One of my profs mentioned that sometimes people formulate theories about some type of object, but then later realize that those objects do not exist. Can someone given me an example of such a theory? I know this is vague, but I hope it makes sense.

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    $\begingroup$ Out of the blue, and without being a scholar at all in these matters, what about the mathematics of string theory? So far it seems to work fine yet the physics world hasn't yet found some physical proof of strings... $\endgroup$ – DonAntonio Dec 7 '12 at 19:08
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    $\begingroup$ I asked a precise variant of this question here. $\endgroup$ – Mariano Suárez-Álvarez Dec 7 '12 at 19:21
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    $\begingroup$ In a weak sense, the set of all sets. $\endgroup$ – André Nicolas Dec 7 '12 at 19:30
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    $\begingroup$ @DonAntonio I believe string theory is both mathematically consistent and consistent with all physical observations so far. However, the acid test for a physical theory is that it yields new predictions which are testable. For example, General Relativity is much more complicated of a theory than Newtonian gravity, but it explains new and testable results (like gravitational lensing). OP's question is different from this (and math differs from physics) because it is deductive not inductive -- we don't use Occam's razor to select "correct" theories in math. $\endgroup$ – orlandpm Dec 7 '12 at 20:04
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    $\begingroup$ There's some widely-told horror story about a student who was up at the board, in the process of defending their thesis, when one of the examiners (Milnor, iirc?) proved that there were no nontrivial examples of whatever objects the thesis had set out to study. $\endgroup$ – Aaron Mazel-Gee Dec 7 '12 at 20:33

This popular article describes something that fits the bill:

Researchers developed what Ravenel calls an entire “cosmology” of conjectures based on the assumption that manifolds with Arf-Kervaire invariant equal to 1 exist in all dimensions of the form 2n – 2. Many called the notion that these manifolds might not exist the “Doomsday Hypothesis,” as it would wipe out a large body of research. Earlier this year, Victor Snaith of the University of Sheffield in England published a book about this research, warning in the preface, “… this might turn out to be a book about things which do not exist.”

Just weeks after Snaith’s book appeared, Hopkins announced on April 21 that Snaith’s worst fears were justified: that Hopkins, Hill and Ravenel had proved that no manifolds of Arf-Kervaire invariant equal to 1 exist in dimensions 254 and higher. Dimension 126, the only one not covered by their analysis, remains a mystery.

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