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.
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