There is an article from 2010 arguing that people may emotionally object to cryonics because cold is metaphorically associated with bad things.
Did the popularity of the Frozen movie change anything about this?
Well, there is the Facebook group "Cryonics Memes for Frozen Teens"...
America is now what anthropologists call a Kardashian Type Three civilisation: more than fifty percent of GDP is in the attention economy.
Stories by Greg Egan are generally great, but this one is... well, see for yourselves: In the Ruins
Do write with contempt.
Uhm, did or did not you miss a "not" here?
What do you mean by "all possible sets that replace some of the empty sets in the structure"? In an actual ZF universe there is only one empty set.
I meant empty sets in the, uhm, description/graph of the set. For example, in the description of the set {{}, {{}}}, there are two instances of empty set: here {{}, {{}}}, and here {{}, {{}}}.
Visually, if you would draw the set as a tree -- the topmost node represents the set itself, each node has below it nodes representing the elements (if two nodes have the same set for a subset, it would be drawn on the diagram below each node separately; we are making a tree structure that only splits downwards, never joins) -- then every set in standard model of ZF is a tree, with some nodes having infinitely many nodes directly below them, but each individual path downwards is finite. And each path downwards ends with the empty set = a node that does not have any nodes below it.
By "replacing some empty sets" I meant replacing some of those nodes at the bottom with the sets from the infinite chain a, b, c... For example, from the set {{}, {{}}}, you would achieve {a, {{}}} by replacing the "first" empty set; {{}, {a}} by replacing the "second" empty set; and {a, {a}} by replacing both empty sets. You would do this for all sets from the chain, even combinations like {a, {b}}.
The only restriction is that when the graph of the set contains an infinite number of empty-set-nodes, for example in {{}, {{}}, {{{}}}, {{{{}}}}... }, you can replace either finite or infinite number of them, but you may only use a finite number of different sets from the descending chain. So for example, you could infinitely many "a"s and infinitely many "b"s; or perhaps two "a"s and infinitely many "b"s; or just two "a"s and three "b"s; but you cannot use infinitely many different letters. So it is allowed to create {{}, {a}, {{{}}}, {{{b}}}... }, or {a, {a}, {{a}}, {{{a}}}... } (with infinitely many "a"s), or {a, {b}, {{a}}, {{{b}}}... } (with infinitely many "a"s and "b"s), but not {a, {b}, {{c}}, {{{d}}}... } (with infinitely many different letters used).
I think that the class of sets created this way satisfies the ZF axioms.
EDIT: I will try to send you an e-mail during this weekend, because this definitely makes more sense with pictures, at least in my head. Thank you for your patience so far!
Sequences: Beisutsukai
One year later: Extreme Rationality: It's Not That Great
I am interested in how learning rationalist techniques could help me navigate making complex life choices.
Have you already found some useful answers?
In my opinion, the greatest value usually comes from a few rather simple techniques, assuming that you actually do them. In other words, the typical failures are "not even trying" and "thinking about what needs to be done (and reading about it), but never actually doing it".
A few simple techniques:
Yes, by "urelements" I meant "elements that are not sets". However, this was just a different way to express the question that if we treat a, b, c... as black boxes, i.e. ignoring the question of what is inside them; whether from the facts that a is a set, b is a set, c is a set... inevitably follows that {a, b, c...} is also a set in given model of set theory.
For example, using the axiom of Pair, we can prove that {a, b} or {g, x} are also sets. Using also the axiom of Union, we can prove that any finite collections, such as {a, b, g, h, x} are sets. But using Pair and Union alone, we cannot prove that about any infinite collection.
The axiom of Infinity only proves the existence of one specific infinite set: ω.
My guess is that using all axioms together, we can only prove existence of sets that involve a finite number of a, b, c...
This needs to be put a bit more precisely, though, because by the very fact of them being an infinitely descending chain, including one of them means involving an infinite amount of them; for example {{}, c} is simultaneously {{}, {d}}, which is also {{}, {{e}}}, etc. But suppose that we stop expanding the expression whenever we hit one of these chain-sets. Thus, the set {{}, c} would involve c explicitly, but d, e... only implicitly.
Now we can rephrase my guess, that using all axioms together, we can only prove existence of sets that involve explicitly a finite number of a, b, c... That means, {a, b, c...} is not among them.
Axioms of Existence, Infinity, Pair, and Powerset do not increase the number of a, b, c... used explicitly. Axioms of Union, Comprehension can only "unpack" these sets one level deeper. -- Union applied to c would only prove the existence of d. Comprehension applied to c would result in either empty set or d. Each of them applied to a structure that explicitly contains a finite number of a, b, c... would result in a structure that also explicitly contains a finite, albeit maybe twice larger, number of a, b, c...
Axiom of Foundation should be okay with such structures, because even if you have a set containing finitely many of a, b, c..., just choose the one furthest down the chain. For example, for set {{{}}, a, c, e}, choose e = {f}, and the intersection is empty.
How to construct the universe: Take a ZF universe, and for each set in that universe also create all possible sets that replace some of the empty sets in the structure by sets a, b, c... under condition that during each replacement you only use a finite number of a, b, c... (but with each of the selected finite few, you can replace an arbitrary, finite or infinite, number of the empty sets in the original structure). My hypothesis is that this, together with the rules a = {b}, b = {c}... is also a ZF universe, and it is one containing an infinite descending chain of sets (a, b, c...).
Yes, you understood my question correctly... and I need to spend some time thinking about your answer. (Mostly because it's past midnight here, so I am leaving my computer for now.)
Thank you! It was a pleasure to be understood -- unlike when I e.g. post a question on Stack Exchange. :D
The main problem with point 2 in my opinion is with not being honest about one's priorities. Otherwise, why would they marry and divorce, instead of staying single, or staying legally married but living separately?
There are rich men who essentially have a deal with their lovers: "I will knock you up, you will take all care of the baby, and I will abandon you at some moment in future, but I will give you enough money to raise the child without needing to get a job". And both sides seem to be happy with the deal: the man keeps his career, and is happy about having reproduced biologically; the woman gets an early retirement.
Problem is with men who want the same deal, without being able/willing to pay enough money to make it a great deal also for the woman. (Assuming, stereotypically, that it's the man who follows his career, and woman who stays with the kids.)
Finer gradations are boring, but there are (1) infrared and ultraviolet colors, and (2) colors that our current system of three types of cone cells perceives as the same, but could be perceived as different by having more types of cones, or even different types of cones. We could use the new qualia for these.