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proper class, and they do not form a set. So in a purely technical sense, they are not a field. But nevertheless, they do satisfy all the field axioms and have all the usual kinds of structure that one would want in a field, and so one can correctly describe them as a proper class field, or Field as Conway writes it, in much the same way that the class of all ordinals is regarded as a proper class well-order. Although this set/class issue may seem mysterious or irritating, in practice it is a routine matter

One

But since you seem particularly interested in what might go wrong, let me suggest on the negative side, one issue that could make a difference is that when dealing with the surreal field No, one will probably want to strengthen the background set theory from ZFC to GBC, which includes the global axiom of choice, the assertion that there is a proper class well-ordering of the universe. It The reason is that it is consistent with ZFC that the surreal numbers do not admit any proper class well-ordering, and actually, the assertion that they have a definable such well-ordering is equivalent to the set-theoretic axiom known as V=HOD, as I proved in my answer to David Feldman's question on a Definable map from all the ordinals to the surreal numbers. So if one wants to undertake algebraic constructions requiring one to have a well-ordering of the field itself, such as finding a proper class maximal ideal inside a particular subring of No, then there could be difficulties undertaking such a construction in ZFC as opposed to GBC. But nevertheless, the theory GBC is conservative over ZFC and one may thereby freely assume the global axiom of choice. (This is used in the various arguments showing that No is universal for class-sized objects, such as the assertion that every class order embeds to a suborder of no.No.) In particular, in GBC one has a well-ordering of the entire universe, including the surreals, and this situation would address such issues. Much of the theory undertaken by Ehrlich on the surreals, for example, works in GBC as the background theory. Beyond this issue, even in GBC one does not have any sense of a well-ordering of the (meta-class) collection of all class-sized subrings of No, if this were desired for any algebraic construction, and this is the kind of issue that would arise with the set/class issue.

But I meanwhile, there is also have a more positive answer. The situation is that if one wants set versions of the surreal numbers, they are abundantly available in increasingly powerful and accurate approximations, which are well-understood and studied. Specifically, we have numerous set-sizedany ordinal $\lambda$, let $\text{No}(\lambda)$ be the set of surreal numbers

Philip Ehrlich mentioned in his recent talk at the CUNY Logic Workshop that truncating at sufficiently powerful ordinals. Indeed, for any particular natural number $n$, there will be a closed unbounded proper class of ordinals $\lambda$ such that $\text{No}(\lambda)$ has all the same $\Sigma_n$-expressible properties as the full class of surreal numbers No. This can be proved as an immediate consequence of the reflection theorem. So in fact, No is the union of a proper class chain of increasingly elementary subfields $\text{No}(\lambda)$.

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One issue that will could make No different from an ordinary field a difference is that in ZFC when dealing with the surreal field No, one doesn't have will probably want to strengthen the background set theory from ZFC to GBC, which includes the global axiom of choice. For example, it the assertion that there is a proper class well-ordering of the universe. It is consistent with ZFC that the surreal numbers themselves do not have admit any proper class well-ordering(, and actually, the assertion that they have a definable such well-ordering is equivalent to the set-theoretic axiom known as V=HOD, as I proved in my answer to David Feldman's question on a Definable map from all the ordinals to the surreal numbers). Thus, So if one wants to undertake a construction algebraic constructions requiring one to have a well-ordering of the field itself, such as finding a proper class maximal ideal inside a particular subring of No, then there could be difficulties undertaking it with No such a construction in ZFC itselfas opposed to GBC. But meanwhilenevertheless, with what is known as the global axiom of choice, which theory GBC is conservative over ZFC and part one may thereby freely assume the global axiom of choice. (This is used in the GBC axiomsvarious arguments showing that No is universal for class-sized objects, such as the assertion that every class order embeds to a suborder of no.) In particular, in GBC one can have has a well-ordering of the entire universe, including the surreals, and this solution situation would completely address such issues. For example, much Much of the theory undertaken by Ehrlich on the surrealssimply , for example, works in GBC as the background theory, so that the global axiom of choice is available. Beyond this issue, even in GBC one does not have any sense of a well-ordering of the (meta-class) collection of all class-sized subrings of No, if this were desired for any algebraic construction, and this is the kind of issue that would arise with the set/class issue.

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Now for

One issue that will make No different from an ordinary field is that in ZFC one doesn't have the global axiom of choice. For example, it is consistent with ZFC that the surreal numbers themselves do not have any proper class well-ordering (and the assertion that they have a definable such well-ordering is equivalent to the set-theoretic axiom known as V=HOD, as I proved in my actual answer to David Feldman's question on a Definable map from all the ordinals to the surreal numbers). Thus, if one wants to undertake a construction requiring one to have a well-ordering of the field itself, then there could be difficulties undertaking it with No in ZFC itself. But meanwhile, with what is known as the global axiom of choice, which is conservative over ZFC and part of the GBC axioms, one can have a well-ordering of the entire universe, including the surreals, and this solution would completely address such issues. For example, much of the theory undertaken by Ehrlich on the surreals simply works in GBC as the background theory, so that the global axiom of choice is available. Beyond this issue, in GBC one does not have any sense of a well-ordering of the collection of all class-sized subrings of No, if this were desired for any algebraic construction, and this is the kind of issue that would arise with the set/class issue.

But I also have a more positive answer. The situation is that if one wants set versions of the surreal numbers, they are abundantly available in increasingly powerful and accurate approximations, which are well-understood and studied. Specifically, we have numerous set-sized