Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 n. The first of the transfinite cardinal numbers. It corresponds to the number of elements in the set of positive integers. Also called alephzero.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 proper n. The first of the transfinite cardinal numbers; corresponds to the number of positive integers, also called natural numbers. Georg Cantor showed that even all the rational numbers could be put in onetoone correspondence with them, and are therefore countable, enumerable or denumerable.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 n. the smallest infinite integer
Etymologies
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Examples

Um... without Jesus you only get the alephnull Eternal Life tm?

If alephnull socks are lost in the laundry, how many pairs of socks does John have?

Martin Gardner, in his review of books by Eli Maor and Rudy Rucker [NYR, December 3, 1987], mistakenly says that "Cantor called the number that counts the real numbers (rational and irrational) alephone, or C," and that "Cantor believed that 2 raised to the power of alephnull is the same as C."

This last designation he applied to whatever cardinal number comes next in order after alephnull; its identity with C, the number of real numbers or "power of the continuum," is a famous conjecture of Cantor's, his "continuum hypothesis."

Meanwhile, Loki's Tricksters defeated the Anubis Crew "c" to alephnull to advance to the semifinals of the Demonic Snowball Tournament in Hell.

Both sets have a count or cardinality equal to the transfinite number א 0  alephnull.

As for the alephnull and alephone: it was proven that the continuum hypothesis essentially whether the cardinality of real numbers is alephone or higher is undecidable in standard set theory, so whether you want to accept it or not, you won’t hit any contradictions.
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qms commented on the word alephnull
Some notions cause my brain unease;
The scale of infinity is one of these.
With weird alephnull
It batters my skull.
Can't we say it's "everything," please?
December 11, 2014
seanahan commented on the word alephnull
Which leads us to my favorite thing, the continuum hypothesis.
December 19, 2007
adoarns commented on the word alephnull
A transfinite number, written ℵ_{0}, which denotes the cardinality of the natural numbers.
In other words, it's a way to express that some infinities are bigger than others.
December 18, 2007