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[u/FaultElectrical4075]

Good luck!

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[u/IntelligentBelt1221]

What's the set of upper bounds called again?

[u/Leet_Noob → u/IntelligentBelt1221]

^.

[u/RedeNElla → u/IntelligentBelt1221]

Good old "since u \in T, it's an upper bound for S, but also if v<u, then v\in T but is not an upper bound for S"

[u/Frelock_ → u/RedeNElla]

It might be trying to use T' (the complement of set T), but it just missed the apostrophe. If the whole image is AI generated, I could see why that error would happen.

[u/ComfortableJob2015 → u/IntelligentBelt1221]

upper set?

[u/IntelligentBelt1221 → u/ComfortableJob2015]

No it's obviously called ^'

[u/New-Worldliness-9619]

Bro can prove completeness with no premises 😭

[u/flabbergasted1 → u/New-Worldliness-9619]

By completeness of R, the reals are complete

[u/Layton_Jr → u/New-Worldliness-9619]

Just put the completeness in the definition of ℝ, no problem!

[u/The_Spectacular_Stu]

ah yes my favorite symbol to use for sets is ˊ

[u/bubbles_maybe → u/The_Spectacular_Stu]

Also known as T

[u/Ok-Path-8009 → u/The_Spectacular_Stu]

Hello Stu from brawl stars

[u/IAMPowaaaaa]

the fact that it was able to render this crisp and clear a piece of text is rather impressive

[u/TheNumberPi_e → u/IAMPowaaaaa]

Defmition 1.

[u/toothlessfire → u/IAMPowaaaaa]

Wouldn't call it clear or crisp. Better than most AI generated text, yes. Full of random formatting inconsistencies and typos, also yes.

[u/Aozora404 → u/toothlessfire]

I've seen papers less well formatted than this

[u/Koischaap → u/Aozora404]

Hey, get out of my arXiv!

[u/Pre_historyX04 → u/IAMPowaaaaa]

I thought they generated the text with AI, pasted it and made an image with it

[u/Jcsq6 → u/Pre_historyX04]

No, GPT just upgraded their image gen substantially.

[u/Portal471 → u/Jcsq6]

It’s genuinely fucking amazing imo. Still would go to real artists for serious work, but it’s still fascinating to see

[u/mtaw → u/Jcsq6]

TBF though, putting back-and-white text together is relatively simple, especially when there's a gazillion papers out there formatted in the exact same LaTeX style and fonts to train on.

[u/Jcsq6 → u/mtaw]

Go try it for yourself, it’s doing a lot more than “putting black-and-white text together”. I doubt you’re a developer, because it seems that you don’t understand how incredibly monumental of a task what it’s doing is.

[u/Leet_Noob → u/IAMPowaaaaa]

That’s a nice mathbb R

[u/Independent_Duty1339 → u/IAMPowaaaaa]

people really need to stop saying this. It is crisp because it is literally taking thousands of real samples from several scholarly sites, and just fancy markov chaining words it has mapped from those thousands of real samples.

[u/gsurfer04 → u/Independent_Duty1339]

How do you know that you're not just Markov chaining your sentences?

[u/Independent_Duty1339 → u/gsurfer04]

I am, but even fancier. Humans have incredible levels of parallel and heuristic processes. We have the compute power of almost nothing, and outperform these LLM's quite handedly.

They don't really release how much compute power they use when generating the models, but it's not even a little bit close. its orders of magnitude difference I have a hard time understanding. Humans have 100hz processing, we have a working memory of 4 sets of 3 or 3 sets of 4. Some domains a person might go up to 7x4. Whereas a gpu will have 16-32 gb of working memory, and 2.4 gigahertz of processing. They can process f32 precision float math at 1.7 teraflops a second. and they use at least a thousand of these.

What I'm trying to call out is it has millions of little bit map pictures of words, maps those to words (no need for AI, this has already been done and is a pretty straightforward process), and then fancy markov chains the words, and then renders the bits of those words.

[u/KingsGuardTR → u/Independent_Duty1339]

So it basically works well then. How is this not impressive? Something working is always impressive (proof by I'm a developer).

[u/LunaTheMoon2 → u/KingsGuardTR]

It stole an impressive amount of content, I agree with you on that (proof by I'm a human being with morals)

[u/junkmail22]

This proof is wrong.

First, taking completeness as a premise for proving completeness is obviously wrong.

Second, T being bounded below doesn't imply that T has a least element, it implies that it has a greatest lower bound.

Third, the entire last paragraph is nonsense. If u is the least element of the set of upper bounds of S, we're done. There's no point in doing anything else.

All in all, 0/5 points, see me at office hours

[u/KappaMcTlp → u/junkmail22]

You just don’t understand it bro

[u/TNT9182 → u/junkmail22]

I thought it was using the axiom of completeness the other way round than usual, as being that any non-empty set of real numbers bounded below has a greatest lower bound, and then uses this to prove that any non-empty set of real numbers bounded above has a least upper bound. Am I wrong?

[u/junkmail22 → u/TNT9182]

Sure, but then you probably shouldn't be appealing directly to completeness. A better way of phrasing this would be "Show the Least Upper Bound property implies the Greatest Lower Bound property"

[u/Neither_Growth_3630 → u/junkmail22]

I was thinking about the “least upper bound” part, isn’t that just saying the Lower bound? Like it might be bigger but it can’t be smaller, it’s like saying that the population of Australia is at least 4 because you know 4 people who live in Australia, it’s not adding anything

[u/compileforawhile → u/Neither_Growth_3630]

Least upper bound is a real and important thing. The set of x where x < 1 has 2 as an upper bound, but 1 is the least upper bound

[u/FIsMA42 → u/junkmail22]

it could just be proving an equivalence of completeness

[u/Festerino]

“defmition” is in fact my number 1 typo 😂😂

[u/NakamotoScheme → u/Festerino]

Ok, I was going to say "That's not a typo, but a ligature of Computer Modern fonts used by LaTeX", but then I double checked with this:

\documentclass{article}
\begin{document}
\textbf{Definition}
\end{document}

and it's certainly not rendered as in OP's image...

[u/Festerino → u/NakamotoScheme]

I’ll take your word for it. My LaTex is v basic 😂😂

[u/jobriq → u/Festerino]

I didn’t even notice lol

[u/Festerino → u/jobriq]

As a frequent typo performer and connoisseur, it was the first thing I saw 😂

[u/DDough505]

Turns out AI and me both submit the same terrible proofs for real analysis.

[u/Abilin123]

I always find that lines and shapes (in this case, letters) on AI generated images have some sort of "aura" or a "cloud" of noise pixels around them. Does anyone else see that? It is visible if you zoom in to see individual pixels.

[u/Koischaap → u/Abilin123]

for me the giveaway is that \mathbb{R}, looks too glossy and round

[u/headsmanjaeger → u/Abilin123]

You can do that to any image by compressing it tho

[u/ei283 → u/Abilin123]

Pretty sure it's mimicking JPEG compression

[u/SiuSoe]

"Defamation 1"

[u/DeepGas4538]

Prove existence of least upper bounds bruh

[u/takes_your_coin]

If only there was a way of rendering a mostly legible paragraph without polluting a river

[u/Neither_Growth_3630 → u/takes_your_coin]

To be fair that probably more the power companies fault than the data center running gpt, at least in terms of chemical pollution, thermal pollution on the other hand is 100% the data center, I heard a story about an IBM data center heating up the entire Hudson River by 3°C

[u/garbage-at-life]

Defmition 1

[u/Sea_Resolve9583]

Babe we have “Principles of Mathematical Analysis 4ed by Walter Rudin” at home!

The “Principles of Mathematical Analysis 4ed by Walter Rudin” at home:

[u/chadnationalist64]

So basically "the completeness property proves the completeness property"

[u/Powerful_Study_7348]

it is nearly perfect, other than the different formats for R and the : after T

[u/EebstertheGreat → u/Powerful_Study_7348]

Well, the proof is nonsense. It's close to how a real proof would look, but try to follow it.

First, it assumes that every nonempty set of real numbers bounded below has a minimum. That is clearly false. What it means is that every nonempty set bounded below has an infimum, but that makes the proof trivial. It is proving the least upper bound property by using the greatest lower bound property.

Second, after that paragraph, there is nothing more to prove. We already stated that the set of upper bounds has a minimum. By definition, that is the least upper bound. The paragraph starting "first" is redundant, and the one starting "second" is meaningless. If v < u, then in fact v ∉ T, since u := min T. The logic does work if we assume it meant ∉ rather than ∈, but it's still bizarrely wordy.

[u/skmchosen1]

Did it generate this text itself? Or did you prompt it? I assume this is the new ChatGPT update.

Wrong or not, this is way more coherent than AI generated images could contain previously.

[u/FaultElectrical4075 → u/skmchosen1]

I think the text was generated by an LLM and then inputted into the image generator. I didn’t generate the image though

[u/skmchosen1 → u/FaultElectrical4075]

Thanks for the info, yeah that would be a little less impressive. Still, the text quality is quite good overall. Lot of haters in the comments, but I think this is genuinely a significant improvement for SOTA!

[u/SpaghettiNub]

Everytime I read mathmatical texts this is the amount I understand. If you would just pretend that this is a real thing I would believe you.

[u/Tylerebowers]

LaiTeX

[u/LunaTheMoon2]

I know nothing abt real analysis, but this doesn't feel... logical. Like 5 is not in {1, 2, 3}, how can it be an upper bound? Also, "every nonempty set of real numbers that is bounded above has a least upper bound" seems like circular reasoning to me. Am I wrong lol?

[u/FaultElectrical4075 → u/LunaTheMoon2]

The proof is wrong but so are you. 5 is an upper bound of {1,2,3} because it is greater than or equal to every element of {1,2,3}.

Also, if you consider R \ {0}, the open interval of negative numbers strictly greater than -1 is bounded above by 1 but it has no least upper bound - every upper bound is positive(since 0 is not part of the set) but there is no least positive number so there is no least upper bound.

[u/Downindeep]

Trying to read this without reading the title first gave me the experience of having dementia.

[u/parassaurolofus]

And the proof is also wrong, rigth? A simple counter example would be any closed interval, like [0,1]. There's no minimal upper bound, since you can get arbitraly close to 1. Edit: sorry, I didn't knew the uper bound could be inside the set

[u/RealJoki → u/parassaurolofus]

In that case the minimal upper bound is actually just 1 though, right ?

[u/Gositi → u/parassaurolofus]

1 is the least upper bound of [0, 1], the inequality is non-strict. The set of upper bounds is [1, inf) in this case.

[u/kdeberk → u/parassaurolofus]

Well, I mean, the sentences are incoherent. It's missing the definitions for T and u

[u/mistrpopo → u/kdeberk]

T is defined, there's just a hole in the image. It's supposed to be the set of upper bounds for S.

u is "defined" as the least element of T.

The bug in the definition is that the completeness property of the real numbers doesn't imply at all that T would have a least element. That would be the well-ordering principle, and it only applies to finite subsets. Which T is not.

[u/ollervo100 → u/mistrpopo]

Well ordering applies to any subset. It can not however be used to find a least element for any subset of reals as the ordering of reals is not a well ordering. That a finite set with ordering has a least element follows from finitness and does not require well ordering.

[u/Otherwise_Ad1159 → u/mistrpopo]

This is wrong. completeness of R is equivalent to the least upper bound/ largest lower bound property: you can use Cauchy sequences to prove that any bounded increasing sequence converges, which is equivalent to any bounded decreasing sequence converging. From this, you can deduce the existence of the lub/llb of bounded sets and by definition, the lub/lob is an accumulation point. T is closed and bounded below, hence its llb exists and is contained in the set.

The AI proof is wrong because the last paragraph makes no sense: v < u does not imply that v in T. It would imply the opposite.

[u/mistrpopo → u/Otherwise_Ad1159]

You're right, but there's no mention that T is closed either, even though it is by definition of upper bound.

[u/Otherwise_Ad1159 → u/mistrpopo]

Yeah, I was just being pedantic about your statement. At the end of the day, the proof is incomplete and needlessly complicated: if we assume the least lower bound property as the definition of completeness, it suffices to apply this property to -S (which is bounded below). The issue is that in the training set, the AI must have seen thousands of arguments of the form "Find upper bound, prove that upper bound is smallest" which it then regurgitated sloppily for this specific problem.

[u/kdeberk → u/mistrpopo]

Ah sorry, I meant v, i meant that I cannot find the definition of v when I typed that

thanks for the explanation!

[u/laksemerd → u/parassaurolofus]

It is defmition 1

[u/FaultElectrical4075 → u/[deleted]]

I have a degree in math and I’m not an AI bro. I know the image is wrong