User blog comment:DontDrinkH20/Some ZFC-Definable very big LCOs - uncomputable?/@comment-1605058-20180830120806

I might be missing something, but I think \(\mathfrak s\) is pretty obviously uncomputable. Indeed, take \(\alpha\) computable and \(\beta<\alpha\). \(\beta\) is also computable, so there is a TM computing a well-order \(\prec\) of length \(\beta\). Then \(\beta\in V_\alpha\) can be defined as the ordinal for which there is an order-isomorphism between \(\prec\) and \(\beta\). I think we need \(\alpha\geq\beta+3\) or something like that for this isomorphism to exist, but by your own argument we may assume \(\alpha\) is limit, so no issues there.