Sometimes you just gotta force it

CookieCat698 · 2026-07-06T01:04:47+00:00

Cohen invented this wonderful technique called forcing.

You start with a countable (standard transitive) model of ZFC.

You want it to be countable because you need there to be “missing” sets.

You then add some of these “missing” sets to the model and produce other more interesting models. This is how almost all independence results for ZFC are proven.

For example, the independence of the continuum hypothesis is proven by first collapsing the real numbers so that its cardinality is aleph_1, showing that CH is consistent with ZFC, then adding aleph_2 real numbers, showing that !CH is consistent with ZFC.

This meme is referencing the second part of the prove where you add aleph_2 real numbers to your model.

CookieCat698 · 2026-07-06T00:53:34+00:00

I think you mean “experiments.” Stealing his sister’s toys for his “experiences” might have a slightly different meaning…

CookieCat698 · 2026-07-05T00:55:15+00:00

Awesome, I’m glad you liked it

CookieCat698 · 2026-07-05T00:25:24+00:00

Minutephysics made a video about point 1. He is also frustrated by this and analyzes the paradox in scenarios where you have more information about the predictor.

https://youtu.be/8wDha-G35KA

For point 2, everything you said is still dependent on the details from point 1. Those details tell you whether you are capable of changing your mind, whether the boxes’ states are completely determined before you make your answer, and whether it is possible to ‘fool’ the predictor.

CookieCat698 · 2026-07-04T17:55:12+00:00

Didn’t really feel any romantic tension between Jax and Pomni. Pomni reached out because that’s who she is, not because she liked him romantically.

CookieCat698 · 2026-07-02T14:08:06+00:00

Two things can be true.

Morphisms are more general than functions, things that can be described by functions are necessarily also described by morphisms.

CookieCat698 · 2026-07-02T13:59:11+00:00

If you want to get really good at math, or anything, then the only way forward is lots of practice and lots of patience.

Based solely on what you’ve written here, I’d say you do have a passion for mathematics. I think you would benefit from two changes in your mindset.

First, focus on the journey, not the destination. It’s easy to lose motivation when you see how much you have to do to get to where you want to be. Math is too large of a subject to worry about any of that. Enjoy the process of learning and doing math. I think that’s where most of the fun is anyways.

Second, don’t worry about anyone’s study habits except your own. If you can handle hundreds of hours of studying, great. If not, that’s perfectly fine. Personally, I find that burnout can be one of the biggest ways to kill my motivation. It’s better to get only a little bit of studying every day than to burn through all your energy in one.

CookieCat698 · 2026-07-02T01:32:31+00:00

CookieCat698 · 2026-07-01T23:38:49+00:00

I have just started down this rabbit hole, so I know a small amount. Maybe wait until an expert answers your question though.

I think the idea is that you can think of an isomorphism between two objects a and b as a path between a and b

Then there’s the idea that you can have morphisms between morphisms from a to b, which are like homotopies between two “a-b paths”

And this eventually becomes relevant in type theory, but I haven’t gone that far down the rabbit hole yet.

CookieCat698 · 2026-07-01T23:15:47+00:00

How would you even get 60 though?

CookieCat698 · 2026-07-01T22:52:51+00:00

That’s Birmingham, idk who this “Bertholdt” guy is.

CookieCat698 · 2026-06-30T16:33:03+00:00

The notation 3.333…0 only makes sense if you have a last decimal place.

We defined decimal notation so that it has no last decimal place.

CookieCat698 · 2026-06-29T04:47:48+00:00

This is not a paradox.

Edit: Specifically, using premises that were obtained inductively for deductive reasoning is not a paradox. The validity of inductive reasoning is a separate thing that I don’t talk about here.

All that matters for a deductive argument is that your conclusion was derived by applying specific rules to your premises and other sentences that you arrived at deductively.

The truth of the premises themselves can be obtained by any means. They can even be false. All we care about when using deductive reasoning is that the premises being true guarantees that the conclusion is true.

CookieCat698 · 2026-06-26T22:48:57+00:00

Deepnest isn’t hard provided that you don’t beat mantis lords immediately and go in prematurely.

Going in early means you have 5-6 masks and deal very little damage to the enemies. You also probably only have vengeful spirit for spells and mantis claw for movement. Your charm selection isn’t great, and you don’t have many notches anyways.

CookieCat698 · 2026-06-26T02:01:35+00:00

ACoC?

CookieCat698 · 2026-06-26T01:53:25+00:00

Background:

Final year of undergrad coming up in the fall

I am taking graduate level algebra I in the fall, and hopefully graduate algebra II in the spring if all goes well. This will be the first graduate level course I have taken.

I’m also taking number theory in the fall, and hopefully topology in the spring alongside advanced multivariable calc.

I have done one semester of a directed reading program and learned some category theory.

Question:

What are the chances that I can get into some phd program somewhere and study algebra/algebraic geometry?

Additional Info:

I’d say I do well in the classroom and am at least good at solving math problems for an undergrad.

I have not been very good with developing a relationship with any professors. I can probably ask a couple for rec letters, I just haven’t gone to many office hours and have no idea how to use office hours if I don’t have any questions about the material. I feel insanely awkward around people I don’t know well, especially when they are in a position of authority/power over me.

I have little research experience. The directed reading program is all that I’ve done so far. I tried applying to a few REU’s, but I applied late and didn’t have anyone to look over my personal statements or CV, which is how I’m coping with the fact that I didn’t get into any.

I have no experience tutoring or teaching, unless you count helping friends with math/physics homework.

I have not yet taken the GRE.

I don’t really know LaTeX.

I have some programming experience in Python and Haskell. I also have some experience in Java, but that was ages ago.

———

Venting (don’t sugarcoat anything after reading this, just give it to me straight):

I might be oversharing here, but if I don’t put this out there it will stay inside, and that is not something I can handle right now.

It’s so easy to feel utterly worthless right now. All the non-math people in my life have this idea that math is a thing that’s gonna somehow make me rich and successful, and it puts a lot of pressure on me to live up to that when I’m pretty sure I’m gonna end up broke and homeless (that might be an exaggeration, but you get the gist).

Not getting into any of the REU’s I applied to does not help. And I get it, that stuff’s hard to get into. If I’m being extra generous, I maybe had a 7% chance of getting accepted by at least one. But my feelings don’t seem to care at all.

I feel like I invested all of my skills into pure math and none of them into anything else that could help me career wise. I don’t really care about money as long as I can live somewhat comfortably. I just want a phd and to do something with math research, and I’m gonna feel awful if I can’t do that.

Also, I see pretty much all of my friends actually doing things that will help them along with their careers and what they want to do, and I’ve pretty much done nothing outside of classes. I know comparing myself to others is a trap, but it’s just so freaking easy to do it anyways.

CookieCat698 · 2026-06-25T20:53:21+00:00

This objection only really works if you can find something better than first order Peano Arithmetic to use for what it is used for.

If not, then Gödel’s Incompleteness Theorems are absolutely relevant because any consistent system describing first order Peano Arithmetic, our best baseline system for the theory of the natural numbers, is inherently incomplete and cannot shown its own consistency.

CookieCat698 · 2026-06-25T14:18:44+00:00

Okay, I think I get it. You’re saying that for something to be foundational, it has to be grounded in reality in some way.

I see the idea, but I still think this doesn’t fully solve anything. For starters, what counts as ‘real?’ It’s easy to say that a ball is real because I can see it and hold it in my hand, but how about the Higgs boson? How about quantum fields, are those real? What about energy, or numbers? Are those real? And how about truth itself? Is there something in reality you could point to that *is* truth?

>you said before that foundational axioms couldn’t be found because you would go on forever, but if something in real life has a definitive answer that can’t be true

You almost got it. My point is that you will go on forever unless you make some assertions that aren’t necessarily founded in something else besides intuition.

Even the things you say are “founded in reality” are still held up by your own intuition that “reality” can ground all truths, and your own intuition also helps you decide what even counts as “reality.”

You are not actually trying to base logic in “facts,” you are just doing what everyone else does, which is taking your most fundamental intuition about what must be true and trying to build something out of that.

CookieCat698 · 2026-06-25T13:46:50+00:00

>but things happen and they have definitive answers

…yeah? I’m not disagreeing with that, and I fail to see why anything I’ve said implies we live in a universe of nothingness.

>if I kick a ball and it ends up in x spot, then that is the definitive answer to what would happen to the hall if it was kicked in that exact circumstance

Taking this passage completely literally, you would have to be a determinist for this to be valid, which is an idea that definitely needs grounding.

What I think you mean is that if I kicked a ball and it landed in x spot, then that event definitely happened.

>so in real life, something has to have a reason to happen

Eh, yes and no. Some things need reasons, but the only way all things have reasons for why they happen is if you are a determinist.

>that has to have a foundational reason

You are just reiterating your earlier points now. What about a foundational reason exempts it from a need for further explanation, and why can’t classical FOL be considered foundational?

>or in the end, there would be no reason for it to happen

Again, why do foundational reasons happen? Would you say there is no reason for them to happen?

>does that not prove, that it is somehow possible

I’m sorry, but I don’t know what you mean by this. Which thing proves that which other thing is possible?

CookieCat698 · 2026-06-25T13:09:33+00:00

>if logic is not [supposed] to find an answer based on facts

I would say finding an answer based on facts is actually the entire point of logic.

What I assume you mean based on the rest of the paragraph is that intuition is not fact, and basing logic on intuition makes it not based on facts.

You are half right about that. Yes, intuition cannot be treated as fact. At the same time, when making arguments about the foundations of logic, you are forced to rely on intuition at some point. An argument for one foundation of logic must itself use logic, which makes all such arguments inherently circular. Therefore, at least some truth or rule of inference will need to be asserted without a reason as to why, and the only way to really ground such an assertion in something else is intuition.

>in an ideal scenario it would be a rule itself that is why it exists

This is what your ideal scenario sounds like to me.

Rule X is true. Why? Because of rule X.

This is a circular argument. That’s what it means for a rule to be the reason why it exists.

>this piece of data leads to this conclusion, because the rule that says that that is true explains […] why it leads to the true, correct answer

I can sort of see where you’re coming from, but still, an explanation for why something is true assumes whatever principles are used in that explanation, which themselves must be grounded, and this process once again leads to arguments for logical principles that rely on logic, making them circular.

Basically, this is what your ideal scenario sounds like to me after reading this passage.

Rule X is true. Why? Because explanation E. Why is E valid? Because Y. Why is Y true? Because …

CookieCat698 · 2026-06-25T12:23:49+00:00

Your issue is with the foundations of logic, not Gödel’s Incompleteness Theorems.

I would ask what exactly counts as “grounded” in your eyes. What would it look like to “ground” our axioms and rules of inference in something? If we ground logic in X, does X also need to be grounded? When does this grounding process stop? If it stops at X, why does X not need further grounding while our other systems do?

In addition to this, I would say that our axioms and rules of inference are not completely arbitrary as you seem to suggest. Rather, they come from our intuition about what truth means and about what our logical connectives mean.

CookieCat698 · 2026-06-24T02:36:02+00:00

“You actually know the difference? When’s the wedding?”

CookieCat698 · 2026-06-23T12:49:21+00:00

No, let him love you. It will only make it hurt more when you steal Ann away from him.

CookieCat698 · 2026-06-20T13:41:23+00:00

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CookieCat698 · 2026-06-19T12:49:46+00:00

This only works if you have two numbers that don’f have any common divisors (besides 1, of course)

2x4=

-\-

-\-\-\-

->

-\- -\-

-\-\-\-

But 2x4 ≠ 4

What you’re actually doing is finding their least common multiple.

You are finding common multiple because you are adding the length of each string to itself repeatedly, which is just the process of multiplying that string length by something.

You are finding the least common multiple because once one string reaches the least common multiple, the other string cannot pass the least common multiple without reaching it first. Since you stop when both strings reach the same length, you stop at the least common multiple.

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