**Definition:** Let $V\subseteq W$ be two transitive models of $ZFC$. A pseudo-Prikry sequence, $s$, at a cardinal $\kappa$ for $(V, W)$ is an $\omega$-sequence, cofinal at $\kappa$ such that for every club $D\subseteq \kappa$ from $V$, $s\setminus D$ is finite. 


In the paper "On squares, outside guessing of clubs and $I\_{<f}[λ]$" by Shelah and Dzamonja and independently in the paper "Some results on the nonstationary ideal II" by Gitik, stronger versions of the following theorem are proved:


**Theorem:** if $V\models \kappa$ is inaccessible, and $2^\kappa = \kappa^+$ and $W\models \kappa^+ = (\kappa^+)^V,\,\text{cf }\kappa=\omega$ then there is a pseudo-Prikry sequence at $\kappa$ in $W$.


I wonder how far can a pseudo-Prikry sequence get from just being the regular Prikry sequence.


For simplicity, let's assume that $\kappa$ is measurable in $V$, $\mathcal{U}$ is a normal measure on $\kappa$ and $W = V[G]$ where $G$ is a generic filter for the Prikry forcing for $\kappa$, using the measure $\mathcal{U}$. 


**Question:** Does $W$ contain a pseudo-Prikry sequence which is not a $\mathcal{U}$-Prikry sequence?



Pseudo-Prikry sequences vs Prikry sequences
The answer is no, there is no such pseudo-Prikry sequence in the
Prikry extension.


To see this, let's first make some observations.


**Lemma 1.** If $j:V\to M$ is the ultrapower by a normal measure
$\mu$ on $\kappa$, then $$\bigcap\{\ j(C)\mid C\subset\kappa\text{
club }\}=\{\kappa\}.$$


**Proof.** Certainly $\kappa\in j(C)$ for any club
$C\subset\kappa$, and by considering final segments of $\kappa$,
it is clear that the intersection is contained in
$[\kappa,j(\kappa))$. So consider $\kappa<\alpha<j(\kappa)$. Since
$\mu$ is normal, it follows that $\alpha=j(f)(\kappa)$ for some
function $f:\kappa\to\kappa$. Let $C\_f\subset\kappa$ be the club
of ordinals $\beta$ with $f''\beta\subset\beta$. It follows that
$\alpha\notin j(C\_f)$, and so any particular $\alpha$ other than
$\kappa$ is cast out of that intersection. **QED**


Now consider the $\omega$-iteration of $\mu$ $$V\to M\_1\to
M\_2\to\cdots\to M\_\omega$$


**Lemma 2.** Similarly, for the $\omega$-iteration embedding $j\_\omega:V\to M\_\omega$, we have $$\bigcap \{\ j\_\omega(C)\mid C\subset\kappa\text{ club
in }V\ \}=\{\ \kappa\_n\mid n\in\omega\ \}.$$


**Proof.** For any particular club $C\subset\kappa$, since it is
in $\mu$, we have that $\kappa\_n\in j\_\omega(C)$ for every $n$.
Now suppose that $\kappa\_n<\alpha<\kappa\_{n+1}$ for some $n$. By
standard representation results, we know that
$\alpha=j\_\omega(f)(\kappa\_0,\ldots,\kappa\_n)$ for some function
$f:\kappa^{n+1}\to \kappa$. Let $C\_f$ be the set of $\beta<\kappa$
such that $f''\beta^{n+1}\subset\beta$. This is club in $\kappa$,
but by design, $\alpha\notin j\_\omega(C\_f)$. So every ordinal not
on the critical sequence is cast out of the intersection. **QED**


It is a standard fact that the critical sequence
$s=\langle\kappa\_n\mid n\in\omega\rangle$ is $M\_\omega$-generic
for Prikry forcing with $\mu\_\omega=j\_\omega(\mu)$. Suppose that
$t=\langle\delta\_n\mid n\in\omega\rangle$ is a pseudo-Prikry
sequence in $M\_\omega[s]$. In particular, if $C\subset\kappa$ is
club in $V$, then $j(C)$ contains a tail of the $\delta\_n$. But
for any particular ordinal $\alpha$ not on the critical sequence,
there is a club $C\subset\kappa$ for which $\alpha\notin
j\_\omega(C)$. Thus, if infinitely many $\delta\_n$ are not on the
critical sequence, we can intersect the corresponding clubs to
find a single club $C\subset\kappa$ such that $j\_\omega(C)$ does
not contain a tail of $t$, which would contradict our assumption
that $t$ is a pseudo-Prikry sequence. Thus, $t$ must eventually be
contained in the critical sequence. It follows that $t$ is a
Prikry sequence over $M\_\omega$, since any almost-subsequence of a
Prikry sequence is a Prikry sequence.


Thus, every pseudo-Prikry sequence in $M\_\omega[s]$ is actually a
Prikry sequence, and then by elementarity, the same holds in the
Prikry forcing extension of $V$. 


