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Philip Ehrlich
Philip Ehrlich
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I suspect the following should address the above stated queries.

Let $A$ be an ordered field whose universe is a proper class of NBG (which I take to include Global Choice).

(i) A real-closure of $A$ exists in NBG.

Proof. In virtue of Global Choice, A is the union of a continuous chain $A_{\alpha}$ of ordered fields indexed over all $\alpha < On$ where each $A_{\alpha}$ is a set. One can now construct a continuous chain $B_{\alpha}$ of ordered fields indexed over all $\alpha < On$, where each $B_{\alpha}$ is a real-closure of $A_{\alpha}$. The union of the latter chain is a real-closure of $A$ in NBG.

(ii) The Dedekindean completion of No does not exist in NBG.

Proof. The Dedekindean completion of No arises by supplementing No (whose elements we may assume to be written in normal form) with all entities of the form $${\sum\limits_{\alpha < \, On } {\omega ^{y_\alpha } .r_\alpha }} $$

where $\left( {y_\alpha} \right)_{\alpha < On } $ is a strictly decreasing sequence of surreals that is coinitial with the negative surreals, and $\left( {r_\alpha } \right)_{\alpha < On } $ is a sequence of nonzero real numbers that are neither ultimately positive nor ultimately negative.

While this real-closed ordered field exists in Ackermann’s set theory (as formulated by Reinhardt in Ackermann’s Set Theory Equals ZF 1970, Annals of Math Log]), it does not exist in NBG since it contains $2^{\aleph_{On}}$ many members.

I suspect the following should address the above stated queries.

Let $A$ be an ordered field whose universe is a proper class of NBG (which I take to include Global Choice).

(i) A real-closure of $A$ exists in NBG.

Proof. In virtue of Global Choice, A is the union of a continuous chain $A_{\alpha}$ of ordered fields indexed over all $\alpha < On$ where each $A_{\alpha}$ is a set. One can now construct a continuous chain $B_{\alpha}$ of ordered fields indexed over all $\alpha < On$, where each $B_{\alpha}$ is a real-closure of $A_{\alpha}$. The union of the latter chain is a real-closure of $A$ in NBG.

(ii) The Dedekindean completion of No does not exist in NBG.

Proof. The Dedekindean completion of No arises by supplementing No (whose elements we may assume to be written in normal form) with all entities of the form $${\sum\limits_{\alpha < \, On } {\omega ^{y_\alpha } .r_\alpha }} $$

where $\left( {y_\alpha} \right)_{\alpha < On } $ is a strictly decreasing sequence of surreals that is coinitial with the negative surreals, and $\left( {r_\alpha } \right)_{\alpha < On } $ is a sequence of nonzero real numbers.

While this real-closed ordered field exists in Ackermann’s set theory (as formulated by Reinhardt in Ackermann’s Set Theory Equals ZF 1970, Annals of Math Log]), it does not exist in NBG since it contains $2^{\aleph_{On}}$ many members.

I suspect the following should address the above stated queries.

Let $A$ be an ordered field whose universe is a proper class of NBG (which I take to include Global Choice).

(i) A real-closure of $A$ exists in NBG.

Proof. In virtue of Global Choice, A is the union of a continuous chain $A_{\alpha}$ of ordered fields indexed over all $\alpha < On$ where each $A_{\alpha}$ is a set. One can now construct a continuous chain $B_{\alpha}$ of ordered fields indexed over all $\alpha < On$, where each $B_{\alpha}$ is a real-closure of $A_{\alpha}$. The union of the latter chain is a real-closure of $A$ in NBG.

(ii) The Dedekindean completion of No does not exist in NBG.

Proof. The Dedekindean completion of No arises by supplementing No (whose elements we may assume to be written in normal form) with all entities of the form $${\sum\limits_{\alpha < \, On } {\omega ^{y_\alpha } .r_\alpha }} $$

where $\left( {y_\alpha} \right)_{\alpha < On } $ is a strictly decreasing sequence of surreals that is coinitial with the negative surreals, and $\left( {r_\alpha } \right)_{\alpha < On } $ is a sequence of nonzero real numbers that are neither ultimately positive nor ultimately negative.

While this real-closed ordered field exists in Ackermann’s set theory (as formulated by Reinhardt in Ackermann’s Set Theory Equals ZF 1970, Annals of Math Log]), it does not exist in NBG since it contains $2^{\aleph_{On}}$ many members.

Source Link
Philip Ehrlich
Philip Ehrlich
  • 6.5k
  • 1
  • 42
  • 37

I suspect the following should address the above stated queries.

Let $A$ be an ordered field whose universe is a proper class of NBG (which I take to include Global Choice).

(i) A real-closure of $A$ exists in NBG.

Proof. In virtue of Global Choice, A is the union of a continuous chain $A_{\alpha}$ of ordered fields indexed over all $\alpha < On$ where each $A_{\alpha}$ is a set. One can now construct a continuous chain $B_{\alpha}$ of ordered fields indexed over all $\alpha < On$, where each $B_{\alpha}$ is a real-closure of $A_{\alpha}$. The union of the latter chain is a real-closure of $A$ in NBG.

(ii) The Dedekindean completion of No does not exist in NBG.

Proof. The Dedekindean completion of No arises by supplementing No (whose elements we may assume to be written in normal form) with all entities of the form $${\sum\limits_{\alpha < \, On } {\omega ^{y_\alpha } .r_\alpha }} $$

where $\left( {y_\alpha} \right)_{\alpha < On } $ is a strictly decreasing sequence of surreals that is coinitial with the negative surreals, and $\left( {r_\alpha } \right)_{\alpha < On } $ is a sequence of nonzero real numbers.

While this real-closed ordered field exists in Ackermann’s set theory (as formulated by Reinhardt in Ackermann’s Set Theory Equals ZF 1970, Annals of Math Log]), it does not exist in NBG since it contains $2^{\aleph_{On}}$ many members.