Transfinite Induction Puns

For example, the statement I proved (hopefully) is as follows:

If a < w_1 (omega_1), then N_a^(N_1) = N_a^(N_0) * 2^(N_1) (N is supposed to be aleph).

My 'proof' proceeds by showing that the statement is true for a = 1, then shows if a < w_1 is a successor and the statement is true for all b < a, then it is true for a. Then I showed if a < w_1 is a limit and the statement is true for all b<a, then it is true for a. I'm like 99% sure this method is valid, but I haven't seen it used or mentioned in my book, so I just wanted to double check.

Are there any reasons to not accept Gentzen's use of transfinite induction in order to prove the consistency of PA? In particular, what is the cost of accepting transfinite induction in order to prove Con(PA), and do mathematical logicians/philosophers of mathematics tend to accept Gentzen's proof?

Thanks.

I'm not well-versed in serious set theory, so I wasn't sure how to deal with this quandary.

Transfinite induction works similar to induction, but for ordinals. I always thought that if we assume the Axiom of Choice and then well-order the reals, we could use use transfinite induction to prove things about the reals.

However, by the continuum hypothesis we don't even know what cardinal the set of reals even corresponds to.

If we don't know what cardinal the set of reals corresponds to in terms of the alephs, why should we even expect transfinite induction to reach the cardinality of the reals?

This leads to another related question. Why should we expect the continuum cardinal to correspond to any of the aleph's in the first place? Everyone entertains the idea that 𝖈 = aleph_1 or aleph_2 or aleph_omega, but is it consistent for the continuum cardinal to be unobtainable by the usual processes that derive all the alephs?

Hi all. I have what seemed to be a very easy problem, but I encountered a roadblock along the way. Here's the full problem text:

> Let B_1 be a non-empty closed and bounded set of real numbers and suppose that {B_i} is an infinite nested sequence of non-empty closed sets. Prove that the intersection of all the B_i's is not empty.

My first thought was to suppose there were only B_1 and B_2. In that case, because the sequence is nested, any points in B_2 must also be in B_1 so their intersection is just B_2. Then if I add a B_3 to the sequence, any points in it must also be in B_2 and thus in B_1, so the intersection is B_3.

I can then use a simple proof by induction to show that, provided there are finitely many B_i's, the intersection cannot be empty because it must be the smallest such B_i. However, where I get stuck is making the leap to show that this property holds for infinitely many B_i's.

I have a very vague idea of something called "transfinite induction," but when I did a quick Google search I'm not sure if that's what I need here, or how to apply it. I think maybe if I could show that the sequence of B_i's converge to some limit function (call it maybe just B, I guess) then the rest of the proof might fall into place.

My other thought was that I'm given B_1 is closed and bounded. I'm assuming that since it doesn't specify otherwise I'm meant to use the standard Euclidean topology on the reals. Operating under that assumption, I know that B_1 is compact. And any closed subset of a compact set is also compact, so that means that every B_i is compact. Though I again got stuck here because I'm not sure how, if at all, this helps me.

Any ideas or hints would be greatly appreciated. Thanks!

I've been told that it is possible to prove that the cardinality of the Borel Sigma-Algebra on R - |B(R)| - is equal to |R| using transfinite induction. This proof has been asked as a bonus on assignments in two different analysis classes I've taken, but Ive never managed to figure it out. Can somebody explain how this works? We never studied ordinals or transfinite induction but Ive done a little research. I have also been briefly introduced to the Borel hierarchy

22:44:52 tj__> so an example of recursion would be something like.....5+5=10,10+5=15,15+5=20.

I don't want to step on anybody's toes here, but the amount of non-dad jokes here in this subreddit really annoys me. First of all, dad jokes CAN be NSFW, it clearly says so in the sub rules. Secondly, it doesn't automatically make it a dad joke if it's from a conversation between you and your child. Most importantly, the jokes that your CHILDREN tell YOU are not dad jokes. The point of a dad joke is that it's so cheesy only a dad who's trying to be funny would make such a joke. That's it. They are stupid plays on words, lame puns and so on. There has to be a clever pun or wordplay for it to be considered a dad joke.

Again, to all the fellow dads, I apologise if I'm sounding too harsh. But I just needed to get it off my chest.

Alot of great jokes get posted here! However just because you have a joke, doesn't mean it's a dad joke.

THIS IS NOT ABOUT NSFW, THIS IS ABOUT LONG JOKES, BLONDE JOKES, SEXUAL JOKES, KNOCK KNOCK JOKES, POLITICAL JOKES, ETC BEING POSTED IN A DAD JOKE SUB

Try telling these sexual jokes that get posted here, to your kid and see how your spouse likes it.. if that goes well, Try telling one of your friends kid about your sex life being like Coca cola, first it was normal, than light and now zero , and see if the parents are OK with you telling their kid the "dad joke"

I'm not even referencing the NSFW, I'm saying Dad jokes are corny, and sometimes painful, not sexual

So check out r/jokes for all types of jokes

r/unclejokes for dirty jokes

r/3amjokes for real weird and alot of OC

r/cleandadjokes If your really sick of seeing not dad jokes in r/dadjokes

Punchline !

Edit: this is not a post about NSFW , This is about jokes, knock knock jokes, blonde jokes, political jokes etc being posted in a dad joke sub

Edit 2: don't touch the thermostat

Do your worst!

How the hell am I suppose to know when it’s raining in Sweden?

Ants don’t even have the concept fathers, let alone a good dad joke. Keep r/ants out of my r/dadjokes.

But no, seriously. I understand rule 7 is great to have intelligent discussion, but sometimes it feels like 1 in 10 posts here is someone getting upset about the jokes on this sub. Let the mods deal with it, they regulate the sub.

They were cooked in Greece.

I'm surprised it hasn't decade.

Now that I listen to albums, I hardly ever leave the house.

Don't you know a good pun is its own reword?

Two muffins are in an oven, one muffin looks at the other and says "is it just me, or is it hot in here?"

Then the other muffin says "AHH, TALKING MUFFIN!!!"

For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.

I said "hey look, an escaPEA"

No one near me but it didn't half make me laugh for a good hour or so!

Edit: I can't believe how much this has blown up. Thank you everyone I've had a blast reading through the replies 😂

It really does, I swear!

And now I’m cannelloni

Because she wanted to see the task manager.

And boy are my arms legs.

But that’s comparing apples to oranges

Heard they've been doing some shady business.

but then I remembered it was ground this morning.

Edit: Thank you guys for the awards, they're much nicer than the cardboard sleeve I've been using and reassures me that my jokes aren't stale

Edit 2: I have already been made aware that Men In Black 3 has told a version of this joke before. If the joke is not new to you, please enjoy any of the single origin puns in the comments

They’re on standbi

BamBOO!

A play on words.

My daughter, Chewbecca, not so much.

Calcium, nickel, neon

Pilot on me!!

Christopher Walken

Put it on my bill

Nothing, he was gladiator.