Modern pedagogy in elementary academic preparation for counting numbers may be incomplete, in fact, insofar as, for example, if a prospective customer approaches a loan officer with the claim of a “number” of properties offered for collateral to secure principal funds, yet one of the following three results arise in terms of this “number” as emerges during the loan application process, i.e., one, zero or infinity, does the chance of securing such a loan decrease in respective order, and/or does the probability of being investigated for violation of, say, USC 18 Section 1344 (bank fraud) proportionately increase, not to mention prompting an examination for sanity of the prospective loan customer?
Abstract as a cursory brief:
Preliminary Note: As the essential property of one generates the elemental number line by initially being added first to zero, and then, next, to itself, to create more than itself in the first plurality, i.e., "number," per se, unity subtends the very idea of the sequence commonly regarded as the number line, and this as its prime function remains unique. No other such singular identity exists in characteristic form. As such, any so extrapolated subsequential plurality becomes a unique iteration of one to such degree as its count extends.
On a practical note, the issue of number seems salient to the recent (1931) effort of Kurt Gödel in solving David Hilbert’s second problem of proving the completeness of arithmetic as a self-consistent system. However, Gödel having reached the diametrically opposed conclusion in his landmark proof of the incompleteness of arithmetic has demonstrated, instead, that to be not self-validated. Many extrapolations in widely ranging degrees of conventional legitimacy have resulted, one of which, for example, has been that no system can be self-validating. So, to begin the following demonstration of what may be missing from arithmetic, namely a count operation, formally defined by an identity element per se, no claim is made that this is a self-validating demonstration.
Ancient Greeks regarded two as the first “number” as logically consistent with the idea of plurality, thus, one, regarded as unity, was not a number, by reason of rational semantic as philosophically sound syntax. (As subsequent Roman numerals carried no symbolic representation of zero more than their Greek predecessors had, could any “hypothetical zero” have been symbolized as H?) Indeed, if zero and one are in fact not functional number pluralities as capable of supporting any base number system, then the argument fails that zero and infinity are not reciprocals because their product may be any algebraic extrapolation as any finite quantity, or number, x divided by infinity or zero, to equal zero or infinity, respectively.*
If one, zero and infinity are really only the identity elements of multiplication, addition and count, as multiplication is fast addition, addition is fast count and count is fast “infinition” by “conveniently” glossing and/or skipping over as past all irrationals between rational numbers as such all whole number ratios between whole numbers, in order to “count” whole numbers, as introduced without any real formality in elementary education, so, could counting all finite numbers be as “convenient” as imply all whole numbers between one and infinity, then what chance may be infintion is fast multiplication?
Identity elements, 1, 0 and א:
x=x×1, x=x+0 and x=x·|·א
x|·|x=א, x-x=0, x÷x=1
(Here, ·|· represents that infinition facilitates count as an arithmetic operation,
|·|represents that “refinition” function facilitates count inverse operation,
and “cardinal Aleph” [which is tantamount to “infinity”] is represented as א.)
Could Gödel count how incomplete arithmetic was, among other things?
*As no two exists in binary base-two, no eight in octal base-eight, no A in base A (9+1=10) and no G in base-G hexadecimal, in base-infinity, no infinity exists, and every number is less than ten, ergo, as a single digit.