I was recently learning about ordinal collapsing functions, which are a systematic way to produce very large countable ordinals. Before I talk about collapsing functions, I will briefly explain some ordinals:
$$\epsilon_0=\omega^{\omega^{\omega^{\dots}}}\bigg\}\omega\text{ powers going up}$$
$...
@AkivaWeinberger So a countably finite amount of operations that always map $\alpha_n$ to $\alpha_{n+1}$ for $\alpha_n<\alpha_{n+1}<\omega_1$ will result in some $\alpha_\beta<\omega_1$?
@SimplyBeautifulArt Well I haven't studied that subject at all so I'm not familiar with any of it. Plus, I've got this large glaring gap in knowledge when it comes to infinite sets. I know the layman definition of aleph null and omega, but that's about it.
As you can see in the playlist, this guy's videos have to cover diagonolization and higher ordinals than basic omega for you to fully use fast growing hierarchy
@Ovi And if you remember my large number coding contest, well, I'll leave it to you to understand what $\epsilon_0$ is and how amazing it is that some people (@user21820 @Deedlit ) are able to come up with functions at that order of FGH
@amWhy Actually mission is not accomplished; the first linked post is not deleted, and I can't even vote to delete it because I'm the only downvoter. Sadly. But if you wish to, please help me check that the argument is irreparable. Thanks a lot! =)
@SimplyBeautifulArt @Ovi: PA is unable to prove a well-ordering of length φ[1](0). My entry used a well-ordering of length φ[2](0). Deedlit's entry would (assuming ZFC is meaningful) use a well-ordering of length so long that it can't be described using only countable things. I currently have a way of getting much further than φ[2](0) but haven't fully analyzed it yet!
well, it's the only way to define the exponential function if you require it to be holomorphic
and holomorphic is very important, as it leads to things like allowing you to use the exponential function in the residue theorem or Cauchy's integral formula
Just curious, if you don't answer any questions for a few days, do you have enough posts that you just get a contionus stream of reputation from your previous posts?
Does there exist a computable function that grows faster than fast growing hierarchy for every 'reducible' ordinal $\alpha$? Or does it follow that fast growing hierarchy grows as fast as any computable function? I honestly can't figure out how to even understand how I could tackle such problem...
Perhaps I shall do a twenty dollar contest of who can make the largest number on a white board (the kind that's big and on the wall) after most testing is over
because I want to see what random people come up with XD
I'm following an exercise book, so when I get to the next section, I use multiple sources to learn about it before solving it and then going on the the next one.
I realized that no book covers everything and I need to use multiple sources anyways, so I might as well attempt to solve problems of increasing complexity and new topics as I go through the book.
I'm going to college in Fall so I'm trying to get a (more-less) solid understanding of Calc/Analysis so I can transition into the rigors of college. I don't want to have the "middle" knowledge when the proffessors will be talking about the "previous".
I'd rather have "previous" first (better yet, "middle" too!).
No, as a homeschooler I can write my diploma earlier than 12th grade. Many public college don't admit students who aren't of college age, so I'm going to a private college.
long long is a large-int type. unsigned means that numbers are only positive. When x = 0, the memory sees this as 00000...000 (I don't recall how many 0s) in binary. If we decrement (--x) it underflows, causing 11111...111 which is as largest number representable with a certain number of bits.
I was recently learning about ordinal collapsing functions, which are a systematic way to produce very large countable ordinals. Before I talk about collapsing functions, I will briefly explain some ordinals:
$$\epsilon_0=\omega^{\omega^{\omega^{\dots}}}\bigg\}\omega\text{ powers going up}$$
$...
This is the Realm of Simply Beautiful…
This is the Realm of Simply Beautiful…