TITLE: Measure of the numbers with length of $n$ for a nonstandard number $n$
QUESTION [7 upvotes]: Is there any nonstandard model of $PA$ with the following properties?

There exists a nonstandard number $n\in M$ such that $M\upharpoonright n$ is countable,

Let $|x|=\lceil\log_2x\rceil$, then $|\{x\in M: M\models |x|=n\}|=|\mathbb{R}|=\aleph_1$, (with assumtion CH or any other axiom that implies $|\mathbb{R}|=\aleph_1$)



If the above question has a positive answer, what can be said about the measure of the numbers with length of $n$?
More precisely, let $f:\mathbb{N}\to M\upharpoonright n$ be a one to one function.

Q1. Is set $A_f=\{\sum_{i=0}^\infty 2^{-i-1}\cdot (v)_{f(i)}:v\in M,M\models|v|=n\}$ measurable? ($(x)_i$ is the i'th bit of $x$ in the binary representation)
Q2. Is there any one to one function $g:\mathbb{N}\to M\upharpoonright n$ such that $A_g$ becomes measure one?

REPLY [7 votes]: The point of this note is to point out that "Uncountable" can be strenghtened to "continuum-many" in Hanson's answer, but first a notation: Given a model $M$ of arithmetic and $a \in M$, I will use $[a]_M$ to denote the set $\{x\in M: x \leq a\}$. 


Theorem (Mills and Paris) There is a model of arithmetic $M$ and some $n\in M$ such that $[n]_M$ is countable but $[2^n]_M$ is of the same cardinality as $\Bbb{R}$.

More detail: Mills and Paris showed---without the use of the continuum hypothesis---that given any countable model $M_0$ of PA, and any any nonstandard $n \in M_0$, there an elementary extension $M$ of $M_0$ such that $[n]_{M_{0}} = [n]_M$ but $[2^n]_M$ has cardinality $2^{\aleph_{0}}$.  This result appears in their paper Closure properties of countable nonstandard integers, Fund. Math. 103 (3) (1979) 205–215. An exposition of this result can also be found in Section 3.5 of the book Structure of Models of Peano Arithmetic by Kossak and Schmerl.