## Problems with infinity

It is often claimed that really existing ensembles involve an infinity of universes. However, Hilbert strongly argues that a really existing infinite set is impossible [30]. He points out that the actual existence of the infinite directly or indirectly leads to well recognized unresolvable contradictions in set theory (e.g. the Russell paradox, involving the set of all sets which do not contain themselves) and thus in the definitions and deductive foundations

3 Note that inflation only changes this speciality requirement to some degree; it does not work for very inhomogeneous or anisotropic conditions and, in any case, a general ensemble will have both inflationary and non-inflationary universes.

4 For example, f (m) = 10100¿(mo), where mo is the universe domain we inhabit.

of mathematics itself. Hilbert's basic position is that 'the infinite is nowhere to be found in reality, no matter what experiences, observations and knowledge are appealed to'. One can apply this to both individual universes and multiverses. If we assume 'all that can happen does happen', then we run into uncountable infinities of actually existing universes, even in the FLRW case. For example, we have to include all values of the matter and radiation density parameters at a given value of the Hubble parameter, as well as all positive and negative values of the cosmological constant -a triply uncountable set of models. Hilbert would urge us to avoid such catastrophes.

Thus it is important to recognize that infinity is not an actual number which we can ever specify or determine - it is simply the code-word for 'it continues without end'. And something that is not specifiable or determinate in extent is not physically realizable. Whenever infinities emerge in physics, we can be reasonably sure there has been a breakdown in our model. It is plausible that this applies here too.

In addition to these mathematically based considerations, there are philosophical problems with spatial infinities [31], as well as physical arguments against the existence of an infinity in cosmological models. On the one hand, quantum theoretical considerations suggest that spacetime may be discrete at the Planck scale; indeed, some specific quantum gravity models have been shown to incorporate this feature. Not only would this remove the real line as a physics construct, but it could even remove the ultraviolet divergences that otherwise plague field theory - a major bonus. On the other hand, there are problems with putting boundary conditions for physical theories at infinity, and it was for this reason that Einstein preferred to consider cos-mological models with compact spatial sections (thus removing the occurrence of spatial infinity). This was a major motivation for his static model, proposed in 1917, which necessarily has compact space sections. Wheeler picked up on this theme and wrote about it extensively [32]. Consequently, the famed textbook Gravitation by Misner, Thorne and Wheeler [33] only considered spatially compact, positively curved cosmological models in the main text - those with flat and negative spatial curvatures were relegated to a subsection on 'Other models'.

This theme recurs in present speculations on higher-dimensional theories, where the further dimensions are often (as in the original Kaluza-Klein picture) assumed to be compact. Unless one has some good physical reason for supposing otherwise, one might also expect this to be true of the three dimensions that expanded to a large size, even though this may necessitate 'non-standard' topologies for these spatial sections. Such topologies are commonplace (indeed they are essential) in M-theory. Thus physics also supports the idea that our universe may have compact spatial sections, which avoids infrared divergences as well.