from Wiktionary, Creative Commons Attribution/Share-Alike License.
- adverb Too many to be
counted(either by reason of being infiniteor for practical constraints).
- adverb grammar In an
- adverb mathematics In a way that is
incapableof being put into one-to-one correspondencewith the natural numbersor any subset thereof.
- adverb Used as a general intensifier of amounts and quantities;
from Wiktionary, Creative Commons Attribution/Share-Alike License
These sets are too big to be put into one-to-one correspondence with the natural numbers; they are called uncountably infinite.
Thus, the one-to-one correspondence between the reals and the naturals fails, as there are simply too many reals—they are "uncountably" numerous—making real infinity somehow larger than natural infinity.
* Eleanor Clift loses one of her uncountably many demerits in agitating for Howard Dean to HHS.
For example, it certainly depends on whether your set of trials is countably infinite or uncountably infinite (in other words the cardinality of your set of trials).
Much to the consternation of marine life advocates and to the relief of the out-of-sight/out-of-mind crowd, most of the damage is uncountably ensconced beneath the surface of the Gulf.
Or how might we have a little private time to tell just one of our sons of our affection for him without sharing the moment with uncountably many of his brothers?
Indeed, because cardinality is permutation-invariant, every cardinality quantifier is included, including “there are infinitely many”, “there are uncountably many”, and others that are not first-order definable.
How can a countable model satisfy the first-order sentence which “says that” there are uncountably many things?
The questions raised by the uncountably infinite number of both unrealistic and implausible alternate viewpoints of diversity demand answers.
Or moving the other way, if we form a third language L³ by adding to L the quantifier Qx with the meaning “There are uncountably many elements x such that ¦”, then trivially L is reducible to L³, but the downward Loewenheim-Skolem theorem shows at once that L³ is not reducible to L.