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Let $M$ be a non-standard model of $PA$, $a\in |M|$ be an arbitrary non-standard number and $T$ be a theory of arithmetic. We want to choose a subset $M'\subsetneq M$ such that:

1. $M'\models PA^-$ (or $Q$)
2. $a\not \in |M'|$
3. there exists $b\in M'$ such that $M\models a<b$
4. $M'\models T$

Q1. For which $M\models PA$, $a\in M$, and $T$ we can find a subset $M'$ satisfies above conditions?

 

Q2. For which $T$ is it true that: for every model $M\models PA$, for every nonstandard $a \in M$, there exists $M'$ satisfies above conditions?

Thanks.

Let $M$ be a non-standard model of $PA$, $a\in |M|$ be an arbitrary non-standard number and $T$ be a theory of arithmetic. We want to choose a subset $M'\subsetneq M$ such that:

1. $M'\models PA^-$ (or $Q$)
2. $a\not \in |M'|$
3. there exists $b\in M'$ such that $M\models a<b$
4. $M'\models T$

Q1. For which $M\models PA$, $a\in M$, and $T$ we can find a subset $M'$ satisfies above conditions?

Q2. For which $T$ is it true that: for every model $M\models PA$, for every nonstandard $a \in M$, there exists $M'$ satisfies above conditions?

Thanks.

Let $M$ be a non-standard model of $PA$, $a\in |M|$ be an arbitrary non-standard number and $T$ be a theory of arithmetic. We want to choose a subset $M'\subsetneq M$ such that:

1. $M'\models PA^-$ (or $Q$)
2. $a\not \in |M'|$
3. there exists $b\in M'$ such that $M\models a<b$
4. $M'\models T$

Q1. For which $M\models PA$, $a\in M$, and $T$ we can find a subset $M'$ satisfies above conditions?

Thanks.

Let $M$ be a non-standard model of $PA$, $a\in |M|$ be an arbitrary non-standard number and $T$ be a theory of arithmetic. We want to choose a subset $M'\subsetneq M$ such that:

1. $M'\models PA^-$ (or $Q$)
2. $a\not \in |M'|$
3. there exists $b\in M'$ such that $M\models a<b$
4. $M'\models T$

Q1. For which $M\models PA$, $a\in M$, and $T$ we can find a subset $M'$ satisfies above conditions?

Q2. For which $T$ is it true that: for every model $M\models PA$, for every nonstandard $a \in M$, there exists $M'$ satisfies above conditions?

Thanks.

Let $M$ be a non-standard model of $PA$, $a\in |M|$ be an arbitrary non-standard number and $T$ be a theory of arithmetic. We want to choose a subset $M'\subsetneq M$ such that:

1. $M'\models PA^-$ (or $Q$)
2. $a\not \in |M'|$
3. there exists $b\in M'$ such that $M\models a<b$
4. $M'\models T$

Q1. For which $T$ we can find a subset $M'$ satisfies above conditions?

Thanks.

Let $M$ be a non-standard model of $PA$, $a\in |M|$ be an arbitrary non-standard number and $T$ be a theory of arithmetic. We want to choose a subset $M'\subsetneq M$ such that:

1. $M'\models PA^-$ (or $Q$)
2. $a\not \in |M'|$
3. there exists $b\in M'$ such that $M\models a<b$
4. $M'\models T$

Q1. For which $M\models PA$, $a\in M$, and $T$ we can find a subset $M'$ satisfies above conditions?

Thanks.