* Aleph Null " Computer recreations: Darwin ", Software: Practice and Experience, Vol.

If κ is an infinite cardinal number, then cf ( κ ) is the least cardinal such that there is an unbounded function from it to κ ; and cf ( κ ) = the cardinality of the smallest collection of sets of strictly smaller cardinals such that their sum is κ ; more precisely

: claims to establish a lower bound of 26 gigaparsecs ( 85 billion light-years ) on the diameter of the whole universe, meaning the smallest possible diameter for the whole universe would be 98. 5 % of the diameter of the last scattering surface ( since this is only a lower bound, the paper leaves open the possibility that the whole universe is much larger, even infinite ).

Aleph-null, the smallest infinite cardinal number

Conventionally the smallest infinite ordinal is denoted ω, and the cardinal number is the least upper bound of

Because the class of ordinal numbers is well-ordered, there is a smallest infinite limit ordinal ; denoted by ω.

This ordinal ω is also the smallest infinite ordinal ( disregarding limit ), as it is the least upper bound of the natural numbers.

The Löwenheim number of first-order logic, in contrast, is ℵ < sub > 0 </ sub >, the smallest infinite cardinal.

The hyperfinite II < sub > 1 </ sub > factor R is the unique smallest infinite

In several earlier works Escher explored the limits of infinitesimal size and infinite number, for example the Circle Limit series, by actually carrying through the rendering of smaller and smaller figures to the smallest possible sizes.

The smallest infinite cardinal number is ( aleph-naught ).

The smallest alkane with such a chemical bond, ethane, exists as an infinite number of conformations with respect to rotation around the C – C bond.

In order to use mathematics, it is not necessary to assume that there are infinite parts of finite lines or any quantities smaller than the smallest that can be sensed.

In this section, Hume first argues that our ideas and impressions of space and time aren't infinitely divisible, one of the arguments is that the capacity of the mind is limited therefore it cannot perceive an object with an infinite number of parts, so it cannot be infinitely divisible, the same for impressions and the proof is that if someone moves a piece of paper with a spot of ink on it until it disappears, the moment before it does, it represents the smallest indivisible impression.

It can be shown, using the axiom of choice, that is the smallest uncountable cardinal number.

Measurable cardinals are Π-indescribable, but the smallest measurable cardinal is not Σ-indescribable.

The concept of a measurable cardinal was introduced by, who showed that the smallest cardinal κ that admits a non-trivial countably-additive two-valued measure must in fact admit a κ-additive measure.

Notice that the ultrafilter or measure which witnesses that κ is measurable cannot be in M since the smallest such measurable cardinal would have to have another such below it which is impossible.

The Lindelöf degree, or Lindelöf number, is the smallest cardinal such that every open cover of the space has a subcover of size at most.

Some authors gave the name Lindelöf number to a different notion: the smallest cardinal such that every open cover of the space has a subcover of size strictly less than.

In this latter ( and less used sense ) the Lindelöf number is the smallest cardinal such that a topological space is-compact.

For a well-ordered set U, we define its cardinal number to be the smallest ordinal number equinumerous to U. More precisely:

The smallest ordinal having a given cardinal as its cardinality is called the initial ordinal of that cardinal.

Also, ω < sub > 1 </ sub > is the smallest uncountable ordinal ( to see that it exists, consider the set of equivalence classes of well-orderings of the natural numbers ; each such well-ordering defines a countable ordinal, and ω < sub > 1 </ sub > is the order type of that set ), ω < sub > 2 </ sub > is the smallest ordinal whose cardinality is greater than ℵ < sub > 1 </ sub >, and so on, and ω < sub > ω </ sub > is the limit of ω < sub > n </ sub > for natural numbers n ( any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the ω < sub > n </ sub >).

It should also be noted that the order of consistency strength is not necessarily the same as the order of the size of the smallest witness to a large cardinal axiom.

The transfinite cardinal numbers describe the sizes of infinite sets.

A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the set of real numbers and the set of natural numbers do not have the same cardinal number.

" ( Church Manual, page 41 ) She also wrote: " The cardinal points of Christian Science cannot be lost sight of, namely — one God, supreme, infinite, and one Christ Jesus.

* Hilbert's paradox of the Grand Hotel, a meditation on strange properties of the infinite, is often used in popular accounts of infinite cardinal numbers.

For example, an equivalence relation with exactly two infinite equivalence classes is an easy example of a theory which is ω-categorical, but not categorical for any larger cardinal number.

Barry Mitchell has shown ( in " The cohomological dimension of a directed set ") that if I has cardinality ( the dth infinite cardinal ), then R < sup > n </ sup > lim is zero for all n ≥ d + 2.

More generally, if κ is any infinite cardinal, then a product of at most 2 < sup > κ </ sup > spaces with dense subsets of size at most κ has itself a dense subset of size at most κ ( Hewitt – Marczewski – Pondiczery theorem ).

Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began.

Using the above given notation, suppose that some infinite cardinal.

Fix an infinite cardinal ( e. g. ).

* Aleph-null,, is defined as the first transfinite cardinal number and is the cardinality of the infinite set of the natural numbers.

Some authors, including P. Suppes and J. Rubin, use the term transfinite cardinal to refer to the cardinality of a Dedekind-infinite set, in contexts where this may not be equivalent to " infinite cardinal "; that is, in contexts where the axiom of countable choice is not assumed or is not known to hold.

* Singular cardinal, an infinite cardinal number that is not a regular cardinal

The generalized Suslin hypothesis says that for every infinite regular cardinal κ every tree of height κ either has a branch of length κ or an antichain of cardinality κ.

If G is infinite, the index of a subgroup H will in general be a non-zero cardinal number.

Assuming the axiom of choice, every other infinite cardinal number is either regular or a ( weak ) limit.

Thus this axiom guarantees the existence of an infinite tower of inaccessible cardinals ( and may occasionally be referred to as the inaccessible cardinal axiom ).

If G is infinite, the division by | G | may not be well-defined ; in this case the following statement in cardinal arithmetic holds: