Zero sharp explained

In the mathematical discipline of set theory, 0# (zero sharp, also 0#) is the set of true formulae about indiscernibles and order-indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the natural numbers (using Gödel numbering), or as a subset of the hereditarily finite sets, or as a real number. Its existence is unprovable in ZFC, the standard form of axiomatic set theory, but follows from a suitable large cardinal axiom. It was first introduced as a set of formulae in Silver's 1966 thesis, later published as, where it was denoted by Σ, and rediscovered by, who considered it as a subset of the natural numbers and introduced the notation O# (with a capital letter O; this later changed to the numeral '0').

Roughly speaking, if 0# exists then the universe V of sets is much larger than the universe L of constructible sets, while if it does not exist then the universe of all sets is closely approximated by the constructible sets.

Definition

Zero sharp was defined by Silver and Solovay as follows. Consider the language of set theory with extra constant symbols

c1

,

c2

, ... for each nonzero natural number. Then

0\sharp

is defined to be the set of Gödel numbers of the true sentences about the constructible universe, with

ci

interpreted as the uncountable cardinal

\alephi

.(Here

\alephi

means

\alephi

in the full universe, not the constructible universe.)

There is a subtlety about this definition: by Tarski's undefinability theorem it is not, in general, possible to define the truth of a formula of set theory in the language of set theory. To solve this, Silver and Solovay assumed the existence of a suitable large cardinal, such as a Ramsey cardinal, and showed that with this extra assumption it is possible to define the truth of statements about the constructible universe. More generally, the definition of

0\sharp

works provided that there is an uncountable set of indiscernibles for some

L\alpha

, and the phrase "

0\sharp

exists" is used as a shorthand way of saying this.

A closed set

I

of order-indiscernibles for

L\alpha

(where

\alpha

is a limit ordinal) is a set of Silver indiscernibles if:

I

is unbounded in

\alpha

, and

I\cap\beta

is unbounded in an ordinal

\beta

, then the Skolem hull of

I\cap\beta

in

L\beta

is

L\beta

. In other words, every

x\inL\beta

is definable in

L\beta

from parameters in

I\cap\beta

.

If there is a set of Silver indiscernibles for

L
\omega1
, then it is unique. Additionally, for any uncountable cardinal

\kappa

there will be a unique set of Silver indiscernibles for

L\kappa

. The union of all these sets will be a proper class

I

of Silver indiscernibles for the structure

L

itself. Then,

0\sharp

is defined as the set of all Gödel numbers of formulae

\theta

such that

L\alpha\models\theta(\alpha1,\alpha2\ldots\alphan)

where

\alpha1<\alpha2<\ldots<\alphan<\alpha

is any strictly increasing sequence of members of

I

. Because they are indiscernibles, the definition does not depend on the choice of sequence.

Any

\alpha\inI

has the property that

L\alpha\precL

. This allows for a definition of truth for the constructible universe:

L\models\varphi[x1...xn]

only if

L\alpha\models\varphi[x1...xn]

for some

\alpha\inI

.

There are several minor variations of the definition of

0\sharp

, which make no significant difference to its properties. There are many different choices of Gödel numbering, and

0\sharp

depends on this choice. Instead of being considered as a subset of the natural numbers, it is also possible to encode

0\sharp

as a subset of formulae of a language, or as a subset of the hereditarily finite sets, or as a real number.

Statements implying existence

The condition about the existence of a Ramsey cardinal implying that

0\sharp

exists can be weakened. The existence of

\omega1

-Erdős cardinals implies the existence of

0\sharp

. This is close to being best possible, because the existence of

0\sharp

implies that in the constructible universe there is an

\alpha

-Erdős cardinal for all countable

\alpha

, so such cardinals cannot be used to prove the existence of

0\sharp

.

Chang's conjecture implies the existence of

0\sharp

.

Statements equivalent to existence

Kunen showed that

0\sharp

exists if and only if there exists a non-trivial elementary embedding for the Gödel constructible universe

L

into itself.

Donald A. Martin and Leo Harrington have shown that the existence of

0\sharp

is equivalent to the determinacy of lightface analytic games. In fact, the strategy for a universal lightface analytic game has the same Turing degree as

0\sharp

.

It follows from Jensen's covering theorem that the existence of

0\sharp

is equivalent to

\omega\omega

being a regular cardinal in the constructible universe

L

.

Silver showed that the existence of an uncountable set of indiscernibles in the constructible universe is equivalent to the existence of

0\sharp

.

Consequences of existence and non-existence

The existence of

0\sharp

implies that every uncountable cardinal in the set-theoretic universe

V

is an indiscernible in

L

and satisfies all large cardinal axioms that are realized in

L

(such as being totally ineffable). It follows that the existence of

0\sharp

contradicts the axiom of constructibility:

V=L

.

If

0\sharp

exists, then it is an example of a non-constructible
1
\Delta
3
set of natural numbers. This is in some sense the simplest possibility for a non-constructible set, since all
1
\Sigma
2
and
1
\Pi
2
sets of natural numbers are constructible.

On the other hand, if

0\sharp

does not exist, then the constructible universe

L

is the core model—that is, the canonical inner model that approximates the large cardinal structure of the universe considered. In that case, Jensen's covering lemma holds:

For every uncountable set

x

of ordinals there is a constructible

y

such that

x\subsety

and

y

has the same cardinality as

x

.

This deep result is due to Ronald Jensen. Using forcing it is easy to see that the condition that

x

is uncountable cannot be removed. For example, consider Namba forcing, that preserves

\omega1

and collapses

\omega2

to an ordinal of cofinality

\omega

. Let

G

be an

\omega

-sequence cofinal on
L
\omega
2
and generic over

L

. Then no set in

L

of

L

-size smaller than
L
\omega
2
(which is uncountable in

V

, since

\omega1

is preserved) can cover

G

, since

\omega2

is a regular cardinal.

If

0\sharp

does not exist, it also follows that the singular cardinals hypothesis holds.[1] p. 20

Other sharps

If

x

is any set, then

x\sharp

is defined analogously to

0\sharp

except that one uses

L[x]

instead of

L

, also with a predicate symbol for

x

. See the section on relative constructibility in constructible universe.

See also

References

Notes and References

  1. P. Holy, "Absoluteness Results in Set Theory" (2017). Accessed 24 July 2024.