What is your field/career?

[BRickAstley]

5) I was briefly placed on the trading desk of one of five leading platinum producers, who dial in for a call at noon GMT to set the price of the metal. Clients of those companies get a chance to trade in this fixing window. It's very similar to this: https://en.wikipedia.org/wiki/Gold_fixing, but platinum and arguably less cool. My "job", befitting a 17 year old, was to listen in to the fix and report to a subset of clients.

So our economy and society is even *more* all based on smoke, mirrors, and whims than skeptical me previously thought?

[Bacchus]

It both is and isn't. Market-makers in major capital assets have a lot of power, but it's mostly used to skin their clients via commission, not to arbitrarily push the price around. However the latter also happens, sometimes almost "by accident", through accumulation of relatively small deviations from normal market practice -- the LIBOR scandal is a good case in point.

The level of smoke and mirrors is nowhere near what the European left tends to imagine, but more extensive than what an average educated person thinks. There is a separate issue in that high finance is just an unhealthy industry.

[Bacchus]

I have always had a problem with Cantor's proof, I never quite understood why we should believe that z_n is a properly defined number, if it is defined in a second-order fashion in terms of f(n). Here is the way I like to imagine this, which replicates the method, but with things that at least appear countable:

Imagine a book made of an infinite number of pages, which go as pages would: 1, 2, ..., n, ... The pages can assume a colour at will of the reader, but let's say the pallete is limited to yellow and blue. Each page itself has written on it a possible colouring of the book, by page number. E.g. 1 - blue, 2 - yellow, 3 - yellow, etc. Now, each such colouring references the page it's written on, and colours it in a certain way. Let's now consider a colouring which colours all the pages to the colour OTHER than the one specified for that page in the colouring described on itself. So if page 3 contains a colouring that specifies 3 - blue, our colouring would read 3 - yellow. This colouring seems perfectly fine until we get to the page on which it's actually written. Our colouring has to specify a colour, but it must be the other color to the one it itself specifies. The colouring becomes self-referential and impossible. OK, so what does that prove? Per Cantor, we've just proved that colourings are uncountable, because they can't be bijected to pages. But then the offending colouring by definition has to exclude itself from its domain, whether it can be bijected or not, so why shouldn't we rather consider that it's the definition of the particular colouring that's causing a contradiction. And a related question, even if we accept that colourings in general are uncountable, are first-order colourings, that is colourings defined with reference to page numbers only, countable?

[RefSteel]

RefSteel, interesting resume! How do you manage to write so much amidst your other responsibilities? And -- oops, I was going to ask which RPG company, but you didn't volunteer that, so I won't. :) That's cool, though.

I consciously and intentionally built my lifestyle (everything from my job to the location of my home to my choices of entertainment; for instance, I have never owned a television, and I don't watch video online) around leaving room in my life for writing. Note this is not advice for a writing career; I haven't made any effort to establish one in more than a decade and a half, and never made a realistic strategic effort to do so at all. It's just a description of how I function. I mean, to the extent I function at all.

I have always had a problem with Cantor's proof

I recently became skeptical of it as well - I suspected it of being mostly a clever semantic trick, relying on the fact that we naturally use whole numbers for counting, to conceal the fact that it's just mapping the complete infinte set of whole numbers to a particular infinite subset of the real numbers between 0 and 1. On consideration though, it does make a kind of sense to me that the infinity of {real numbers between 0 and 1} differs in an important way from the infinity of {whole numbers} ... though I don't know for sure if this is what is meant by cardinality:

A whole number can have an arbitrarily large (finite) number of digits and still be a unique whole number; in fact, infinitely many of these numbers are possible. A number with an infinite number of digits can't be a unique whole number though, because any whole number with an infinite number of digits (except in the special case of "all but the last [finite number] before the decimal point are zeroes") would just be infinity, not its own unique number, regardless of what the specific digits were. The real numbers, on the other hand, do include (infinitely many) unique numbers with infinitely many digits apiece.

In other words, with irrational numbers, you have to resort to infinity just to describe each member of the infinite set, which cannot be said of whole numbers. Arguably, Cantor's proof is just a clever way to illustrate this (or demonstrate it mathematically, if it's a formal proof; I'm no mathemetician, so I don't know how you define things like that) by kind of subsuming the infinity of whole numbers within the infinity of the irrational numbers' length. Dunno....

Also of course, my explanation/exploration of the thing may be technically incorrect in any number of ways. I don't pretend to be an actual mathemetician, and may be entirely wrong.

[The Black Sword]

Generally before you see the proof you'd see a definition of a 'real number'. Something like I define all the real numbers between [0,1] as the set of all infinite decimal strings 0.a_1a_2a_3...a_n... where a_i is in {0,1,2,3,4,5,6,7,8,9}.

The you have Couerva's proof and it's pretty clear that z satisfies this definition. It seems you have a different definition in mind Bacchus?

1) BSc in Mathematics
2) My undergrad was pretty varied with introductions to applications in as many fields as they could think of(at the cost of some depth of course).  I'm working on some completely non-applicable pure maths stuff for my postgrad.
3) Student / I teach some classes 
4) Not really
5) Nope
6) Turin Turambar, Dor-Lomin. Should meet Huinesoron soon.

[Bacchus]

Well, z as defined is not yet a decimal string, it's a way to construct such a string, and to construct it, z references an existing mapping and its elements. If z is itself included in the mapping, the construction fails, as we cannot write out the string at all places (specifically it's undefined at z_n).

Z is like that barber that shaves all the people that don't shave themselves. The definition seems fine until we ask whether he shaves himself.

[Coeurva]

Sorry for sloppiness in presenting the real-number part. TBS tactfully omitted that the restriction to "non-periodical" isn't necessary, either.

As I understand it, Bacchus' argument is that, by defining z, one presupposes that any number matching it cannot be on the list, and that therefore, such a thing was never a decimal string in the first place. That's an interesting point, which is of course why I'll try to refute it vigorously. Germans work that way.

First of all, let's be clear on what a real number is (for the purpose of this proof): a decimal string of at most countably many digits.

Let's assume that the bijective mapping f : N -> (0, 1) exists. It follows that the construction of a decimal string (matching the pattern of) z is well-defined at each step, because f(n) must be well-defined for every n (it's a function), and particularly, so is its n-th digit. Constructing z differs from building a completely arbitrary real number only in that we choose from nine digits, rather than ten, at each step. The end result, however, is a well-defined real number. That it cannot be on a countable list by definition is precisely the point.

I think that the argument combines a general methodical objection (of how to construct patterns that don't presuppose what is to be proven) with a specific one (that the construction used is faulty in this sense).

Re the former, take a proof that directly mirrors Russell's antinomy: that a naive "set of all sets" cannot exist. Let M be the set of all sets. Let U := {x is a set in M | x is not an element of x}. (For instance, the set of natural numbers isn't a natural number itself.) Now U is, itself, a set in M; does it contain itself? Assuming it does, it doesn't; assuming it doesn't, it does. -- Should the conclusion be that
(a) U isn't a well-defined set, despite appearances,
(b) giving a construction for U presupposes that M was never complete (but how much more inclusive could the definition of M be)
© the assumption that M exists allows the construction of paradoxical sets, and therefore M cannot exist.

Re the latter: The construction of z could easily be applied to, say, a finite list (a mapping f : {1, ... , n} -> (0, 1)) of infinite decimal strings in (0,1) such as this, where z_m for m > n could be chosen freely:
0.2222...
0.3333...
0.4444...
0.5555...

We could construct z = 0.123455555.... (or 0.88888...) -- This isn't a number on the list; in fact, the pattern of z is essentially a simple algorithm to ensure that the resulting string is never one on the list. However, x is obviously a decimal string in (0, 1).

Now Cantor's point is essentially that, if we append z to the list to make it "more complete", and even if we repeat this process countably many times for countably many numbers matching z to build an ostensibly-complete list, we still never arrive at a complete list of the real numbers, because the algorithm to construct a number matching z remains well-defined throughout.

[Ruff_Hi]

1. BEc ... but that was just one of the options.  With a bit of tweaking to my studies I could have had BSc, B Maths (do they have those?)
   Post grade studies landed me FIAA (fellow of the institute of actuaries of australia) and then travel added FCAS (fellow of the casualty actuarial society)
   
2. Married 2/3 of the way through 2nd year at uni ... still married to the same girl some [cough] years later
   making sure my set lectures / tutorials were only on Tue, Wed of Thu so I had a 4-day weekend for my whole 3rd year
   
3. Insurance.  Property & Casualty (US) / General Insurance (rest of world)
   
4. I get involved in a ton of things; computing, management, capital modeling, answering questions like 'How much is Irma going to cost us' 2 days before it makes landfall in Florida
   
5. Actuarial science has been fantastic to me.  I heard about it when I was 15 and that set the course of my life / career.  It has enabled me to work in multiple countries around the world.
   
6. Ruff Hi
   

Speaking of books, my wife just published her 5th and last book in a series.  And 100k words is about 250 pages.  I know ... because I type set KE's books for her.

http://www.lulu.com/spotlight/KEStapylton

[TheHumanHydra]

Adrien, I think I understand. Sorry if I made your eyes roll.

MJW, completely understand, was just joshing you.

I had been trying to respond to everyone, but now I think I will just say to you all that you have some exceptionally interesting careers and admirable accomplishments. Especial congratulations to those who (or whose wives) have written, as that is something for which I have an evident affinity. Thank you to RefSteel, Ichabod, suboptimal, Adrien, Dreylin, Dark Savant, and anyone I missed for your answers to my particular questions.

Looking forward to hearing from further members of the site! 

[Dreylin]

Thank you to RefSteel, Ichabod, suboptimal, Adrien, Dreylin, Dark Savant, and anyone I missed for your answers to my particular questions.

I note that no-one has asked /you/ any questions yet, so I want to know: which historic site, and what costume?

I've only been to Canada once (to the true side of Niagara), but I do want to go back. Calgary is our business location, but where would you recommend?