Editing Categorical theory (section)

==History and motivation==
[[Oswald Veblen]] in 1904 defined a theory to be '''categorical''' if all of its models are isomorphic. It follows from the definition above and the [[Löwenheim–Skolem theorem]] that any [[first-order theory]] with a model of infinite [[cardinal number|cardinality]] cannot be categorical. One is then immediately led to the more subtle notion of {{math|''κ''}}-categoricity, which asks: for which cardinals {{math|''κ''}} is there exactly one model of cardinality {{math|''κ''}} of the given theory ''T'' up to isomorphism?  This is a deep question and significant progress was only made in 1954 when [[Jerzy Łoś]] noticed that, at least for [[complete theory|complete theories]] ''T'' over countable [[formal language|languages]] with at least one infinite model, he could only find three ways for ''T'' to be {{math|''κ''}}-categorical at some&nbsp;{{math|''κ''}}:

*''T'' is '''totally categorical''', ''i.e.'' ''T'' is {{math|''κ''}}-categorical for all infinite [[cardinal number|cardinal]]s&nbsp;{{math|''κ''}}.
*''T'' is '''uncountably categorical''', ''i.e.'' ''T'' is {{math|''κ''}}-categorical if and only if {{math|''κ''}} is an [[countable|uncountable]] cardinal.
*''T'' is [[Omega-categorical theory|'''countably categorical''']], ''i.e.'' ''T'' is {{math|''κ''}}-categorical if and only if {{math|''κ''}} is a countable cardinal.

In other words, he observed that, in all the cases he could think of, {{math|''κ''}}-categoricity at any one uncountable cardinal implied {{math|''κ''}}-categoricity at all other uncountable cardinals. This observation spurred a great amount of research into the 1960s, eventually culminating in [[Michael D. Morley|Michael Morley]]'s famous result that these are in fact the only possibilities.  The theory was subsequently extended and refined by [[Saharon Shelah]] in the 1970s and beyond, leading to [[Stability (model theory)|stability theory]] and Shelah's more general programme of [[spectrum of a theory|classification theory]].