## On Transfinite Chains in the Lipophysical Multiverse

It has been theorized that our universe may in fact be contained within a higher universe, and that this higher universe may be contained within an indefinite number of even higher universes. Likewise, our universe may house within it lower levels of universes. A lipophysical multiverse is envisioned, built up out of universes nested within each other.

But how many universes are there? How can we represent this sequence? We must assume that there is a lowest universe, at a level where there are no further universes that exist within it. This is because such a multiverse exists as a non-empty well-ordered set of universes - this is a set in which there is a least member. For example, the natural numbers features the least member 1 and hence is a well-ordered set, unlike the integers which extend infinitely to the two extremes of +∞ and to -∞. If a well-ordered set of universes were the case, then no minimal universe would exist.

One thing we must consider is ordinals. Ordinal numbers were devised by Georg Cantor in 1883 in order for us to be able to comprehend infinite sequences in a more fathomable light by assigning each infinite member a value that can represent it. A universe in itself is an infinite construct, and so the ordering would need to take this into account. We have finite ordinals, recognisable as our natural numbers like 31, but we also have infinite ordinals, key to the task at hand. The smallest infinite ordinal value is denoted by the Greek letter omega, ω, and is followed by ω+1, ω+2, ω+3 et cetera. We may denote the least universe as universe omega U_ω1 – no smaller infinite ordinal numbers exist, so this must be the lowest universe.

It becomes apparent that our own universe can be stated as U_ω+n, where n is an integer greater than or equal to zero. If n=0, then our universe would be U_ω, the lowest universe, otherwise it would lie somewhere along the indefinitely long sequence of universes. Hence we have a long line of universes starting at Uω. This would be all and well if there were indeed an infinite succession of universes, each universe contained within the next, but we now take into account the possibility of a Super Massive Progenitor (SMP). Such a universe would transcend all other existing universes: an ultimate universe; the being of pure fat at the apex of reality, if you will.

One may argue that there is no SMP, and that the sequence of universes continues infinitely. However, one may also note that ω, ω+1, ω+2 and so on go on for only one kind of infinity, these numbers being transfinite ordinals, larger than all finite numbers but not absolutely infinite. Our SMP would come not from the continuation of the universes of U_ω, U_ω+1, U_ω+2 to infinity (U_ω+ω) but as a new level of infinity altogether. It has been shown that there are more real numbers (e.g. √2≈1.414…) than rational numbers (which must be expressible as the fraction a/b with both a,b ∈ℤ), despite there being an infinite quantity of both kinds of number. This is because rational numbers are “countable”, where a one-to-one mapping with the natural numbers can be achieved. Real numbers, on the other hand, have been proved uncountable by means of a “Cantor Slash”, also devised by Georg Cantor, essentially showing how a new real number can be formed that is different from any other value on a given list of real numbers. The value ω₁ represents the smallest uncountable ordinal, and hence we may denote the SMP with the label U_ω₁.

Since ω₁ as an uncountable value exceeds any value ω+n where n is an integer greater than or equal to zero, we know this new universe to truly surpass the rest. Of course, as with all matters regarding infinity, one is uncertain whether universes can be represented by the ordinal infinities represented by a scale starting from ω and culminating at ω₁. The assumption has been made that universes are infinite in complexity, but if universes are in fact finitely so then it may simply be a case of ordering the universes as U1, U2, U3 ad infinitum. Before we can shed light on this, we may need to better understand the nature and biology of Cosmic Progenitors. Also, the use of cardinal numbers has not been discussed, though these could give insights as to the number of universes that are in being.

Another key aspect that requires more thought is of the transition between the transfinite universes U_ω+n and the SMP U_ω₁, since this marks a jump between a countable infinity and an uncountable infinity that cannot be easily bridged; it may be of worth considering the properties of ω that may allow for a continuation into ω₁ rather than an abyss of discontinuity. And of course, all matter discussed is purely theoretical. No observational evidence has been collected that either supports or disputes the content of this article. Any attempts of doing so appear to be beyond the level of current scientific techniques, and no method of observing universes whether outside of or within our own have been proposed – it is not even known what exactly we would be looking for.

However, from the foundational principles of Fatercism, it is a solid proposition that universes within universes is a true aspect of our lipophysical multiverse, and that in order to grasp a more sophisticated view of reality we must seek for a greater understanding of our existence based on the few axiomatic models such as that discussed of universes found in greater universes.

But how many universes are there? How can we represent this sequence? We must assume that there is a lowest universe, at a level where there are no further universes that exist within it. This is because such a multiverse exists as a non-empty well-ordered set of universes - this is a set in which there is a least member. For example, the natural numbers features the least member 1 and hence is a well-ordered set, unlike the integers which extend infinitely to the two extremes of +∞ and to -∞. If a well-ordered set of universes were the case, then no minimal universe would exist.

One thing we must consider is ordinals. Ordinal numbers were devised by Georg Cantor in 1883 in order for us to be able to comprehend infinite sequences in a more fathomable light by assigning each infinite member a value that can represent it. A universe in itself is an infinite construct, and so the ordering would need to take this into account. We have finite ordinals, recognisable as our natural numbers like 31, but we also have infinite ordinals, key to the task at hand. The smallest infinite ordinal value is denoted by the Greek letter omega, ω, and is followed by ω+1, ω+2, ω+3 et cetera. We may denote the least universe as universe omega U_ω1 – no smaller infinite ordinal numbers exist, so this must be the lowest universe.

It becomes apparent that our own universe can be stated as U_ω+n, where n is an integer greater than or equal to zero. If n=0, then our universe would be U_ω, the lowest universe, otherwise it would lie somewhere along the indefinitely long sequence of universes. Hence we have a long line of universes starting at Uω. This would be all and well if there were indeed an infinite succession of universes, each universe contained within the next, but we now take into account the possibility of a Super Massive Progenitor (SMP). Such a universe would transcend all other existing universes: an ultimate universe; the being of pure fat at the apex of reality, if you will.

One may argue that there is no SMP, and that the sequence of universes continues infinitely. However, one may also note that ω, ω+1, ω+2 and so on go on for only one kind of infinity, these numbers being transfinite ordinals, larger than all finite numbers but not absolutely infinite. Our SMP would come not from the continuation of the universes of U_ω, U_ω+1, U_ω+2 to infinity (U_ω+ω) but as a new level of infinity altogether. It has been shown that there are more real numbers (e.g. √2≈1.414…) than rational numbers (which must be expressible as the fraction a/b with both a,b ∈ℤ), despite there being an infinite quantity of both kinds of number. This is because rational numbers are “countable”, where a one-to-one mapping with the natural numbers can be achieved. Real numbers, on the other hand, have been proved uncountable by means of a “Cantor Slash”, also devised by Georg Cantor, essentially showing how a new real number can be formed that is different from any other value on a given list of real numbers. The value ω₁ represents the smallest uncountable ordinal, and hence we may denote the SMP with the label U_ω₁.

Since ω₁ as an uncountable value exceeds any value ω+n where n is an integer greater than or equal to zero, we know this new universe to truly surpass the rest. Of course, as with all matters regarding infinity, one is uncertain whether universes can be represented by the ordinal infinities represented by a scale starting from ω and culminating at ω₁. The assumption has been made that universes are infinite in complexity, but if universes are in fact finitely so then it may simply be a case of ordering the universes as U1, U2, U3 ad infinitum. Before we can shed light on this, we may need to better understand the nature and biology of Cosmic Progenitors. Also, the use of cardinal numbers has not been discussed, though these could give insights as to the number of universes that are in being.

Another key aspect that requires more thought is of the transition between the transfinite universes U_ω+n and the SMP U_ω₁, since this marks a jump between a countable infinity and an uncountable infinity that cannot be easily bridged; it may be of worth considering the properties of ω that may allow for a continuation into ω₁ rather than an abyss of discontinuity. And of course, all matter discussed is purely theoretical. No observational evidence has been collected that either supports or disputes the content of this article. Any attempts of doing so appear to be beyond the level of current scientific techniques, and no method of observing universes whether outside of or within our own have been proposed – it is not even known what exactly we would be looking for.

However, from the foundational principles of Fatercism, it is a solid proposition that universes within universes is a true aspect of our lipophysical multiverse, and that in order to grasp a more sophisticated view of reality we must seek for a greater understanding of our existence based on the few axiomatic models such as that discussed of universes found in greater universes.

*This article was submitted by Fatercist follower Tom Wang*