Zero sharp
$0^{\#}$ is a $\Sigma_3^1$ real number which cannot be proven to exist in $\text{ZFC}$. It's existence contradicts the Axiom of constructibility, $V=L$. In fact, it's existence is somewhat equivalent to $L$ being completely different from $V$. [1]
Contents
Definition
$0^{\#}$ is defined as the set of all Gödel numberings of first-order formula $\varphi$ such that $L\models\varphi(\aleph_0,\aleph_1...\aleph_n)$ for some $n$. Because of the stability of $\aleph_\omega$, $0^{\#}$ is equivalent to the set of all Gödel numberings of first-order formula $\varphi$ such that $L_{\aleph_{\omega}}\models\varphi(\aleph_0,\aleph_1...\aleph_n)$. This definition implies the existence of Silver indiscernibles. Moreover, it implies:
- Given any set $X\in L$ which is first-order definable in $L$, $X\in L_{\omega_1}$. This of course implies that $\aleph_1$ is not first-order definable in $L$, because $\aleph_1\not\in L_{\omega_1}$. This is already a disproof of $V=L$ (because $\aleph_1$ is first-order definable).
- For every $\alpha\in\omega_1^L$, every uncountable cardinal is $\alpha$-iterable, $\geq$ an $\alpha$-Erdős, and totally ineffable in $L$.
- There are $\mathfrak{c}$ many reals which are not constructible (that is, $x\not\in L$).
The existence of $0^\#$ is equivalent to:
- $(\aleph_\omega)^V$ being regular in $L$.
- $\Pi^1_1$-determinacy (lightface form).
- The existence of a nontrivial elementary embedding $j:L\to L$.
- The existence of a nontrivial elementary embedding $j:L_\alpha\to L_\beta$ with $\text{crit}(j)<|\alpha|$.
And is implied by:
- Chang's Conjecture.
- Both $\omega_1$ and $\omega_2$ being singular (requires $\neg\text{AC}$).
- The negation of the singular cardinal hypothesis ($\text{SCH}$).
- The existence of an $\omega_1$-iterable cardinal or of a $\omega_1$-Erdős cardinal.
- The existence of a weakly compact cardinal $\kappa$ such that $|(\kappa^+)^L|=\kappa$.
- The existence of some uncountable regular cardinal $\kappa$ such that every constructible $X\subseteq\kappa$ either contains or is disjoint from a closed unbounded set.
Silver indiscernibles
More information to be added here.
$0^{\#}$ exists iff there is a nontrivial elementary embedding $j:L\rightarrow L$ (by a theorem of Kunen). If any such embedding exists, then a proper class of those embeddings exists and every Silver indiscernible (in particular every uncountable cardinal) is the critical point of a such embedding. In $L$, every Silver indiscernible is totally indescribable, totally ineffable, and much more.
E.M. blueprints and sharps of arbitrary sets
Consequence of the nonexistence of $0^\#$
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References
- Jech, Thomas J. Set Theory. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) www bibtex