I think you're taking this point a little too forcefully; this is meant to informally motivate Russell's paradox, in my reading - which is exactly the title of the section you're referencing.
The point here is a little more subtle; category theory doesn't necessarily rely on sets; the definitions of categories that you often see (involving sets of objects and sets of morphisms) is more axiomatically forceful than the more general definition, which uses the notion of classes; category theory can use set theory, but does not depend on it.
The point here is that type theory offers just such another way to design in an avoidance of Russell's paradox.
You might also want to read about e.g. Grothendieck universes - they're quite relevant here.
> Yes -- in set theory sets can contain themselves
Which set theory? ZFC doesn't permit this.
Non-well-founded set theories are so non-standard that I think it's wrong, or at least misleading, to claim that unqualified "set theory" permits this.
> Yes -- in set theory sets can contain themselves
Hrbacek and Jech would like a word. It is very much not the case that in standard axiomatic set theory sets can contain themselves, precisely because this leads to things like Russell’s paradox. Sets containing themselves is generally prevented by the axiom of regularity. (Every non-empty set S contains an element wihch is disjoint from S) https://en.wikipedia.org/wiki/Axiom_of_regularity
> types are not sets and sets are not types
This is also not true. All types can be expressed as sets but not all sets are types in the standard definitions.
to justify my claim with an excerpt from the article:
““
What is type theory
“Every propositional function φ(x)—so it is contended—has, in addition to its range of truth, a range of significance, i.e. a range within which x must lie if φ(x) is to be a proposition at all, whether true or false. This is the first point in the theory of types; the second point is that ranges of significance form types, i.e. if x belongs to the range of significance of φ(x), then there is a class of objects, the type of x, all of which must also belong to the range of significance of φ(x)” — Bertrand Russell - Principles of Mathematics
In the last section, we almost fell in the trap of explaining types as something that are “like sets, but… “ (e.g. they are like sets, but a term can only be a member of one type). However, while it may be technically true, any such explanation would not be at all appropriate, as, while types started as alternative to sets, they actually ended up being quite different. So, thinking in terms of sets won’t get you far. Indeed, if we take the proverbial set theorist from the previous section, and ask them about types, their truthful response would have to be:
“Have you seen a set? Well, it has nothing to do with it.” [<=== important bit]
So let’s see how we define a type theory in its own right.
””
The modern formulation of functions as sets doesn’t require type theory but is entirely congruent with Russell’s definition, just much less cumbersome. In this view, φ is a relation on the set (D X C) where D and C are the domain and codomain of the function (which he calls the “range of significance of x” and the “range of significance of φ(x)” respectively). So since he’s talking about propositional functions, here C is the set {true, false} and D is all the things that are like whatever x is ie the set {x’: x’ is of the same type as x}.
Now a relation is just a particular type of predicate (ie it too is a set) so here we have x ~ y if φ(x) = y for all (x,y) in (D X C).
Notice here both the propositional function and the type are sets.
Parents do not have to be "experts in chilhood development" to know what is best for their children. Especially experts in their fields like the manufacturing of alcohol, guns or other products universallly considered dangerous.
So, if parents can rely on a a century of more of science showing the negative impacts of guns, tobacco, and alcohol on children… they can rely on vibes and politicians for evidence of harm from screens?
I’m not even arguing with you. I’m just disappointed in how quickly so many on HN throw out all pretense of being interested in data as soon as a personal hot button issue comes up. It’s human nature I guess, but still depressing.
Screens are harmful for adults too. Everyone knows this through the personal experience of doomscrolling hours of one's own life away. Why would they be any better for children?
Or do you imagine that there's a study out there that will reveal that arguing on Twitter with someone called Catturd2 is good for your mental health?
You feel pain? Doctor says it's probably in your head because statistically you shouldn't. -- Based on countless true stories.
Data is map, not terrain. It can explain some of the quantifiable world, not all of it. Common sense can also fill some of the gaps, some of the time. And there remains plenty still that's too entropic for our grasp. Waiting for data to speak is not always the best move. Heck, it might even sometimes be the worst. It seems this is a lesson we collectively keep forgetting over and over, despite the endless list of data-backed "facts" that, in hindsight, it turns out we were wrong or short-sighted about. Apparently, that too is human nature.
The existence of science does not obligate us to either receive a double-blind study of massive statistical significance on the exact question we're thinking about or to throw our hands up in total ignorance and sit in a corner crying about the lack of a scientific study.
It is perfectly rational to rely on experience for what screens do to children when that's all we have. You operate on that standard all the time. I know that, because you have no choice. There are plenty of choices you must make without a "data" to back you up on.
Moreover, there is plenty of data on this topic and if there is any study out there that even remotely supports the idea that it's all just hunky-dory for kids to be exposed to arbitrary amounts of "screen time" and parents are just silly for being worried about what it may be doing to their children, I sure haven't seen it go by. (I don't love the vagueness of the term "screen time" but for this discussion it'll do... anyone who wants to complain about it in a reply be my guest but be aware I don't really like it either.)
"Politicians" didn't even begin to enter into my decisions and I doubt it did for very many people either. This is one of the cases where the politicians are just jumping in front of an existing parade and claiming to be the leaders. But they aren't, and the parade isn't following them.
No, but I believe that science and quantifying the specific danger leads to better policies than going on vibes. For instance, laws to require safe storage are based on data quantifying reductions in harm to children [1]
Data beats vibes, even when vibes are qualitatively correct. I’m surprised this is surprising.
Quite often clients were more powerful than servers. Hell, at one point a CPU embedded into a printer could be faster than, say, 8088. An X server (running on the client side) often required a more powerful machine than one running X clients (i.e. a server). A web browser is not an exception.
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