Editing Zero sharp (section)

== Consequences of existence and non-existence ==
The existence of <math>0^\sharp</math> implies that every [[Uncountable set|uncountable]] [[Cardinal number|cardinal]] in the set-theoretic universe <math>V</math> is an indiscernible in <math>L</math> and satisfies all [[large cardinal]] axioms that are realized in <math>L</math> (such as being [[Ineffable cardinal|totally ineffable]]).  It follows that the existence of <math>0^\sharp</math> contradicts the ''[[axiom of constructibility]]'': <math>V=L</math>.

If <math>0^\sharp</math> exists, then it is an example of a non-constructible <math>\Delta^1_3</math> set of natural numbers. This is in some sense the simplest possibility for a non-constructible set, since all <math>\Sigma^1_2</math> and <math>\Pi^1_2</math> sets of natural numbers are constructible.

On the other hand, if <math>0^\sharp</math> does not exist, then the constructible universe <math>L</math> 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 <math>x</math> of ordinals there is a constructible <math>y</math> such that <math>x\subset y</math> and <math>y</math> has the same [[cardinality]] as <math>x</math>.

This deep result is due to [[Ronald Jensen]]. Using [[forcing (mathematics)|forcing]] it is easy to see that the condition that <math>x</math> is uncountable cannot be removed. For example, consider [[List of forcing notions#Namba forcing|Namba forcing]], that preserves <math>\omega_1</math> and collapses <math>\omega_2</math> to an ordinal of [[cofinality]] <math>\omega</math>. Let <math>G</math> be an <math>\omega</math>-sequence [[cofinal (mathematics)|cofinal]] on <math>\omega_2^L</math> and [[generic filter|generic]] over <math>L</math>. Then no set in <math>L</math> of <math>L</math>-size smaller than <math>\omega_2^L</math> (which is uncountable in <math>V</math>, since <math>\omega_1</math> is preserved) can cover <math>G</math>, since <math>\omega_2</math> is a [[regular cardinal]].

If <math>0^\sharp</math> does not exist, it also follows that the [[singular cardinals hypothesis]] holds.<ref>P. Holy, "[https://www.dmg.tuwien.ac.at/holy/dip.pdf Absoluteness Results in Set Theory]" (2017). Accessed 24 July 2024.</ref><sup>p. 20</sup>