I mean, the artifact in LotR is magical. Reading too much into the name would be a mistake.

You’re welcome!

The succinct way of defining the topology (on NxN) is the product topology of the discrete topology(/ies). Maybe that’s the discreteness you’re feeling?

Axiom of Choice is not regarded as a big deal by most set theorists, but it’s interesting when it comes up. The diagonalisation proof that there are undetermined games uses choice to well-order the set of strategies, so it’s actually the other way around: without choice it is consistent (assuming consistency of some other stuff) that all games in this formulation are determined. This is called the Axiom of Determinacy.

The axioms in question are power set and replacement: to prove full Borel determinacy you need to apply the power set axiom infinitely many times (using the replacement axiom). These two axioms are what gives the ZFC axioms their power, really.

Set theorists nowadays I don’t think debate about axioms per se. Set theory nowadays is at once somewhat pluralistic and somewhat settled (paradoxically). I’ll explain a little: set theorists are basically agreed that the ZFC axioms are natural, intuitively “true” (many set theorists would not put scare-quotes there, but I would), powerful enough to do all ordinary mathematics and more, and probably consistent. They also generally agree that many large-cardinal axioms are natural and probably consistent, though there is a wide variation in whether people think they are “true”; there is not nearly so much intuition that such huge objects could exist. This is different than our intuition behind the axiom of infinity, because that axiom is actually needed to do some ordinary mathematics (though you can do without it for a lot!)

The Projective Hierarchy continues the stratification of the Borel Hierarchy even further. If you assume infinitely many woodin cardinals, then you can prove Projective Determinacy. I have heard Tony Martin being quoted as saying that “if Projective Determinacy were found to be inconsistent” (and hence infinitely many Woodin cardinals is inconsistent) “then I’d be having serious doubts about [the axiom of] Replacement.” This gives you a flavour of how people think about the relationship between these concepts.

Steam already has an FPS monitor, right? I think that’s enough for 95% of everything (and it’s very unobtrusive).

As a field of study, it’s the study of two-player games of perfect information (so think chess, not football or poker) in which each player may make countably many moves (you can also look at uncountable-length games but it’s not common). I’ll give you more detail than I would a child :P

Each player takes turns to move. You can encode the moves they make as coming from some set - for example they might just play numbers. The rules of the game are imposed by a winning set, which is a set of countable-length sequences of moves, and we say that player I wins if the infinite sequence of her first move followed by player II’s first move followed by her second move, etc, is in the winning set. Otherwise player II wins. (There are no draws, which technically means chess falls outside the scope of this setup, but it turns out not to be a big deal)

(This allows you to encode what moves are allowed by the rules - you just say that any sequence which contains a move where that player broke a rule is a loss for that player, regardless of what comes afterwards.)

Each winning set defines a different game. The property of determinacy is a property of sets of infinite sequences which says that there is a winning strategy for either player. A strategy is just a function which takes the finite sequence of moves up to that point in the game and tells the player (the player for whom the strategy is) what to do. A winning strategy is one which, if followed, always results in a win for that player.

If we modify the rules of noughts and crosses (tic-tac-toe) so that draws are arbitrarily decided to give a win to player I, we know that this (finite) game has a winning strategy. In fact, any finite game has a winning strategy (or, if there are draws, this means there is a non-losing strategy). The outline of the proof is that if player I does not have a strategy to get to one of the (finitely many) winning states, then we can find a strategy for player II which avoids those winning states. (Remember, winning states are winning for I).

So, which games are determined? Are all games determined? Well, it’s actually easy (through a diagonalisation argument, same as proving uncountability of the reals) that not all infinite (countable-length, that is) games played with natural numbers (as moves) are determined. But you can create a way of categorising the sets of countable sequences of natural numbers (i.e. the possible winning sets) by a kind of complexity. This is the basis of descriptive set theory. It starts with topology: you can define basic open sets in this space as those sets consist of all infinite sequences which share a common finite prefix. Closed sets are the complements of open sets, as usual. But then you can define a hierarchy of complexity where the next level are countable unions of closed sets, then the next level are countable unions of complements of countable unions of complements of open sets. (An introduction to descriptive set theory will say more about this).

It’s quite easy to prove that all open sets and all closed sets in this hierarchy of complexity are determined. It’s a little harder to prove that the second level is determined, and harder still to prove that the third level is. Eventually a guy named Tony Martin (D. A. Martin) proved that all Borel sets in this hierarchy are determined. If you know your analysis, the Borel sets are exactly what you’re thinking: they’re the sets formed by all arbitrary countable unions, intersections and complements of open sets.

The interesting thing about this proof was that it needed a huge amount of set theoretic “power”. Most ordinary mathematics like analysis doesn’t need all the axioms of set theory, but this needed a massive chunk of them. This makes it interesting to set theorists because it tells us something about the relationship between something quite concrete: complexity of sets and strategies for easily-defined games on the one hand, and something quite abstract: the axioms of set theory. This pattern continues higher up: more determinacy can be proved if you assume even stronger axioms, going beyond what is typically included in set theory.

How much sugar do you add?

How long do you boil it for? (You don’t know; you have to monitor the temperature)

How long does putting it all in jars take you? (ages)

It’s not difficult, but it is time consuming and not trivial.

Making jam is not trivial but it I think that makes it rewarding! My dad has made jam and marmalade for as long as I’ve known and it’s always an event. My parents have hundreds of jars (for some reason my dad calls them bottles? Only in a jam context though!) and every so often he cooks up a giant pot of jam with an old-fashioned sugar thermometer, testing the batch on a piece of baking paper, then bottling everything up. He often did it with my sister, who now also makes her own jam.

He labels all the jars, and we’ve opened jars that were… I dunno, a decade old I’m sure, and they were totally fine. So they will definitely keep for a long time!

I have a PhD in mathematics (set theory). It would be pretty tough to explain to a child. Specifically my PhD is in determinacy, which is way easier to explain than most branches of set theory, but you do need a decent understanding of infinity to really get anywhere.

Yeah this is something that has got way worse over time. It used to be that most forums would default to “no politics” and then there were discussion areas set aside for that. And now if you criticise someone for bringing politics into something where it doesn’t belong, you will get angry responses declaring that you’re burying your head in the sand. No, I just don’t want lowest-hanging-fruit political comments on every cat picture.

JAQing off and Whataboutism are not those things.

Yeah, but if you go in saying that this is the inevitable result of having conservatives discuss politics here, I am suspicious that your threshold for those terms is waaayy lower than mine.

The person I replied to originally wasn’t talking about trolling or toxic behaviour, they were talking about conservative viewpoints (likening them to cannibalism, I might add) so, if you want to chip in that trolling isn’t welcome then I’ll certainly agree with that, but there’s a reason I’m not really talking about that.

It used to be, in the early days of mass social media (and it was widespread on forums)

Moderation isn’t easy but it also needn’t be fraught - set standards of civility (strict or loose) and basic rules about hate speech, and let people take themselves out of discussions that are within the rules that they nevertheless don’t like.

It works a lot better in small communities where you talk to the same people - you can ignore people you don’t like and not have the same conversation over and over.

Yeah, I am on… two forums. I think the issue is that it’s not where the people are any more.

I actually think the problem with voting is that the way it’s used to promote threads instead of doing it on a time basis. It’s part of a way of engaging with a site which punishes long-running conversations.

The fact that you characterise natural ways of engaging in a discussion negatively doesn’t mean it’s not genuine, and it doesn’t mean you’re forced to look at it if it’s available.

NOTHING is stopping a conservative from coming here and making cogent, factual arguments, aside from their own fragility.

The structure of vote-based social media makes it difficult, and the people who, rather than remove themselves from places where arguments happen, shout down the people having the arguments, stops this from happening.

You’d be right to point out that conservative-majority spaces are just as, if not more guilty of this, but that doesn’t make it less true.

The conservatives who “can’t” post to Lemmy are the ones who don’t know how to have an actual conversation and get banned.

That’s true but it’s not the only thing that’s going on.

It’s probably an issue with vote-based discussions full stop. Post something funny and it’ll get votes because of the laughs; post something everyone in your echo-chamber agrees with and it’ll get votes because it’s right-on.

Maybe I just want to go back to forums.

You don’t have to subscribe to political communities if you don’t want to see political discussion. But the dearth of genuine political discussion here is a problem for the people who do want it, that can’t be fixed by individual action.

I got a chef’s knife as a gift. I was a bit put out that came from Wilko (a very budget brand) and it turned out to be absolutely excellent. I think it won’t have cost any more than £20. I am astounded that anyone apart from professionals pays more than £100 for a knife, never mind the even more insane prices you can pay

Yep, it’s a big problem in audio and other subjective areas, because you have no way of knowing what the anonymous reviewer’s point of reference is, and most professional reviewers’ reference points are not suitable. It’s worse too, because purchaser-reviewers self-select into their category, so you expect most people to be satisfied with the subjective aspects of a product they’ve purchased, even though most people would not be satisfied with a random cheap product. This is all not helped by the fact that, in audio when differences are so minute, virtually no-one is conducting blind reviews so confirmation bias probably accounts for huge amounts of the final score. Sure, any professional reviewer is going to be able to identify a bum product that costs thousands, but I bet most of them will rate an identical product more highly if they’re told it costs 10x as much and comes from a fancier brand.

I’ve ended up crowdsourcing my recommendations from places like reddit where people tend to make tiered recommendation lists so you at least know they have the goal of producing the best products at each price level.

Yes? That’s why loans with collateral charge a lower interest rate than unsecured loans.

There’s a mile of difference between saying “consumers need to get comfortable not owning their games” and “we want consumers to get comfortable not owning their games (but using subscription services instead)”.

The former statement is extremely arrogant. The latter is just obvious. And it’s reasonable even if you or I personally don’t want to get our games on a subscription model - millions of people get their music through Spotify and it suits them just fine even though other people don’t want that. So it’s a way of straw-manning the people pushing subscriptions so you can hate them.

The point of the dishonest article is to make you believe the CEO feels entitled to gamers becoming OK with subscription models. What he actually feels is a hope that subscription models will take off. It’s rage-bait. Did it work?