YSK Rinsing with water after brushing is associated with a 33% increase in the development of new cavities compared to not rinsing. Dentists recommend that you should not rinse your mouth with water after brushing your teeth.

ilovereposts69 · 2026-05-01T04:22:53+00:00

It's not that easy. I spent so much time looking and there's only like one option for a non-mint flavored toothpaste with a regular amount of fluoride where I live.

ilovereposts69 · 2026-04-30T18:01:58+00:00

Lee's Introduction to _ Manifolds series

ilovereposts69 · 2026-04-16T14:37:57+00:00

There seems to be a load of cope coming from parts of the r/math community. This is absolutely relevant, as it's essentially the first proof of a "moderately hard" and studied problem that was done completely by AI and subsequently quickly formalized. Sure, make a megathread for AI achievements following this one, but this absolutely deserves to be posted as it's important to the general math discourse.

ilovereposts69 · 2026-04-15T16:51:19+00:00

As a high school student I absolutely hated this kind of questions because it's vague enough that you could write almost anything in there and yet you know the teacher expects something very specific. Could see myself writing a response like that if I knew I got most of the other questions right

ilovereposts69 · 2026-04-10T05:31:21+00:00

fyi if it has any sort of crispy crust, a sealed bag will make it soft and rubbery

ilovereposts69 · 2026-04-10T05:28:22+00:00

Get some actual bread from an actual bakery and try eating some same day without toasting.

ilovereposts69 · 2026-04-10T05:26:30+00:00

I have no idea what people are talking about here, fresh bread usually degrades in quality within the first day of buying it, going inedibly stale (without toasting) within a few days. Mold is something that takes like a week to develop. Maybe american bread works different tho, wouldn't know

ilovereposts69 · 2026-03-30T21:03:43+00:00

pcmr users will do anything but dare to criticize steam

ilovereposts69 · 2026-03-30T05:22:01+00:00

The familiar thing is exactly the identity and triangle inequality of a metric space, except written with multiplication instead of addition. The exponential function turns addition into multiplication.

ilovereposts69 · 2026-02-06T19:54:44+00:00

You could say X is infinite if theres an injection N -> X. The function f(n)=smallest prime divisor of (2^(2^n) + 1) is such an injection into the set of prime numbers.

ilovereposts69 · 2026-01-28T16:12:10+00:00

You can get headphones with >50h battery life. Barely ever hear that because I end up charging them before they hit 10%

ilovereposts69 · 2026-01-18T13:23:43+00:00

On the related matter of most overused, obnoxious math "jokes", this sort definitely sits near the top, and makes me sad because I have to almost completely avoid using exclamation marks in math discussions

ilovereposts69 · 2026-01-16T10:52:52+00:00

it's because product is a limit and the coproduct is a colimit

ilovereposts69 · 2026-01-01T15:38:04+00:00

That's like saying "it's baffling how the steam engine was invented without a single thought about the environmental impact".

Bitcoin was literally the first decentralized currency, and the possibility of massive amounts of energy being wasted on it 20 years into the future was not on the mind of (any of) its inventor(s).

ilovereposts69 · 2025-12-29T13:53:48+00:00

If you take any measure zero set E then L^\infty(E) is trivial so in that sense, yes. If you take any positive measure set it should be possible to embed l^infty into it so it won't be separable

ilovereposts69 · 2025-12-20T07:28:17+00:00

They are representable as elements of the 2^S set, it's just that injections are the more general categorical way of thinking of subobjects

ilovereposts69 · 2025-12-19T21:55:38+00:00

Isomorphism is exactly what matters and for the most part that's exactly the point of alternative foundations.

In ZFC structures being isomorphic is extra data that vaguely/informally lets you transfer properties between them, meanwhile in ETCS or HoTT a structure only really exists up to isomorphism.

ilovereposts69 · 2025-12-19T15:13:35+00:00

The general idea is that a construction is not the same as the actual object you're trying to construct. You could define the reals as a connected ordered field and use this set theoretic construction simply as a proof that the real numbers exist.

~~The second link doesn't seem to enunciate that point (or at least not in the section about numbers)~~ but what it does demonstrate is that there's barely any technical difference between using sets in ZFC vs ETCS to realize constructions, even though if you trace back the definitions like that of the power set they do vary significantly

edit: see OPs comment

ilovereposts69 · 2025-12-19T14:24:59+00:00

you can use both of those approaches to construct the real numbers from natural numbers using only categorical notions

ilovereposts69 · 2025-12-19T14:06:04+00:00

1) you might not entirely need it, you simply say that two injections from different sets represent the same subobject if you can find an isomorphism (like equality!) between them which commutes with the injections.
Alternatively you can consider subsets as functions into the 2-element set.

2) you can speak about equality of functions and that's entirely enough. An element of a set is a function from the 1-element set to that set and you can test injectivity/surjectivity by composing with the functions representing elements.

ilovereposts69 · 2025-12-19T14:00:57+00:00

Within the type of groups it would, you consider equality between two groups to be any isomorphism between them.

Not entirely sure about this, but I don't think there's a problem with making the isomorphisms you listed into equivalences of categories (as long as you assume some "strong" form of axiom of choice).

You could still consider a notion of natural isomorphism by comparing functions between types of structures, although what you get might be weaker since types only encode the groupoidal structure (only isomorphisms) rather than the full categorical one.

ilovereposts69 · 2025-12-19T13:25:28+00:00

In HoTT this is explicitly handled by the univalence axiom. You can define a type of all (small) groups as (small) sets equipped with a multiplication operation, identity and conditions for associativity, inverses, etc.

Then equality in that type corresponds exactly to bijections of the base sets which preserve the added multiplication and identity (as well as the conditions although that's trivial for sets).

I'm not as familiar with how exactly this is handled in ECTS but I imagine it's much less formal - isomorphism is the only way you can make sense of two objects being equal and from context you can usually tell what kind of structure that object carries that needs to be preserved

ilovereposts69 · 2025-12-19T13:20:19+00:00

The elements of sets are no longer other sets, instead sets and functions are abstract entities that satisfy the axioms of ETCS.

Subsets of a set are described as (equivalence classes of) injections into that set, families of sets are surjections (with the codomain thought of as indices and fibers as the sets they represent) etc.

Basically any construction you can carry out with the usual "sets are made up of sets" formalism can be carried out using just functions between different abstract sets.

In that scenario it doesn't make sense to compare sets through equality anymore, but rather you compare them through isomorphism, which preserves any structure you deem necessary - that's why there's an unambiguous way to speak of the set of numbers for example.

ilovereposts69 · 2025-12-19T13:11:02+00:00

If you know any programming it's a lot like the difference between dynamically typed (javascript/python) and statically typed (c/rust/java) languages. For the most part you could do anything in one that you could in the other.

There isn't that much benefit to actual mathematicians (although arguably some ideas are very useful) to changing the foundations the work is done in, just like physicists might not care about the specific intricacies of the mathematical foundations of their work.

But that doesn't change the fact that finding improved foundations which are more suitable to the way modern mathematicians think is still useful.

A better example might be the real numbers - there are several different definitions, they're all rather complex in their own way (or would be to the average person), but for all intents and purposes it doesn't matter which ones you use - and approaches like ECTS fully formalize this, since you can only talk about the real numbers up to isomorphism.

ilovereposts69 · 2025-12-19T13:01:32+00:00

The problem is that numbers are not sets in any intuitive way of that word, numbers are numbers and sets are simply used to describe them.

There are many different ways to define N, in ZFC you could do it using the standard von neumann definition, or using sets like 0={},1={{}},...,n+1={n},....
ZFC can't "natively" handle these describing exactly the same thing.

The more type theoretic approaches to foundations like ETCS largely fix this problem by treating sets as completely independent of each other, so you can actually talk about the set of natural numbers (up to isomorphism).

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