"under the pseudonym ℵ₀"

http://www.zen224037.zen.co.uk/

You can really get a sense of his humor from his website: "55 years ago I first used Occam's Razor. Since then I have used no other."

Why am I not surprised it was Richard Parkin?

Charles Lang, is of course, a good candidate for "Archimedes" but I'm suspecting it may be David Wheeler, an incredible polymath that seemed to be beyond the cutting edge on every frontier of computer science for his entire career.

Apparently (per a comment in the article) Parkins was a member of a group called the Archimedeans and "Archimedes" is a composite character representing questions asked or points raised by his fellow Archimedeans.

These stories fascinate me; I remember how giddy I was when the famous "Mel" programmer's identity was discovered, even though I have zero need to know, nor even any relationship with any of the story. Other than said fascination.

That guy 'Mel' has amazing name recognition. On par with Mitnick (but for much better reasons).

If you don't know how Mel is:

https://en.wikipedia.org/wiki/The_Story_of_Mel

The story of Mel explained (2015):

https://jamesseibel.com/the-story-of-mel/

  Aleph one bottles of beer on the wall!
  Aleph one bottles of beer!
  Take aleph null down
  Pass them around
  Aleph one bottles of beer on the wall!"

https://math.bu.edu/people/jeffs/aleph-null.html

For a second I was thinking of Aleph One of this site from way back when: https://web.archive.org/web/19980214102549/http://undergroun...

Interesting read regardless.

I was thinking of another Aleph One who made both ARM processor upgrades [0] and biofeedback boxes [1]

[0] http://chrisacorns.computinghistory.org.uk/32bit_UpgradesA2G...

[1] http://www.aleph1.co.uk/about-us.html

That's a cool site. How did you discover it? What happened to it?

It's the homepage of the author of the phrack article. See my other comment.

I thought this was going to be about the stack smashing guy.

Aleph One wrote "Smashing The Stack For Fun And Profit." [1] I found this Aleph Null through a citation in "The UNIX Time-Sharing System" by Dennis Ritchie and Ken Thompson [2]

In case anyone doesn't know, Aleph-null is the cardinality of the natural numbers and Aleph-one is the cardinality of the countable ordinal numbers. On the continuum hypothesis, aleph-one would equal the cardinality of the continuum (i.e. reals.)

1. http://phrack.org/issues/49/14.html

2. https://news.ycombinator.com/item?id=28826870

I think you have a mistake in your aleph-one summary, if I recall correctly aleph-zero is the cardinality of countably infinite set (like natural numbers), and aleph-one is for uncountably infinite sets. It's been a while though so I may be misremembering.

Edit: PEBKAC, parent is correct.

Unless guerrilla's comment was edited and you're referring to an older version, guerrilla's comment is completely correct.

Aleph_1 doesn't refer to uncountable cardinals in general, it's a specific uncountable cardinal. More specifically, it is, as guerrilla says, the number of countable ordinals.

Aleph-0 is the naturals (countable). (And many other things, like integers, rationals, algebraic numbers)

Aleph-1 is the reals (uncountable), aka 2^Aleph-0. (Also any multi-dimensional (finite dimensions) complex space.)

Aleph-2 is 2^Aleph-1 &c

The continuum hypothesis is that there isn't an infinite cardinality between 0 and 1.

This is incorrect, and a common misconception.

Aleph_1 refers to the second-smallest infinite cardinal, whether that's equal to 2^(aleph_0) or not. (Aleph_1 is also equal, as has been mentioned, to the number of countable ordinals.)

The continuum hypothesis, in stating that there are no infinite cardinals inbetween aleph_0 and 2^(aleph_0), is equivalent to the statement that 2^(aleph_0) = aleph_1.

The series you refer to is denoted by the Hebrew letter bet, not aleph. So bet_0 = aleph_0, bet_1 = 2^(bet_0), bet_2 = 2^(bet_1), etc. But that is not how the aleph numbers work.

Some of us prefer to assume the continuum hypothesis just for the sake of this notation... but it turns out that the continuum hypothesis is still pretty hot.

https://www.quantamagazine.org/how-many-numbers-exist-infini...

"Assuming the continuum hypothesis for the sake of this notation" is being deliberately obfuscatory, encouraging confusion, and requiring other mathematicians to do extra work to translate your results to the more general context. There's perfectly good notation for the bet series, and no need to use aleph as a substitute when that simply isn't what it means.

Checks Yes, you're right.

I am also rusty, but I believe both statements are true but mine may be more precise as there is more than one uncountable infinite set and the set of all countable ordinal numbers is an uncountable infinite set.

The cardinality of Real numbers is 2^Aleph_Null. Continuum hypothesis is equivalent to saying that it is equal to Aleph_One. But that is debated :) [Debated in the sense that you could accept one or the other, but it is independent of ZF and there should be a good reason to accept one or the other]

https://en.wikipedia.org/wiki/Aleph_number#Aleph-one

> but it is independent of ZF

Yeah, if I recall that's why forcing was invented, to specifically prove that C and CH were independent of ZF. Yeah: [1]

1. https://en.wikipedia.org/wiki/Paul_Cohen#Continuum_hypothesi...

You're right, I got thrown off by the "countable" part in the "set of countable ordinal numbers", which is indeed uncountably infinite.

Do we know who the stack-smashing Aleph One was?

FTA: “ℵ0 is also listed as the author of a 1973 book review in SP&E. The book under review is Games Playing with Computers, by A. G. Bell.”

I tried finding out what those initials stand for, but couldn’t find an answer. If it happens to be Alexander Graham Bell, I give it a decent chance that’s a pseudonym, too.

Actually if you look through the comments at the foot of that page, John Francis makes it clear that the author was Richard Parkins of Cambridge, UK: http://www.zen224037.zen.co.uk/

Nice catch, I didn't see that while skimming the comments.

OK why did I get downvoted for this one?