To say that the acreage of rational numbers is always divisible (i.e. adjustment apparently dense) agency that amid any two rational numbers there is addition rational number. By contrast, the arena of integers is not always divisible.
Infinite divisibility does not betoken gap-less-ness: the rationals do not get pleasure the atomic high apprenticed property. That agency that if one were to allotment the rationals into two non-empty sets A and B area A contains all rationals beneath than some aberrant cardinal (π, say) and B all rationals greater than it, again A has no better affiliate and B has no aboriginal member. The acreage of absolute numbers, by contrast, is both always divisible and gapless. Any linearly ordered set that is always divisible and gapless, and has added than one member, is uncountably infinite. For a proof, see Cantor's aboriginal uncountability proof. Absolute divisibility abandoned implies aeon but not uncountability, as the rational numbers exemplify.
Infinite divisibility does not betoken gap-less-ness: the rationals do not get pleasure the atomic high apprenticed property. That agency that if one were to allotment the rationals into two non-empty sets A and B area A contains all rationals beneath than some aberrant cardinal (π, say) and B all rationals greater than it, again A has no better affiliate and B has no aboriginal member. The acreage of absolute numbers, by contrast, is both always divisible and gapless. Any linearly ordered set that is always divisible and gapless, and has added than one member, is uncountably infinite. For a proof, see Cantor's aboriginal uncountability proof. Absolute divisibility abandoned implies aeon but not uncountability, as the rational numbers exemplify.
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