possible to work abstractly with spectra also, but I'm not aware of any work in this direction. $\endgroup$
    – Valery Isaev
    Commented Aug 14, 2017 at 9:28
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Non-trivial $1$-dimensional representations of simple algebraic groups.

The way this turns up in many places is that one can often by various means show that certain Hom-spaces are $1$-dimensional, so if these are Hom's of restrictions of modules for a normal subgroup, the Hom-space must be trivial as a module for the original group, which often simplifies things when applying the LHS spectral sequence.

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