I can do my bank sort code & account number from memory - but I wouldn't have a chance at my credit card number. I suspect it's something to do with the length of the numbers - the human brain is better at remembering two shorter sequences than one sixteen digit number.

I'm pretty good with memorising numbers in general. One Christmas my family gave me one of those "Improve Your Memory" books containing tricks for memorising long lists of things and (for example) recalling the item at a particular numbered position on demand. The book started off with a quick test - "see how good your memory already is so you know what to try to improve" - and while my short-term recall of word-oriented things was about average, my score on number memory was near-perfect (and my score on matching names to faces was actually perfect, which definitely doesn't seem to reflect my real-life experience).

I've had some trouble with my credit card number since I changed credit cards - I could do the old one with no problem, but I've only just managed to memorise the new one and I still can't remember the expiry date offhand.

But normally, I'd expect to be more familiar with numbers I use a lot, and I think I've used my bank details about three or four times ever.

I happen to have a very memorable account number, and sort codes are only three chunks (I rememebr it as three two-digit numbers, not a six-digit number).

foolishly picked a fight with the Axiom of Choice, which wiped the floor with me

Dammit, I’m now intrigued, despite assuming that the reason you weren’t more specific was that it’s uninteresting otherwise.

I was having another try at inventing some useful cryptographic primitives for use on the Infinity Machine.

I have an unbreakable (one-time-pad-based) combined cipher+MAC, which is a decent start, but I lack a viable hash function, key exchange algorithm or signature algorithm. While attempting to invent a hash function I ran headlong into the AoC, and stupidly didn't immediately slink away with my tail between my legs.

Hmmm, that's a nice mental toy..

/me wanders off to cogitate

Oh and here was me thinking you were going to make the sun the size of a pea (http://en.wikipedia.org/wiki/Banach-Tarski_Paradox).

I'm surprised your page on the Infinity Machine doesn't mention Godel's theorem, even if only to say "this machine always halts within two seconds even given an Godel-unsolvable question".

I can't work out whether machine at the moment supports real numbers or just infinite floating point, and whether it's memory capacity is Aleph-0 or Aleph-1. If it has integer addressing I guess it must be Aleph-0.

I think a hash function may be impossible if you want to hash infinite messages to a finite hash, because there are infinitely many messages that hash to the same thing and they can all be found in a single run. Hashing infinite messages to infinite-length hashes is possible but I can't see it being useful.

I think signature algorithms are also impossible, because I can always generate all possible signatures and check to see if that is a valid signature for the message I want to sign.

I think the way to useful crypto may be to find problems which have Aleph-1 possible solutions, which the infinity machine can only generate and try Aleph-0 attempts at.

Everything in the Infinity Machine is at most countably infinite (aleph-0). Memory is countable since memory locations are directly bijectable on to the natural numbers; processing power is countable too, since every instruction occupies a non-zero time interval, and we can therefore show that at most countably many instructions are ever executed by observing that each such interval contains at least one rational.

There's no difference between real numbers and "infinite floating point" (modulo the occasional number with redundant encodings, such as 0.9 recurring being equal to 1). The reals are the same size as 2^aleph-0. (Be careful saying "aleph-1"; that's not a terribly useful concept owing to the undecidability of the Continuum Hypothesis. For the infinity of the reals we generally say C.)

A signature algorithm generating an infinitely long signature is not impossible (or at least can't be shown to be impossible by this type of argument), since there are uncountably many such signatures and hence even the Machine can't generate and test them all. Likewise a public/private key scheme and a Diffie-Hellman-like key exchange: anything which involves generating an infinitely long bit string using a secret key is secure in principle, because another Machine can't possibly generate all the possibilities to reverse the algorithm.

I agree that a finitely long hash wouldn't be secure for most purposes, but even an infinitely long one would have uses. For example, consider the cryptographic scissors-paper-stone protocol: I tell you the hash of my move, you tell me your move, I reveal my move and you check the hash. The virtue of a hash is not just that it reduces the size of the data; its trapdoor-ness is also useful.

Is there a connection between the Infinity Machine and Turing's 'oracles' (http://www.livejournal.com/users/acheron_hades/44711.html)?

I can remember my bank account and sort code with no problems, however I do have good reason to - they are used to log into my online bank account.

More worryingly I can remember my credit card number, start and expiry date... and the additional three digits on the back of the card which a couple of places ask for these days.

It's funny, if I intentionally set out to rote memorise something, I almost always fail, but things that I use fairly regularly do seem to stick without my noticing. Off the top of my head I know:

my NI number
one of ccard numbers
about 10 years worth of passwords for various computer systems
my library card number (it's a long bastard) + PIN

I also used to know the account numbers + sort codes for my two Natwest accounts, before I closed them, but for some reason I've never memorised any of my current bank account numbers. Or my times-tables <g>

I understand very little about how brain works :/

http://www.axiomofchoice.com/

Today, by contrast, seems unusually disjointed.

There has to be a joke in there about disjoint sets and having to choose what to do, but I can't find it.

What a strange-sounding band. I wish they had some demo MP3s for download; I'd be intrigued to hear more, but not nearly intrigued enough to risk real money on it...

I can remember my bank account number and sort code for my main account, and my NI number, which is good because I think I've lost the card again. I even remember the PIN occasionally.

I also know all the phone numbers my mother has ever had except the one she has now. This is very annoying. Her mobile phone number is too much like Andrew's for me to bother remembering his with any degree of assurance that I won't ring my mother by mistake. I therefore have every phone number I've ever been given written down in my diary.

(The Infinity Machine is really interesting, hope you don't mind me picking at it...)

Possible hash algorithm:

Find a mapping from integers to irrational numbers in [0,1). For example, find the Nth prime, take its square root, subtract the integer part. Compute an infinite number of bits of this number and send them as your hash. The recipient does the same hashing his copy of the message. As more bits arrive, the probability that you both have the same message approaches 1. However, no matter how many bits have arrived, the number of possible integers that could have hashed to the same value is still countably infinite.

It's still useless in the face of infinite active attackers, all of whom can send you an infinite number of copies of all possible messages in all possible orders in a finite amount of wall time such that you can't distinguish them from "legitimate" messages. Sooner or later I'm going to send the correct "transfer $1,000,000" to Mallet's bank account" message and you've lost.