Editing Multiverse (section)

==== Level IV: Ultimate ensemble ====
The ultimate [[mathematical universe hypothesis]] is Tegmark's own hypothesis.<ref name="Tegmark2014">{{cite book |first=Max |last=Tegmark |date=2014 |title=Our Mathematical Universe: My Quest for the Ultimate Nature of Reality |publisher=Knopf Doubleday Publishing Group |isbn=9780307599803|title-link=Our Mathematical Universe: My Quest for the Ultimate Nature of Reality }}</ref>

This level considers all universes to be equally real which can be described by different mathematical structures.

Tegmark writes: {{quotation|text=[[Abstract mathematics]] is so general that any [[theory of everything|Theory Of Everything (TOE)]] which is definable in purely formal terms (independent of vague human terminology) is also a mathematical structure. For instance, a TOE involving a set of different types of entities (denoted by words, say) and relations between them (denoted by additional words) is nothing but what mathematicians call a [[Set theory|set-theoretical]] model, and one can generally find a [[formal system]] that it is a model of.}}

He argues that this "implies that any conceivable parallel universe theory can be described at Level IV" and "subsumes all other ensembles, therefore brings closure to the hierarchy of multiverses, and there cannot be, say, a Level V."<ref name="X0302131">{{Cite journal |arxiv=astro-ph/0302131 |bibcode = 2003SciAm.288e..40T |doi = 10.1038/scientificamerican0503-40 |pmid=12701329 |title = Parallel Universes |year = 2003 |last1 = Tegmark |first1 = Max |journal = Scientific American |volume = 288 |issue = 5 |pages = 40–51 }}</ref>

[[Jürgen Schmidhuber]], however, says that the set of mathematical structures is not even [[well-defined]] and that it admits only universe representations describable by [[constructive mathematics]]—that is, [[computer programs]].

Schmidhuber explicitly includes universe representations describable by non-halting programs whose output bits converge after a finite time, although the convergence time itself may not be predictable by a halting program, due to the [[Undecidable problem|undecidability]] of the [[halting problem]].<ref>[[Jürgen Schmidhuber|J. Schmidhuber]] (1997): A Computer Scientist's View of Life, the Universe, and Everything. Lecture Notes in Computer Science, pp. 201–208, Springer: [http://www.idsia.ch/~juergen/everything/ IDSIA – Dalle Molle Institute for Artificial Intelligence].</ref><ref>{{Cite arXiv|eprint=quant-ph/0011122|last1=Schmidhuber|first1=Juergen|title=Algorithmic Theories of Everything|year=2000}}</ref><ref>[[Jürgen Schmidhuber|J. Schmidhuber]] (2002): Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit. ''International Journal of Foundations of Computer Science'' 13 (4): 587–612. [http://www.idsia.ch/~juergen/kolmogorov.html IDSIA – Dalle Molle Institute for Artificial Intelligence].</ref> He also explicitly discusses the more restricted ensemble of quickly computable universes.<ref>[[Jürgen Schmidhuber|J. Schmidhuber]] (2002): The Speed Prior: A New Simplicity Measure Yielding Near-Optimal Computable Predictions. Proc. 15th Annual Conference on Computational Learning Theory (COLT 2002), Sydney, Australia, Lecture Notes in Artificial Intelligence, pp. 216–228. Springer: [http://www.idsia.ch/~juergen/speedprior.html IDSIA – Dalle Molle Institute for Artificial Intelligence].</ref>