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 that so-called "number" infinity, zero, and/or one, was more or less than plural, does the chance of securing such a loan merely diminish, or tailspin in a nosedive to crash and burn? As may be led by further regard for zero, for example, risk of being investigated for violation of USC 18 § 1344 (bank fraud) bodes ill, especially if the prospective customer ends up but prompting an examination for his sanity by court order.
On a practical note, number issues seem salient to the recent (1931) effort of Kurt Gödel to solve David Hilbert's second problem, demonstration that arithmetic is complete as self-consistent. However, Gödel having reached the diametrically opposite conclusion in his landmark proof of the incompleteness of arithmetic has shown instead that arithmetic is not a self-validating system. Various and sundry extrapolations in widely ranging degrees of conventional legitimacy have resulted, one of which, for an example of generalization, has been the universal notion that no system self-validates.
Ancient Greeks regarded two as the first number, as bankers may so be practical as predisposed as well to this day. One, regarded as "unity" was therefore not a number in terms of rational semantic as philosophically sound syntax consistent with plurality. One is added to itself as many times as any whole number generation reiterates. How Greek numerals preceded Roman begs how practical who count. Review suggests H may have been regarded as of zero value, so, seldom merit abstract symbol value, if that much.
If zero and one are not functionally plural as numbers capable of any countable number system support base, then the argument fails that, because their false "number" product can be substituted by any true number product, zero and infinity are not reciprocal.
If one, zero and infinity are real identity elements of multiplication, addition and count, respectively, then multiplication is fast addition, addition is fast count, and count is fast multiplication, by infinition, counting all finite numbers by skipping over them as easily as past all real numbers between any two adjacent rational numbers, all rational numbers between any whole number adjacent pair and/or all whole numbers between zero and infinity.
Count operator, ·l·, for infinition, and for the inverse operation, l·l, refinition, employ Cardinal Aleph, א, to identify any quantity, x, such that:
x·l·א=x, as x+0=x and x×1=x, and
xl·lx=א, as x-x=0 and x÷x=1.
Emile Litella, LL