Transfinite in a sentence

Transfinite recursion Transfinite recursion is similar to transfinite induction; however, instead of proving that something holds for all ordinal numbers, we construct a sequence of objects, one for each ordinal.

First, from the highest perfection of God, we infer the possibility of the creation of the transfinite, then, from his all-grace and splendor, we infer the necessity that the creation of the transfinite in fact has happened.

Each set in this hierarchy is assigned (by transfinite recursion ) an ordinal number α, known as its rank.

For example, many results about Borel sets are proved by transfinite induction on the ordinal rank of the set; these ranks are already well-ordered, so the axiom of choice is not needed to well-order them.

However, if the relation in question is already well-ordered, one can often use transfinite induction without invoking the axiom of choice.

It contained Cantor's reply to his critics and showed how the transfinite numbers were a systematic extension of the natural numbers.

Once the later types are allowed to accumulate the earlier ones, we can then easily imagine extending the types into the transfinite just how far we want to go must necessarily be left open.

One can describe the cumulative hierarchy into which Zermelo developed his models as the universe of a cumulative TT in which transfinite types are allowed.

One may define by transfinite recursion : : and for a limit ordinal α, : if this limit exists.

Over the next twenty years, Cantor developed a theory of transfinite numbers in a series of publications.

Results Using this definition, the following holds: Even the most trivial-looking of these equalities may involve transfinite induction and constitute a separate theorem.

The paper attempted to prove that the basic tenets of transfinite set theory were false.

There is a transfinite sequence of cardinal numbers: : This sequence starts with the natural numbers including zero (finite cardinals), which are followed by the aleph numbers (infinite cardinals of well-ordered sets ).

These controversies are strongly linked as the logical methods used by Cantor in proving his results in transfinite arithmetic are essentially the same as those used by Russell in constructing his paradox.

The uniqueness of the sequence satisfying these properties can be proved using transfinite induction.

This involves an infinite union of infinite sets, which is a "stronger" set theoretic operation than the previous transfinite induction required.

This union will include transfinite terms if the UCS does not stabilize at a finite stage.

This was done by a direct proof of the unprovability of the principle of transfinite induction, used in his 1936 proof of consistency, within Peano arithmetic.