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Corrected formatting in Theorem 2.
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Ali Enayat
Ali Enayat
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The theorem below answers question 4, and the corollary answers question 3. See also Theorem 2.

Theorem 1. For every pair of infinite cardinals $(\kappa$,$\lambda)$, there is a model $M$ of $PA$ (Peano arithmetic) that has an initial segment $I$ such that $cf(I) = \kappa$ and $cf(M$$dcf(M$ \ $I) =\lambda$, where $dcf$ is "downward cofinality".

The model $M$ can be constructed using the McDowell-Specker-Gaifman machinery of "minimal types"; which an be thought of as Ramsey ultrafilters over the Boolean algebra of the parametrically definable subsets of an ambient model of $PA$. The rough idea is as follows: start with a countable model $M_{0}$ of $PA$ and use the McDowell-Specker theorem to build an elementary end extension $M_{\kappa}$ of $M_0$ such that $cf(M_{\kappa})=\kappa$. Then build a minimal type/Ramsey ultrafilter $\cal{U}$ over $M_{\kappa}$, and let $M$ be $L$-th iterated ultrapower of $M_{\kappa}$ modulo $\cal{U}$, where $L$ is the REVERSE of $\lambda$. Then the desired $I$ is $M_{\kappa}$.

Corollary. For every pair of infinite cardinals $\kappa$ and $\lambda$, there is a real closed field $F$ that has a Dedekind cut $(J,K)$ such that $cf(J) = \kappa$ and $dcf(K)=\lambda$.

Explanation: Given a model $M$ of $PA$, $M$ can define the real closure $F$ of itself (by using the same arithmetical recipe that defines the field of algebraic real numbers within the standard model of arithmetic). Since every positive element of $F$ is less than one away from some member of $M$, no gap of $M$ is filled by an element of $F$. Hence given $I$ as in the theorem, the cut $J$ of $F$ defined as the set of all $x \in F$ such that $x < i$ for some $i \in I$ does the job.

Let me also point out another results which shows that a real closed field constructed as above as the real closure of a model of $PA$ always has a "matched cut". This follows from the following result of Shelah which also appears as Theorem 11.1.1 (p.281) of the Kossak-Schmerl text on models of $PA$.

Theorem 2. For every nonstandard model $M$ of $PA$ there is an infinite cardinal $\kappa$, and there is an initial segment $I$ of $M$ such that $cf(I) = \kappa = dcf(M$ \$I)$\ $I)$.

The theorem below answers question 4, and the corollary answers question 3. See also Theorem 2.

Theorem 1. For every pair of infinite cardinals $(\kappa$,$\lambda)$, there is a model $M$ of $PA$ (Peano arithmetic) that has an initial segment $I$ such that $cf(I) = \kappa$ and $cf(M$ \ $I) =\lambda$, where $dcf$ is "downward cofinality".

The model $M$ can be constructed using the McDowell-Specker-Gaifman machinery of "minimal types"; which an be thought of as Ramsey ultrafilters over the Boolean algebra of the parametrically definable subsets of an ambient model of $PA$. The rough idea is as follows: start with a countable model $M_{0}$ of $PA$ and use the McDowell-Specker theorem to build an elementary end extension $M_{\kappa}$ of $M_0$ such that $cf(M_{\kappa})=\kappa$. Then build a minimal type/Ramsey ultrafilter $\cal{U}$ over $M_{\kappa}$, and let $M$ be $L$-th iterated ultrapower of $M_{\kappa}$ modulo $\cal{U}$, where $L$ is the REVERSE of $\lambda$. Then the desired $I$ is $M_{\kappa}$.

Corollary. For every pair of infinite cardinals $\kappa$ and $\lambda$, there is a real closed field $F$ that has a Dedekind cut $(J,K)$ such that $cf(J) = \kappa$ and $dcf(K)=\lambda$.

Explanation: Given a model $M$ of $PA$, $M$ can define the real closure $F$ of itself (by using the same arithmetical recipe that defines the field of algebraic real numbers within the standard model of arithmetic). Since every positive element of $F$ is less than one away from some member of $M$, no gap of $M$ is filled by an element of $F$. Hence given $I$ as in the theorem, the cut $J$ of $F$ defined as the set of all $x \in F$ such that $x < i$ for some $i \in I$ does the job.

Let me also point out another results which shows that a real closed field constructed as above as the real closure of a model of $PA$ always has a "matched cut". This follows from the following result of Shelah which also appears as Theorem 11.1.1 (p.281) of the Kossak-Schmerl text on models of $PA$.

Theorem 2. For every nonstandard model $M$ of $PA$ there is an infinite cardinal $\kappa$, and there is an initial segment $I$ of $M$ such that $cf(I) = \kappa = dcf(M$ \$I)$.

The theorem below answers question 4, and the corollary answers question 3. See also Theorem 2.

Theorem 1. For every pair of infinite cardinals $(\kappa$,$\lambda)$, there is a model $M$ of $PA$ (Peano arithmetic) that has an initial segment $I$ such that $cf(I) = \kappa$ and $dcf(M$ \ $I) =\lambda$, where $dcf$ is "downward cofinality".

The model $M$ can be constructed using the McDowell-Specker-Gaifman machinery of "minimal types"; which an be thought of as Ramsey ultrafilters over the Boolean algebra of the parametrically definable subsets of an ambient model of $PA$. The rough idea is as follows: start with a countable model $M_{0}$ of $PA$ and use the McDowell-Specker theorem to build an elementary end extension $M_{\kappa}$ of $M_0$ such that $cf(M_{\kappa})=\kappa$. Then build a minimal type/Ramsey ultrafilter $\cal{U}$ over $M_{\kappa}$, and let $M$ be $L$-th iterated ultrapower of $M_{\kappa}$ modulo $\cal{U}$, where $L$ is the REVERSE of $\lambda$. Then the desired $I$ is $M_{\kappa}$.

Corollary. For every pair of infinite cardinals $\kappa$ and $\lambda$, there is a real closed field $F$ that has a Dedekind cut $(J,K)$ such that $cf(J) = \kappa$ and $dcf(K)=\lambda$.

Explanation: Given a model $M$ of $PA$, $M$ can define the real closure $F$ of itself (by using the same arithmetical recipe that defines the field of algebraic real numbers within the standard model of arithmetic). Since every positive element of $F$ is less than one away from some member of $M$, no gap of $M$ is filled by an element of $F$. Hence given $I$ as in the theorem, the cut $J$ of $F$ defined as the set of all $x \in F$ such that $x < i$ for some $i \in I$ does the job.

Let me also point out another results which shows that a real closed field constructed as above as the real closure of a model of $PA$ always has a "matched cut". This follows from the following result of Shelah which also appears as Theorem 11.1.1 (p.281) of the Kossak-Schmerl text on models of $PA$.

Theorem 2. For every nonstandard model $M$ of $PA$ there is an infinite cardinal $\kappa$, and there is an initial segment $I$ of $M$ such that $cf(I) = \kappa = dcf(M$ \ $I)$.

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Ali Enayat
Ali Enayat
  • 17.7k
  • 2
  • 63
  • 105

The theorem below answers question 4, and the corollary answers question 3. See also Theorem 2.

Theorem 1. For every pair of infinite cardinals $(\kappa$,$\lambda)$, there is a model $M$ of $PA$ (Peano arithmetic) that has an initial segment $I$ such that $cf(I) = \kappa$ and $cf(M$ \ $I) =\lambda$, where $dcf$ is "downward cofinality".

The model $M$ can be constructed using the McDowell-Specker-Gaifman machinery of "minimal types"; which an be thought of as Ramsey ultrafilters over the Boolean algebra of the parametrically definable subsets of an ambient model of $PA$. The rough idea is as follows: start with a countable model $M_{0}$ of $PA$ and use the McDowell-Specker theorem to build an elementary end extension $M_{\kappa}$ of $M_0$ such that $cf(M_{\kappa})=\kappa$. Then build a minimal type/Ramsey ultrafilter $\cal{U}$ over $M_{\kappa}$, and let $M$ be $L$-th iterated ultrapower of $M_{\kappa}$ modulo $\cal{U}$, where $L$ is the REVERSE of $\lambda$. Then the desired $I$ is $M_{\kappa}$.

Corollary. For every pair of infinite cardinals $\kappa$ and $\lambda$, there is a real closed field $F$ that has a Dedekind cut $(J,K)$ such that $cf(J) = \kappa$ and $dcf(K)=\lambda$.

Explanation: Given a model $M$ of $PA$, $M$ can define the real closure $F$ of itself (by using the same arithmetical recipe that defines the field of algebraic real numbers within the standard model of arithmetic). Since every positive element of $F$ is less than one away from some member of $M$, no gap of $M$ is filled by an element of $F$. Hence given $I$ as in the theorem, the cut $J$ of $F$ defined as the set of all $x \in F$ such that $x < i$ for some $i \in I$ does the job.

Let me also point out another results which shows that a real closed field constructed as above as the real closure of a model of $PA$ always has a "matched cut". This follows from the following result of Shelah which also appears as Theorem 11.1.1 (p.281) of the Kossak-Schmerl text on models of $PA$.

Theorem 2. For every nonstandard model $M$ of $PA$ there is an infinite cardinal $\kappa$, and there is an initial segment $I$ of $M$ such that $cf(I) = \kappa = dcf(M$ \$I)$.