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On page 183 of “Fields of Surreal Numbers and Exponentiation”, Fundamenta Mathematica 167 (2001), pp. 173-188, Lou van den Dries and Philip Ehrlich prove the following result that I believe provides an answer to the author’s queries. The result is an improvement of a result the author attributes to Norman Alling, which was established independently by Ehrlich in “An Alternative Construction of Conway’s Ordered Field No,” Algebra Universalis 25 (1988), pp. 7-16. Ehrlich’s proof actually slightly predates Alling’s’s, but it appeared later.

Proposition. Let $\lambda$ be an epsilon number $\le On$ and let $\tau =1/\omega $. Then

(i) $${\bf{No}}\left( \lambda \right) = \bigcup\nolimits_\mu {\left( {\left( {\tau ^{{\bf{No}}\left( \mu \right)} } \right)} \right)_\lambda } $$ where $\mu$ ranges over the additively indecomposable ordinals $ < \lambda $;

(ii) ${\bf{No}}\left( \lambda \right)$ is real-closed;

(iii) $${\bf{No}}\left( \lambda \right) = \left( {\left( {\tau ^{{\bf{No}}\left( \lambda \right)} } \right)} \right)_\lambda $$ if and only if $\lambda $ is a regular cardinal;

(vi) For all $y \in {\bf{No}}$, $y \in {\bf{No}}\left( \lambda \right) $ if and only if $\omega ^y \in {\bf{No}}\left( \lambda \right)$.

On page 183 of “Fields of Surreal Numbers and Exponentiation”, Fundamenta Mathematica 167 (2001), pp. 173-188, Lou van den Dries and Philip Ehrlich prove the following result that I believe provides an answer to the author’s queries. The result is an improvement of a result the author attributes to Norman Alling, which was established independently by Ehrlich in “An Alternative Construction of Conway’s Ordered Field No,” Algebra Universalis 25 (1988), pp. 7-16. Ehrlich’s proof actually slightly predates Alling’s’s, but it appeared later.

Proposition. Let $\lambda$ be an epsilon number $\le On$ and let $\tau =1/\omega $. Then

(i) $${\bf{No}}\left( \lambda \right) = \bigcup\nolimits_\mu \mathbb{R}{\left( {\left( {\tau ^{{\bf{No}}\left( \mu \right)} } \right)} \right)_\lambda } $$ where $\mu$ ranges over the additively indecomposable ordinals $ < \lambda $;

(ii) ${\bf{No}}\left( \lambda \right)$ is real-closed;

(iii) $${\bf{No}}\left( \lambda \right) = \mathbb{R}\left( {\left( {\tau ^{{\bf{No}}\left( \lambda \right)} } \right)} \right)_\lambda $$ if and only if $\lambda $ is a regular cardinal;

(vi) For all $y \in {\bf{No}}$, $y \in {\bf{No}}\left( \lambda \right) $ if and only if $\omega ^y \in {\bf{No}}\left( \lambda \right)$.

On page 183 of “Fields of Surreal Numbers and Exponentiation”, Fundamenta Mathematica 167 (2001), pp. 173-188, Lou van den Dries and Philip Ehrlich prove the following result that I believe provides an answer to the author’s queries. The result is an improvement of a result the author attributes to Norman Alling, which was established independently by Ehrlich in “An Alternative Construction of Conway’s Ordered Field No,” Algebra Universalis 25 (1988), pp. 7-16. Ehrlich’s proof actually slightly predates Alling’s’s, but it appeared later.

Proposition. Let $\lambda$ be an epsilon number $\le O$ and let $\tau =1/\omega $. Then

(i) $${\bf{No}}\left( \lambda \right) = \bigcup\nolimits_\mu {\left( {\left( {\tau ^{{\bf{No}}\left( \mu \right)} } \right)} \right)_\lambda } $$ where $\mu$ ranges over the additively indecomposable ordinals $ < \lambda $;

(ii) ${\bf{No}}\left( \lambda \right)$ is real-closed;

(iii) $${\bf{No}}\left( \lambda \right) = \left( {\left( {\tau ^{{\bf{No}}\left( \lambda \right)} } \right)} \right)_\lambda $$ if and only if $\lambda $ is a regular cardinal;

(vi) For all $y \in {\bf{No}}$, $y \in {\bf{No}}\left( \lambda \right) $ if and only if $\omega ^y \in {\bf{No}}\left( \lambda \right)$.

On page 183 of “Fields of Surreal Numbers and Exponentiation”, Fundamenta Mathematica 167 (2001), pp. 173-188, Lou van den Dries and Philip Ehrlich prove the following result that I believe provides an answer to the author’s queries. The result is an improvement of a result the author attributes to Norman Alling, which was established independently by Ehrlich in “An Alternative Construction of Conway’s Ordered Field No,” Algebra Universalis 25 (1988), pp. 7-16. Ehrlich’s proof actually slightly predates Alling’s’s, but it appeared later.

Proposition. Let $\lambda$ be an epsilon number $\le On$ and let $\tau =1/\omega $. Then

(i) $${\bf{No}}\left( \lambda \right) = \bigcup\nolimits_\mu {\left( {\left( {\tau ^{{\bf{No}}\left( \mu \right)} } \right)} \right)_\lambda } $$ where $\mu$ ranges over the additively indecomposable ordinals $ < \lambda $;

(ii) ${\bf{No}}\left( \lambda \right)$ is real-closed;

(iii) $${\bf{No}}\left( \lambda \right) = \left( {\left( {\tau ^{{\bf{No}}\left( \lambda \right)} } \right)} \right)_\lambda $$ if and only if $\lambda $ is a regular cardinal;

(vi) For all $y \in {\bf{No}}$, $y \in {\bf{No}}\left( \lambda \right) $ if and only if $\omega ^y \in {\bf{No}}\left( \lambda \right)$.

On page 183 of “Fields of Surreal Numbers and Exponentiation”, Fundamenta Mathematica 167 (2001), pp. 173-188, Lou van den Dries and Philip Ehrlich prove the following result that I believe provides an answer to the author’s queries. The result is an improvement of a result the author attributes to Norman Alling, which was established independently by Ehrlich in “An Alternative Construction of Conway’s Ordered Field No,” Algebra Universalis 25 (1988), pp. 7-16. Ehrlich’s proof actually slightly predates Alling’s’s, but it appeared later.

Proposition. Let $$\lambda $$ be an epsilon number $$\le On$$ and let $$\tau =1/\omega $$. Then

(i) $${\bf{No}}\left( \lambda \right) = \bigcup\nolimits_\mu {\left( {\left( {\tau ^{{\bf{No}}\left( \mu \right)} } \right)} \right)_\lambda } $$, where $$\mu $$ ranges over the additively indecomposable ordinals $$ < \lambda $$;

(ii) $${\bf{No}}\left( \lambda \right)$$ is real-closed;

(iii) $${\bf{No}}\left( \lambda \right) = \left( {\left( {\tau ^{{\bf{No}}\left( \lambda \right)} } \right)} \right)_\lambda $$ if and only if $$\lambda $$ is a regular cardinal;

(vi) For all $$y \in {\bf{No}}$$, $$y \in {\bf{No}}\left( \lambda \right) $$ if and only if $$\omega ^y \in {\bf{No}}\left( \lambda \right)$$.

On page 183 of “Fields of Surreal Numbers and Exponentiation”, Fundamenta Mathematica 167 (2001), pp. 173-188, Lou van den Dries and Philip Ehrlich prove the following result that I believe provides an answer to the author’s queries. The result is an improvement of a result the author attributes to Norman Alling, which was established independently by Ehrlich in “An Alternative Construction of Conway’s Ordered Field No,” Algebra Universalis 25 (1988), pp. 7-16. Ehrlich’s proof actually slightly predates Alling’s’s, but it appeared later.

Proposition. Let $\lambda$ be an epsilon number $\le O$ and let $\tau =1/\omega $. Then

(i) $${\bf{No}}\left( \lambda \right) = \bigcup\nolimits_\mu {\left( {\left( {\tau ^{{\bf{No}}\left( \mu \right)} } \right)} \right)_\lambda } $$ where $\mu$ ranges over the additively indecomposable ordinals $ < \lambda $;

(ii) ${\bf{No}}\left( \lambda \right)$ is real-closed;

(iii) $${\bf{No}}\left( \lambda \right) = \left( {\left( {\tau ^{{\bf{No}}\left( \lambda \right)} } \right)} \right)_\lambda $$ if and only if $\lambda $ is a regular cardinal;

(vi) For all $y \in {\bf{No}}$, $y \in {\bf{No}}\left( \lambda \right) $ if and only if $\omega ^y \in {\bf{No}}\left( \lambda \right)$.