But just calling something "infinitesimal" is not a property that pins down a number to bespecific. You have always acted like there must be one such smallest positive number, but that is not a general property of numbers less than 1. Therefore it follows if there is one such positive numbersmaller than any positive rational numberthere must be an unending progression of numbers with the same properties.

This follows because the square of a positive number is a positive number and the product of anytwo non-zero numbers ofmagnitude less than 1is a third number with magnitude smaller than either of the first two.

That is one of the problems in the current axiomatic interpretation of "1". You have to realize that the infinitesimal QUANTUM is in EFFECT physically and logically the ORIGINAL "1" QUANTUM that starts the whole cascade of aggregation to higher level 'units' which are comprized at root BY whatever number of infinitesimal quantums make any particular aggregate number 'unit' construct.

Quickly then because I'm rushed and have to go again, once the '1' is THE infinitesimal, then that above logics used by rpenner, implying that 'infinitesimal is ASSUMED TO BE LEES THAN '1'....and then using that arbitrary designation as THE '1' concept/value to justify the argument that any product of any two infinitesimals must be less than either infinitesimal is patently circuitous 'proofing' argument....simply because of the initial assumption unwittingly 'defining' an infinitesimal quantum as less than '1' in the first place!

If one logically and physically effectively STARTS from the 'entity' of infinitesimal quantum as THE starting '1' unitary from which all higher level aggregations labeled '1' are CONSTRUCTED in the first place, then the 'product between infinitesimals CANNOT BE LESS than '1' infinitesimal quantum.

Problem solved. By starting with new axioms treating the infinitesimal as THE starting '1' unitary entity and going from there to all the higher level maths '1' entities constructed therefrom.

Sorry if that looks rushed. It was. But I think you got the drift of what a NEW and more reality-sensible 'number system' might look like soon if the axioms are changed for the better?

Gotta go. See ya round. Good luck and enjoy your discussions, guys!