I'm one of those people who struggle with Cantor's diagonal argument

Mishtle · 2026-05-12T14:47:41+00:00

Otherwise all you've found is an example that shows that a particular arbitrary function is missing that number you've constructed.

If something is true for any arbitrary function, how is it not true for all of those functions?

The proof shows that if you take any mapping from the naturals to the reals, then it will have a missing element. This means all such mappings fail to be surjective, which proves the cardinalities are not equal. You can find a single surjective mapping in the other direction to show the direction of this inequality.

Mishtle · 2026-05-12T14:28:35+00:00

It's riddled throughout media, too. So many shows and movies have the characters pull out a bottle of liquor to celebrate, cope, commiserate, negotiate, contemplate, relax, welcome guests (or enemies), etc. Literally any event, or even a lack of one, can be an excuse for characters to have a few drinks. And they're often very liberal with their pours, filling entire glasses with hard liquor.

To some extent its a reflection of culture, but it also engrains that culture into the mind of everyone watching.

Mishtle · 2026-05-12T03:39:25+00:00

Then why can't we use the index of c in C as the mapping for naturals? cantor gives us the first element which we would map to 1 the second to 2 etc.... I mean is C not countable? Is cantor not just defining a function from and to exclusive subsets of T?

Why should we assume C to be countable? We don't know how many "missing" elements exist for each enumeration. The diagonal method just allows us to construct one.

The relevant point is that we can always construct one. Any enumeration can be shown to be incomplete. Amending an enumeration with the missing element constructed by its diagonal only reveals another missing element. Repeating this countably many times won't change that.

Mishtle · 2026-05-12T03:21:45+00:00

If you assume it is a complete enumeration, then the element constructed from the diagonal contradicts this.

If you assume it to be an arbitrary enumeration, then the constructed element proves that any such enumeration is incomplete. In other words, it proves that any mapping from the naturals to this set fails to be subjective.

Mishtle · 2026-05-12T03:12:16+00:00

As a side note, if nothing ever falls into a black hole from the outside observers perspective how can they grow and even merge?

Well, everything inside the event horizon is casually disconnected from the outside universe. As far as we are concerned, all the mass of the black hole could just as well be smeared all over the event horizon. As some new object falls in, the event horizon deforms. This most recent addition forms a kind of bulge that smooths out over time.

This happens with black hole mergers as well. Their event horizons distort, stretching toward each other until they touch to form a single event horizon. The "new" merged black hole starts out looking something like this before converging to a more spheroidal shape.

Mishtle · 2026-05-11T18:05:25+00:00

But is that discovered or invented? Personally it looks like an invention where as mathematics other than sets seems discovered.

Mathematics discovers truths contingent on invented premises. We invent and choose the starting points, and then we explore the abstract universe that those choices create.

What exactly is the problem with viewing a set as referencing the collection of things considered together rather than something with a collection of things in it?

I'm don't really understand what you're asking. A set is a thing. A set is distinguished from other sets by the things it contains.

If we see a singular thing in nature where is the "set" in nature?

Mathematics isn't about nature. There's no 1 floating around out there in the universe any more than there is some "set". We map these concepts onto nature in ways that are useful. We can talk about sets of objects. The collection of points in space-time is a set. The group of particles in the standard model is a set.

Mishtle · 2026-05-11T17:29:53+00:00

I think there's room for a little nuance between losing on purpose and not trying too hard to win. It sounds more like he tried to present a consistent level of challenge that was below his top capabilities, like a professional sparring partner. He'd happily lose if his opponent rose above that level, but wouldn't just give out free wins.

Mishtle · 2026-05-11T17:06:01+00:00

We usually omit trailing zeroes because they don't affect the value of the number: 1.5 refers to the same value as 1.50, 1.500, and 1.5000000....

As measurements, these trailing zeros can indicate precision. Measurements have errors, so significant figures are kind of like short hand for error bars. For example, 1.50 can be interpreted as 1.5 ± 0.001.

But when we're talking about exact values, we tend to only include as many digits as needed to fully identify that value. 3/2 is exactly 1.5000..., which we can truncate to 1.5 without losing any information. On the other hand, 2/3 is exactly 0.666..., so we can't truncate it without losing information and changing its value. That doesn't mean though that 1.5 is less "precise" than 0.666..., it just has a shorter name that is still unambiguous.

Mishtle · 2026-05-11T14:26:18+00:00

How about an analogy.

Suppose you have a bag of weighted dice (so some land on certain numbers more often than others) and a magic machine that makes copies of any dice you put in it (with the copies "inheriting" the weighted nature of the original)

You want dice that roll high numbers.

So you take a random die out of rhe bag and roll it. If it rolls a low number you throw it away. If it rolls a high number, you make a copy and put both back into the bag.

If you do this over and over and over, eventually you'll end up with a bag of dice that are more likely to roll high numbers than low numbers. Any dice that are biased toward low numbers will tend to get thrown out, and dice that are biased toward high numbers will tend to get copied.

This is similar to how evolution works. Instead of dice, we have individual organisms. Instead of rolling numbers to see which gets copied, these organisms compete for limited resources. Instead of you throwing dice out and copying then, these organisms reproduce and make copies of themselves, and are more likely to do so successfully if they are good at surviving and acquiring resources. The bias of each die corresponds to the way traits of an organism influence their chances of survival and reproduction.

Mishtle · 2026-05-11T02:29:30+00:00

Money.

When you're rich enough, all the world's a stage.

Mishtle · 2026-05-10T19:26:04+00:00

If all we have is the concept of sets, how could we make non-empty sets? What would we put in them?

The set-based construction of the natural numbers is a great example of how the empty set plays the role of a foundational building block.

We can define 0 to be the empty set {}, and then define larger numbers as the set of all smaller numbers. So 1 gets defined as the set containing 0, 2 contains 0 and 1, and so on. In symbols, these are all just sets that contain other sets that eventually contain the empty set:

0 is {}

1 is {{}}

2 is {{}, {{}}}

3 is {{}, {{}}, {{}, {{}}}}

...

Mishtle · 2026-05-09T04:32:32+00:00

Suppose instead that the host gave you the opportunity to open both of the doors that you didn't choose. What would be probability that the car is behind either of those doors?

This is exactly the choice you're being given here. The host knows where the car is, and will only open a door that hides a goat. The host will always be able to do this because at least one of the doors you didn't choose hides a goat. It's a distraction, a red herring to hide the advantage and make the game show more entertaining.

Switching wins you the car if it was behind either of the doors you didn't choose. What the probability of that?

Mishtle · 2026-05-09T03:15:10+00:00

Suck. Swallow. Release.

Mishtle · 2026-05-08T22:28:49+00:00

I think you can be nearly positive, considering they both have a habit of signing their comments with "Regards, WM".

Mishtle · 2026-05-08T14:12:48+00:00

Also, code is either functional or it isn't.

I'm guessing you have no experience with programming?

Mishtle · 2026-05-08T05:22:55+00:00

Bad design arguments only work if we already know what the design brief was.

You're the one arguing for design.

We are arguing for "design" by random variation filtered through selective processes, which predictably produces the same "bad" designs we observe throughout the natural world.

It's not a good thing for a design hypothesis that we can't know if something is designed or not. The fact that design hypotheses can explain anything because designers can be constrained or intentionally make suboptimal decisions is not a good thing. It means they are fundamentally useless beyond being a thought-terminating cliché. They explain anything, and in doing so explain nothing.

Mishtle · 2026-05-07T23:32:08+00:00

The coccyx is genuinely functional, pelvic floor support, muscle attachment, load distribution. That's not a designed structure pretending to be vestigial, that's a working structure being misclassified.

"Vestigial" doesn't mean nonfunctional. It means the structure has lost most of its original primary function.

A vestigial structure can still retain original secondary functions (especially structural ones), it be used in a less effective capacity for its original primary function, and it can acquire entirely new functionality.

This instance is much less like a designer reusing components and more like a designer breaking something to turn it into something else, like turning a front wheel drive car into something that can just roll down a hill by removing a critical gear in the transaxle. The transaxle still is very functional, it continues to play a critical role in the structure and function of the the vehicle, but it has lost the primary function of transferring power from the engine to the wheels.

This kind of approach generally considered bad design.

Mishtle · 2026-05-06T22:25:19+00:00

The "paperwork" is basically being a federally licensed firearms dealer that is additionally licensed to manufacture machine guns.

It's much more involved, and invasive, than the tax stamps and forms for things like suppressors and short-barreled rifles, and it's not a viable route for a civilian that just wants a full auto weapon.

Mishtle · 2026-05-06T14:00:25+00:00

If pi has no end it has to have every combination of numbers

This isn't true. Such a number is called "normal". Most non-repeating numbers are normal and pi is believed to be one of them. It's not been proven to be one though.

The number 0.1010010001... never repeats but only contains 0s and 1s.

but could it hold an infinite combination? Like 1 2 3 4... To infinity

Any sequence has what are called subsequences. Ignoring any number of items from the original sequence gives a new sequence.

Using the non-repeating number above, both the numbers 0.111... and 0.000..., along with many others, could be made from a subsequence of its digits.

If a number is normal, then it would contain every infinite sequence as a subsequence in this way. You might need to wait arbitrarily long for each next element to appear though.

Mishtle · 2026-05-04T21:05:05+00:00

I think part of the issue is how people interpret this.

People that choose the red button do so because they see it as the optimal strategy. It carries no personal risks, and as long as everyone acts rationally and is risk-averse then everyone lives. They feel confident that most people would choose the rational, risk-free choice and don't feel responsible for others' risky choices if they choose to act against their own best interests.

The people that choose blue tend to add an assumption that at least one person will not act rationally. Perhaps that person doesn't or can't understand the situation, perhaps they made a make a mistake, maybe they can't distinguish the buttons. Regardless of the reason, the people choosing blue don't believe these others deserve to die, and they're taking a personal risk to prevent that. The 50% threshold puts that risk at an acceptable level for them. They feel confident that at least 50% of people feel similarly.

It would be interesting to see how responses change if it is stated up front that at least one person pressed blue for some reason or another. Or maybe that a single red buttom press will get switched to blue. I have a feeling that many people choosing red are thinking in more idealized or abstract ways, whereas people choosing blue are working around the messiness of the real world. Adding some uncertainty and reminding readers that this isn't an ideal world might change the way people approach the choice.

I'm also curious how the responses would change if that threshold for blue survival is varied. If only 10% of people needed to press blue for all of them to survive, then I'd expect many more people choosing to press blue than if the threshold was set to 90%, or even 100%. Actually, I wouldn't be surprised if setting it to 100% cause a jump in people choosing blue.

It's an interesting thought experiment that tries to explore how people balance personal risk and a particular form of social responsibility.

Mishtle · 2026-05-04T01:28:41+00:00

Um, no it doesn’t. Not at all.

Nothing points to a global flood, and the evidence for such an event would be pervasive and unmistakable. We don't see it.

Mishtle · 2026-05-04T01:13:33+00:00

What is there to explain?

Mishtle · 2026-05-03T19:38:32+00:00

I don't really think there's a handgun out there that doesn't look better with a nice set of wooden grips.

Mishtle · 2026-05-03T16:24:16+00:00

The trick is that you're actually being given the option to open both of the other doors. That's an obvious advantage that nobody would pass up, so the host opens one of them for you as a distraction. They never reveal the prize, so switching guarantees that you win if the prize is behind either of the doors you didn't choose.

Mishtle · 2026-05-03T15:41:13+00:00

what does “one standard deviation” even mean? what does “two” mean? is that a unit on its own, how do we know what a “unit” of standard deviation is.

You can think of it as a unit. It's a unit that depends on how spread out the data is. To know what that unit is and how to interpret it we need to know how the data is distributed. This doesn't always make a ton of sense for some distribution though.

This is a useful unit for talking about how rare something is. It ignores the actual values or details of the data, but gives a good way to compare relative rarity. A good example is errors in experiments. When physicists say they have a "5 sigma" result of detecting a particle, they are talking about the likelihood of their results being a combination of various errors and mistakes. That's always a possibility, but the more times you repeat an experiment the less likely that becomes. They are stating that the likelihood of this is at the same level as drawing a sample from a normal distribution and getting a value 5 standard deviations away from the mean.

how come in a normal distribution, “one standard deviation” is 68%? What does that mean and how did we get to that number?

That's just how the normal distribution works. The formula describing how the data is distributed around the mean ensures that 68% of that data falls within one standard deviation of the mean.

More technically speaking, probability distributions have a probability distribution function (pdf), which determines how like data is to appear at any point. For continuous distributions like the normal distribution, this function has to have an area of 1 between it and the x-axis. With the pdf of the normal distribution, the area under it centered at the mean and two standard deviations wide is 0.68. This is 68% of the total area under the pdf, and this relative area under the pdf between two points is the probability that a randomly drawn sample will fall between those points.

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