In the chart above, each turn of the spiral represents one power of ω --or a 3 that has fallen asleep on its back-- and represents ordinal numbers up to ω

^{ω}, or a 3 that is both asleep and dreaming that it is asleep.

In set theory, an

**ordinal number**, or just

**ordinal --**if you are on a first name buddy-buddy basis with it

**--**is a type of a well-ordered set. It is most often identified with hereditarily transitive sets --born that way. Ordinals are an extension of

**natural**numbers (numbers captured wild, then trained and sometimes even civilized) different from

**integers**(Latin,

*interdigitus --*numbers you can count on your fingers) and

**cardinals**(numbers who dress like hand puppets and can vote for new Popes).

Like other numbers, ordinals can be added, multiplied or exponentiated. Exponentiatiaton, being the most expensive of the three, is an arithmetical status reserved for royalty. Thus did the child of Henry the Eighth (ordinal number) and his third wife (exponent 3), Jane Seymour, produce an heir to the throne (8 cubed) known as King Edward The Five Hundred And Twelfth, who shortened his name to Edward VI (his cube root minus 2 parents) so he could get a fake I.D. and buy beer.

Ordinals were introduced by mathematician, Georg Cantor, in 1883 to accommodate infinite sequences and classify certain

**orderly**sets apart from sets that used foul language and started barroom brawls. He derived them by accident while working on a problem concerning trigonometric series when Sir Isaac Newton fell on his head. Although principal characters in this account were not contemporaries and lived centuries apart, such minor differences were set aside in view of the problem and its immense gravity. Would that the world do likewise.

Cardinals dress like hand puppets. (Snicker, snicker) Yeah, they do!

ReplyDeleteThanks Squid. Yours is the only comment I've got so far; maybe ordinal numbers are just too controversial.

DeleteLooks like you've gotten plenty of love since. Isn't it interesting, though, how many of the comments express a fear of numbers? Granted, your post goes quite a bit beyond basic arithmetic but math anxiety is epidemic in our society. Numbers are our friends, people!

DeleteI know. It's understandable though. In 1962 I got a kit from Radio Shack, a soldering gun and a pamphlet on Boolean algebra. My 1st thought was: Now here's a system that's going places! My comprehension, however, remained in that decade.

DeleteI'm sorry....this is just waaaay over my head....I'm doing well to count to 100 and please don't suggest 'new math' ...just ordinary numbers for me in orderly sequences with no mystical ordinals.

ReplyDeleteOh I'm sorry too. I just got real excited about arithmetic at midnight and left bed to write about it --probably a poor decision.

DeleteDefinitely from a different species than myself....I have NEVER gotten excited about arithmetic lol.

DeleteI'm glad I can write because I'd be in trouble if my living relied on my math skills!

ReplyDeleteAs a retired gardener, I'm similarly limited but sometimes calculations make the world seem more reliable. I too am glad you write!

DeleteYou remind me of just why it was that I was happy for my mother and one of my brothers to fight over which of them did my maths (and physics) homework.

ReplyDeleteNumbers are a mystery to me.

Mind you every now and then I can see (some of) the fascination. Did you know that if you add the digits in any multiple of nine the result is also nine?

Yes, a favorite of recreational insomniacs from 9 to 90!

DeleteWould.

ReplyDeleteGood.

DeleteThe power of w is actually a 3. It is on its back because it was knocked out during a barroom brawl. When it hallucinated that it was an exponent of itself it took another look at its disorderly ways. That was when it decided to turn over a new leaf and become an ordinal exponent.

ReplyDelete"Exponent of itself", there's something cosmological there, a new enigma! I like it.

DeleteEh, it's all like the Collatz Collateral, you never get to the end of it, hence it's unproven. But it's a good thing to go to sleep to; the journeys you'll take.....

ReplyDeleteThanks! I have used the Collatz Sequence to identify odd or even numbers and try to further simplify it. I usually divide by 2 and, if anything greater than zero appears right of the decimal, I conk out.

DeleteWell, if you ever have a breakthrough, or perhaps solve in set theory the problem of the core model let me know, a Fields Medal might be out there somewhere......

DeleteTonight I'm working on the Russell Paradox: can a set of numbers that don't belong to any set exist?

DeleteArithmetic was always my very worst subject, I nearly flunked algebra, and to this day I can hardly add a simple column. I was following you perfectly until we got to the incomprehensible chart and then I took an unexpected and irreversible detour.

ReplyDeleteThe ordinal numbers hopelessly confused me - - but your explanation of them provided entertaining and delightful confusion.

Thanks Jon, but I don't understand the chart either. Best I can grasp ordinals is that they are numbers who decided to become adjectives --first lady, second thoughts, third time around, fourtha July, fifth amendment etc.

DeleteMaybe it isn't understanding the chart that matters: it's what is described thereof that is much more interesting. Your final sentence is certainly one I do understand, and with which I wholeheartedly concur.

ReplyDeleteThanks Tom, Would AND Good!

DeleteGoodness gracious Geo...I haven't had enough coffee this morning to process this post. I'm a little dizzy trying to. But then again, math has never been my strong suit :)

ReplyDeleteKeith, math is HARD. I'm not sure the IS enough coffee to process all of it.

DeleteI did NOT suck at math, and yet I read this not from the point of one who does not suck at math but as one who does not suck at writing.

ReplyDeleteAnd it was hilarious.

Pearl

A kind equation. Thanks, Pearl.

DeleteMy husband (who is a mathematician) read this aloud to me, giggling at times and guffawing at others. While I could not completely understand the stream of thought, we both enjoyed this. He went on to read about The Prisoner, and agreed that the balloon were completely creepy.

ReplyDeleteGlad you both enjoyed it. I seem to be thinking about numbers a lot this month!

DeleteWhen I saw your post title, I pictured the yurt that holds my brain collapsing. Then, hahahahahahahahahahahah!!!!!! Thank you Geo., I needed that. I love your wit!

ReplyDeleteMy pleasure, Austan, and my priviledge.

DeleteSo how long did it take you to write this post, Geo? My fuzzy mind (and getting fuzzier) who loves numbers cannot comprehend your brilliant mind. You are more than a treat to read.

ReplyDeleteKind Arleen, I think I spent an hour or two on the post, which includes snacks. Glad you have fun with numbers too.

DeleteDear Geo.,

ReplyDeletecoming back from a short holiday in Sylt, the stormy weather made me too dizzy to think about numbers, sorry. Another time fully alert again.

Dear Brigitta,

DeleteIf a beautiful island made you dizzy, avoid looking at the spiral-shaped graph in my post. It makes me dizzy!

Over my head Geo ...my mom was the math expert in the family ...

ReplyDeleteYou have a good mind ...

Be well and take care ..

Ah, my mother was the perfect speller and grammarian of the family. I could've used a math mom too. Thanks and good wishes!

DeleteNot sure, dude, but this post may be my all-time favorite of yours. I always enjoy your posts, of course, but this one had me laughing out loud. Brilliant. (Yes... my name is Susan, and I'm a math nerd.)

ReplyDeleteThanks Susan. Dudes can be math nerds too! Admission is the 1st step, what would the remainder be?

DeleteYou should have been a math teacher--your students would have laughed and there is no better way to learn. Thanks for sharing.

ReplyDeleteKind Sage, in and out of school, real communication is often accompanied by laughter.

DeleteHaving seen your comments on Words for Wednesday, I decided this morning it was time to visit your blog! Your post made me laugh. Just minutes before, I had been cleaning off my desktop, and trashed a graph my son had left there at the end of the school year. (Yes - some computer organization needed to be done!) The graph was a circle - similar enough to yours to make me do a double take - but full of degrees, fractions and Pi symbols.

ReplyDeleteLife is an enigma - and the coincidence I just experienced has convinced me I should be following your blog :)

Thanks, Susan! In view of this synchronicity, I added you to my blogroll.

DeleteI don’t relate well with numbers – in any languages. My husband has a strange relationship with numbers – he will look up at the clock and it will say, for example, 11:11 am, then another time he will tell me, look at the clock, and it says 12:34, or at the clock on the radio in the evening and it will show 4:56, or 5:55, and that happens almost every other day. It is very strange and I don’t know how he knows to check the clock then…

ReplyDeleteI know, sometimes time just doesn't seem very orderly.

DeleteYou really are brilliant.

ReplyDeleteAnd you are really kind.

Delete