RudyRucker book finally finished:
1. A possible form of physical infinity is the indefinite divisibility of spacetime, which in theory is hard to experimentally verify
2. Another sign of physical infinity is if our universe exhibit a recurrence in structure when scaled small enough or enlarged enough, then the universe has to be infinite for that to happen. Simply put, a closed topology of the universe place heavy constraint on its volume
3. Thought process can be infinitely nested, such that it is either incomplete, or that its completion can only be perceived all at once with mystical methods
4. The Absolute infinite is inconceivable, and the reflection principle holds
5. $\omega_1$ is the smallest cardinality of all eventually dominating functions
6. An absolutely continuous line will have absolute infinite number of points, meaning all cardinals and proper classes will be smaller than it
7. Definability is indefinable
8. The Euler spiral can be used to construct a length of $\pi$. Many other transcendentals can be constructed using other curved shapes that are not ellipses
thus it is possible to get some transcendentals in finite number of steps provided you use the correct shape
9. A number that encodes all possible integer sequence cannot be defined using said encoding and hence look random to that encoding
10. Even if some perfectly accurate description U of the universe exists, it seems likely that if we represented U by the clumsy expedient of putting numbers in books, then this representation of U would not fit in the universe. Of course, the most efficient representation of U is the universe itself, so at least one representation of U exists. But could we ever hope to have a desk-top or pocket-sized model of the universe? Only if matter is indefinitely divisible.
This is an interesting point because it is proved in a paper that if our universe is a simulation, and hence have bounded memory and CPU time, the accuracy of simulated black holes in a computer in this simulation will get less accurate the more it is nested
11. If every possible universe exists, then there is no need to account for the special peculiarities of this universe (e.g., the facts that there is an ant crawling up my screen right now, or that there are 79 clover blooms in my backyard, or that sentient beings exist in this universe, or that space has three dimensions). If every possible universe exists, then there is no need to explain any peculiarity.
But anyone who has ever savored the endless diversity of nature must feel–and even hope–that the universe can never be fully captured by any finite schema, and that the pattern of the universe is, in a formal sense, random and unnameable.
12. ii) It is a mistake to let everyday reality condition possibility, and only to imagine the combinings and permutations of physical objects—the mind is capable of directly perceiving infinite sets. iii) The ultimate goal of such thought, and of all philosophy, is the perception of the Absolute.
13. There is no reason why R could not print out his program for us, perhaps in some extremely compact and coded form (compare sperm cells!). But, as was discussed above, we would not be able to understand this program, nor would we be able to prove to our satisfaction that R was consistent. The interesting thing is that even though we cannot understand the program of R, we are able to set up the physical conditions that lead to R’s coming into existence.
I am looking for an example of an incomplete measure space with a measure that is not sigma-finite.
All the measures which are not sigma-finite which I have come across so far are the following:
counting measure on a set that is not countable (e.g. on the measurable space $(\mathbb{R},\mathca...
From a vertex of a n-sided convex polygon we draw the diagonals. I want to show that the angles that are created between two consecutive diagonals are all equal.
For that do we have to consider the triangles that created and to show that these have the angles?
