Max Tegmark is a professor of physics at MIT specializing in cosmology. He has dramatically extended Hugh Everett, III's theory of many worlds to describe four possible levels of what Tegmark calls the "multiverse."
Like many mathematical physicists, he thinks the universe "Is" mathematics. His popular 2014 book is called Our Mathematical Universe..
Tegmark thinks consciousness is a mathematical pattern.
The Four Levels of the Multiverse
A Level I multiverse is based on the infinite nature of our universe (as was David Layzer's).
They are both close to the "possible worlds" of philosopher David Lewis, who said any possible world must exist somewhere.
There is room in an infinite universe for an infinite number of merely "observable" universes like ours, which has a finite distance to our "horizon," beyond which we will never see.
Tegmark's Level I also depends on Alan Guth's theory of an inflationary origin of our universe and the so-called "eternal inflation" of other "universes." The law of physics and the natural constants are the same in all these Level I "universes." But different universes may have different histories.
The idea of another universe with a "doppelganger" of Tegmark (or Layzer) who up until a certain moment in time is identical, but then does something different is mathematically absurd.
A Level II multiverse changes the fundamental laws and constants of physics to be merely "effective laws" that can vary from place to place. Fine-tuning may provide evidence for Level II.
A Level III multiverse is based on Hugh Everett's basic idea that the quantum wave function never collapses. Possibilities do not exist. Randomness is only an illusion.
A Level IV multiverse includes universes that have different fundamental laws of physics.
All the other "universes" in our Level I multiverse are completely and forever unobservable (by definition).
All multiverses above Level I violate the cosmological principle, that laws of nature, physical constants are identical everywhere. Layzer's "strong cosmological principle" assumes that statistical properties are the same everywhere. "Eternal inflation" denies that.
Definitions of Tegmark's Multiverse Terminology.
Physical Reality
Everything that exists; Tegmark says this equals the Level IV multiverse
Space
The part of physical reality that’s continuously connected to what we can observe; with eternal inflation, this equals the Level II multiverse
Our Universe
The part of physical reality we can in principle observe; quantum complications aside, this is the spherical region of space from which light has had time to reach us during the 14 billion years since our Big Bang
Parallel Universe
A part of physical reality that can in principle be observed from somewhere else but not from here—parallel universes are not a theory, but a prediction of certain theories
Multiverse
A collection of universes
Level I Multiverse
Distant regions of space that are currently but not forever unobservable; they have the same effective laws of physics but may have different histories
Level II Multiverse
Distant regions of space that are forever unobservable because space between here and there keeps inflating; they obey the same fundamental laws of physics, but their effective laws of physics may differ
Level III Multiverse
Different parts of quantum Hilbert space; same diversity as Level II
Level IV Multiverse
All mathematical structures, corresponding to different fundamental laws of physics
Fundamental Laws
The mathematical equations that govern physics
Effective Laws
Particular solution to the mathematical equations that describe physics; can be mistaken for fundamental laws if the same solution is implemented throughout universe
Fine-Tuning
Physical constants in the effective laws having values in a very narrow range allowing life; observed fine-tuning is arguable evidence for the Level II multiverse
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