The Smallest Thing Takes Forever and a Day

Infinity. In my life many people have often told me that i cannot grasp infinity, but you can hear the doubt in their voice, almost begging you not to challenge the dictate. And like so many things in my life, i railed against this, it chaffed on some kind of primitive level, so i stubbornly refused to accept this and worked for many years to grasp the infinite. Well, i am happy to say that i can grasp infinity, for some time now, and it's quite liberating, almost easy in hind site.

What i was still struggling with was the infinitesimal. Put another way: What's the smallest number not 0? The simplest graphical representation would be the rectangular hyperbola. As X approaches infinity, Y approaches zero, but what happens at infinity? Now there is no "at infinity" but there is "an infinity" so there should be a corresponding and reciprocal thing for infinity, right? What is one over infinity? Is it zero? No, for several reasons, the best being that all of mathematics would come crashing down if it were :). So if it is not zero then it must be a number, an actual entity capable of being understood just as infinity is.

Infinity was an accepted concept long before the infinitesimal, in fact the belief that the 'tesimal (as i will call it) didn't even exist was the largest roadblock to the acceptance of calculus, and it wasn't until the introduction of the limit that a satisfactory evidence of the viability of calculus became available. However, the idea that the 'tesimal is defined as "a quantity which yields 0 after the application of some limiting process " didn't sit well with me. It seems ugly, or clumsy somehow. Infinity, zero, one (and other integers), negatives, fractions, irrationals, transcendentals, complex numbers, even the transfinite numbers all seamed to have positions of greater stature in the hierarchy of numbers.

In 1966, Abraham Robinson, wrote a classic work call, Non-standard Analysis, which used "genuine infinitesimals" (as well as "infinitely large numbers") as part of Hyperreal numbers. I have not read this book yet, as i have to brush up on my set theory and predicate logic, so this is just what i have gleaned from piecemeal readings. Hyperreal numbers define the 'tesimal as a number that is smaller than every positive real number and bigger than every negative real number (note: this includes zero, along with other non-zero numbers). Non-standard analysis though has quite a few critiques though, and while the definition of 'tesimals is now on it's way my intuition tells me that it has not arrived.

Enter, circa 1973, John Conway, and his Surreal numbers. Surreal numbers are generated by starting with nothing and ending with of all the real numbers plus the transfinite and infinitesimals, plus strange new numbers like the roots of transfinite numbers and infinitesimals! Using Conway's notation the "equivalent class" for the first infinitesimal is ε = {0 | ... 1/4, 1/2, 1}, and is a number larger than zero and less than any real number. Interesting to note, using Conway's recursive system every number can be assigned a "birthday" representing the number of the iteration in which it was first created, and the birthday of ε is the day after infinity.

With the discovery of surreal numbers, my quest is essentially at an end. If the answer to "what is the largest number?" is ω, the first ordinal number (it represents the counting numbers: 1, 2, 3, 4...) then the "the smallest number not zero" is ε. I say essentially because just as there are larger ordinals than ω, there are smaller infinitesimals than ε, but likewise just as all of the transfinites have been defined (and can there for be claimed as known) so too can all of the 'tesimals now be defined (and known).

In part two i will attempt to explain, in simple terms, the construction of 'tesimals and surreal numbers in general, if there is any (and i mean any) interest.

Comments

[Ojo Rojo]

Even this much (ε) interest?

[MathJames]

Yes, ε is greater than zero, so you are all going to pay now, MWHAHAHAHA! I will thrill you to sleep with mathematical tales of high dullness... as soon as i return from the "Face of Victoria's Secret" Model search which i was forced to judge.

[tom]

This talk of infinite sets and whatnot causes me to recall something I learned in high school Algebra II.

Having programmed computers since age 7, I was pretty sure that any positive real number divided by 0 was infinity, which my Apple ][ obviously couldn't handle.

But per my teacher, not so. He insisted that it was undefined.

To which I said "Ah, bullshit. I'm already learning some calculus, and I can tell you that as 1/.01 is big. 1/.001 is bigger. 1/.000000000001 is even bigger. Continue, and as it approaches zero, you get larger and larger results, with no upward bound. Thats infinity, bro."

To which Mr. Loewer responded: "Nice, but remember for something to be =, the inverse must be maintaied. In other words, for 1/0 = infinity to be true, then 0*infinity = 1 must also be true.

"Which it ain't. Anything zero times is zero." Duh.

UNDEFINED == PWNED

(all quotes above from a malt liquor-addled memory of high school and may not reflect actual contents of discussion)

[MathJames]

Ah, but with Mr. Conway's generous new set, omega is not only be defined in a new way, but can be used in addition/negation, multiplication/division, powers/roots. It so happens that 1/ω = ε, and can proved. But not only can you divide by infinity you could, say, take the infinite root of a number!!!

[MathJames]

Surreal Numbers == RoXoRz VVi7h 1+5 c0x0rZ

[tom]

Hey! Are you coming to my birthday party? Did you get the evite?

[Pembrose]

It's a bit OT but, I just read this interesting probability study....

www.montyhallproblem.com

P.

[tom]

Ahhh, the Lets Make A Deal problem where the probability shifts after one solution is revealed.

I've seen this argued at bars over beers, on blogs and newsgroups, in classrooms, and even in the pages of Parade magazine (in the Marilyn Vos Savant column, who allegedly has the world's highest IQ).

To sum it up:

If you've got three doors, and one of them has a prize behind it, you win 1 out 3. If you pick 1, but then I reveal 3, and give you the option to change your choice, you always should as your new choice will be right 1/2 times instead of 1/3.

I'm not an genius or a hyperintelligent shade of blue or anything, so I'll give you my layman's perspective on this one. In the immortal words of Geddy Lee, however, "If you choose not to decide you still have made a choice."
If I have an option to choose again, and choose the original door, I've got a 1 in 2 chance. Therefore, not changing your pick when given an option gives you the same 1/2 that changing your choice would have given you.

P.S. Erik "Schwarzburg" wrote a c++ program during his EE degree at UT to test this problem, and oddly enough it indicated that I'm wrong. Go figure.