leslie townes

people who study that stuff eventually go nuts, or are nuts to begin with. no exceptions

it's nice for my sense of humor to be appreciated

which is why you combine it with the krein milman theorem.

nah, every time they do that they are secretly applying the krein milman theorem and the boolean prime ideal theorem.

it would also be helpful to see a concrete example of something that this principle would be applicable to :)

so it is likely that any difficulty of proving something by normal induction on the natural numbers will not be removed by some space age general induction principle

its maybe worth keeping in the background, all of these schema for proving things, are just schema for proving things. there are difficult and open problems that might conceivably be provable by induction on natural numbers, or maybe not. a problem generally is not simplified, or made more complicated, by the availability of approaches to solving it

it might help to eliminate any reference to P or other abstraction and give an example of something you would like to prove

and let that be a lesson to you

yeah, people are just objecting to lang's treatment of the axiom of choice.

i put all of the blame on serge lang

i could not tell the difference.

from the POV of someone who cannot see deleted posts, it very much looks the same to post a new amended question, vs. amending a previously posted question.

i do think that MSE orthodoxy is to amend something rather than repost something, but i do not have high enough reputation to see what the originally posted now deleted thing was.

oh, just ignore ted. he is salty.

this all strikes me as unusual, as far as it goes with selecting a route from AC to zorn's lemma. although i do not remmeber how it is normally done. i am guessing that it is not done in this way

it is the hat from which we can produce a great many rabbits

in his algebra book i think he takes it as fundamental. it might be how he presents the axiom of choice in the book. i could be misremembering things. but that is how it is often used in algebra.

at a high level, what is lang proving zorn's lemma from? is it a choice principle? is it something like a krein milman theorem (e.g. a selection theorem pulling elements from sets with certain properties)

i chatted with lang a couple of times, never about math. he was a very unusual person

okay, yeah, that seems kind of weird to me. i would want to know more about what f is. this probably isn't your fault. it is lang's fault.

full disclosure, i am not fully devoted to this. my son is running around yelling as i type this.

is "extreme point" here relative to a particular f, or is it some absolute notion?

and also makes me think that "extreme point" is not the usual thing but some weird fever dream from the mind of serge lang

what's kind of weird about all of this is that maybe lang is working in a slightly less than familiar context. if you said to be "zorn's lemma" and "extreme point" i would be thinking something like the krein milman theorem, where zorn's lemma or something equivalent to it is what you are using to prove the theorem. here, we are proving zorn's lemma, which raises the spectre of what we are assuming.

:)

anyway, i salute you for attempting to read a textbook by serge lang.

although there is no satisfying some people :)

but to be clear i think it is already fine as written, as far as meeting the threshold, for "i would not vote to close this."

ben it looks fine to me as written although it feels out of order. if the statement is that "every element of M is an extreme point" the next thing my mind wants is what M is. not, like, four bullet points later. i might also want to know if an "extreme point" in an "inductively ordered poset" is the same thing as what i might be familiar with from the vector space context, or what an "extreme point" is.

i was trying to get my daughter to appreciate the number line the other day (she was doing some subtraction problems) and she just refused to hear it. she didn't want to be bothered with "weird pictures." i didn't either, believe me. but i grew up.

e.g. the = set is roughly the boundary of either of the > or < sets, and the > set can be obtained from the < set (or vice versa) by complementation and removing boundary points.

before diving into any algebraic specifics. people will always mess those up anyway

yeah, there is a fundamental thing, which is that the mutually exclusive conditions f(x) < g(x), f(x) = g(x), f(x) > g(x) always divide the real line into disjoint regions, and under normal highschool type circumstances the set f(x) = g(x) is a set of "points," and the other sets are intervals separated by those points, and you can use information about any one of these sets to sort of guess at the structure of the others.

we might be well into a regime where sane people might use a continuous approximation. :)

and i love that we now have 10^5.

interesting. it hadn't occurred to me to use a formula involving binomial coefficients, mainly because i was working in an environment that did not have them built in, but also because that (-1)^n stuff suggests maybe the formula involves cancellation of numbers that are larger than they "need to be" to get the result. i was iterating a single function over a list, which only ever dealt with accumulation of nonnegative numbers.

you need a really big abacus

i think

eventually you run into just needing to do arithmetic with really big numbers, and that becomes the limiting thing

the only thing stopping me from doing it in assembly is feeling like i can't do really big numbers better than haskell

mine was like 40 minutes but i was gaming and shilling crypto in other windows while it was going

no optimization, just zipping a list with translates of the list and +

anank how did you implement your s(n,k)? i had mine go by n, computing the full list s(n,k) by iterating that pascal recursion thing over s(n-1,k) on a list that got 6 things longer with each iteration

it might be more productive than what society is currently doing, at least we'd learn about coin flips

hahah let's boil the oceans and put AI levels of compute into s(10^k,3*10^k)

yeah, that's basically where i am with mine. let's declare victory

@anankElpis i dont think my setup can do s(100000,300000), can yours?

mod 100,000, anyway

we agree!