The paper that Saibal Mitra cited brings up problems with infinity. The idea that "everything that can exist, exists" is, to my understanding, essential to the multiverse of Tegmark and others. Yet a part of "Defining Multiverses" (quoted below) says that there is NOT "an infinite set of really existing universes". The paper speaks of "profound difficulties" with infinite universes, but merely alludes to them. What are these difficulties? What is wrong with the idea of infinite space and infinite time, with universes popping out of rare quantum events, or singularities, which randomly appear?
----- Original Message -----
From: Saibal Mitra
To: FoR
Cc: everything
Sent: Monday, May 19, 2003 2:16 AM
Subject: Defining Multiverses
http://arxiv.org/abs/astro-ph/0305292
Defining Multiverses
Authors: G.F.R. Ellis, U. Kirchner, W.R. Stoeger
Comments: 28 pages, 2 figures
The idea of a multiverse - an ensemble of universes - has received increasing attention in cosmology, both as the outcome of the originating process that generated our own universe, and as an explanation for why our universe appears to be fine-tuned for life. Here we carefully consider how multiverses should be defined, stressing the distinction between the collection of all possible universes, and ensembles of really existing universes or universe domains (essential for anthropic arguments). We show that such realised ensembles are by no means unique. A measure on the space of all universes or universe domains is needed so that probabilities can be calculated. We examine these issues in the case of the set of Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) universes, and consider scenarios like chaotic inflation which propose how ensembles of universe domains may be generated. Finally, we discuss philosophical and meta-scientific issues raised by the concept of a really existing ensemble.
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2.4 Problems with Infinity
When speaking of multiverses or ensembles of universes - possible or realized - the issue of infinity often crops up. Researchers often envision an infinite set of universes, in which all possibilities are realized. Can there really be an infinite set of really existing universes - whether countably infinite or non-countably infinite? We shall not go into these questions in depth here, but we suggest that, on the basis of well-known philosophical arguments, the answer is No. It is important to be aware that the issue of realized infinity is a controversial and uncertain one in philosophy and in the philosophy of mathematics.
There is no conceptual problem with an infinite set - countable or uncountable - of all possible universes. A potentially infinite set is not a problem. Nevertheless, if the set of possible models is uncountably infinite and there is only a countably infinite (or finite) number of universes in the ensemble then almost all possible models are not realized - the ensemble represents a set of measure zero in the set of possible universes.
However, as stressed by the mathematician David Hilbert (1961), a really existing infinite set is probably not possible - there are strong philosophical arguments against it. Others (Spitzer 2000, Stoeger 2003) have elaborated aspects of this issue. Whenever infinities emerge in physics - such as in the case of singularities - we can be reasonably sure, as is usually recognized, that there has been a breakdown in our models. An archived infinity in any parameter (temperature, density, spatial curvature) is almost certainly not a possible outcome of any physical process - simply because it means traversing, in actuality, an interval of values which never ends. Infinity is not an actual number we ever can specify or reach - it is simply a code word for "it continues without end." The same may be true in terms of the supposed ensembles of universes. It is difficult enough conceiving of an ensemble of many 'really existing' universes that are totally causally disjoint from our own, and how that could come into being, particularly given that many universes in the ensemble may themselves be claimed to have infinite spatial extent and to contain an infinite amount of matter, with the paradoxical conclusions that entails (Ellis and Brundrit 1979). To conceive of physical creation of an infinite set of universes (many of which will themselves be spatially infinite) is at least an order of magnitude more difficult. The phrase "everything that can exist, exists" might well imply such an infinitude, but glosses over all the profound difficulties implied.
Received on Mon May 19 2003 - 15:34:05 PDT