Set Theory – What’s Wrong with the Surreals?

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Of all the constructions of the reals, the construction via the surreals seems the most elegant to me.

It seems to immediately capture the total ordering and precision of Dedekind cuts at a fundamental level since the definition of a number is based entirely on how things are ordered. It avoids, or at least simplifies, the convergence question of Cauchy sequences. And it naturally transcends finiteness without sacrificing awareness of it.

The one "rumor" I've consistently heard is that it is hard to naturally define integrals and derivatives in the surreals, although I have yet to see a solid technical justification of that.

Are there known results that suggest we should avoid further study of this construction, or that show limitations of it?

Best Answer

At a recent conference in Paris on Philosophy and Model Theory (at which I also spoke), Philip Ehrlich gave a fascinating talk on the surreal numbers and new developments, showcasing it as unifying many disparate paths in mathematics. The abstract is available here, on page 8, and here his article on the Absolute Arithmetic Continuum. The principal new technical development is a focus on the underlying tree.

Philip expressed his frustration that Conway often treated his creation of surreal numbers as a kind of game or just-for-fun project---an attitude reinforced by the excellent Knuth book---whereas they are in fact a profound mathematical development unifying disparate threads of mathematical investigation into a single unifying structure. And he made a very strong case for this position at the conference.

Meanwhile, perhaps exhibiting Philip's point, at a conference on logic and games here at CUNY, I once heard Conway describe the surreal numbers as one of the great disappointments of his life, that they did not seem after all to have the profound unifying nature that he (and many others) thought they might. Philip Ehrlich strove to make the case that Conway was his own worst enemy in promoting the surreals, and that they actually do have the unifying nature Conway thought they did, but that Conway scared people away from this perspective by treating them as a toy. I encourage you to read Philip's articles.

So my answer, supporting Philip, is that nothing is wrong with the surreals---please have at them! Of course they have their own issues, which will need to be surmounted, but we shall all benefit from a greater investigation of them.

Related Solutions

[Math] Who discovered the surreals

Norman Alling's Conway's field of surreal numbers (1985) gives full credit to Conway:

Conway introduced the Field No of numbers, which Knuth has called the surreal numbers. No is a proper class and a real-closed field, with a very high level of density, which can be described by extending Hausdorff's $\eta_\xi$ condition. In this paper the author applies a century of research on ordered sets, groups, and fields to the study of No.

References to Alling's own 1962 paper appear only in passing, on page 373 and 377–378. The 1987 book referred to in the OP follows up on this 1985 exposition, which makes it clear that Alling in no way claims independence from Conway.

Set Theory – Consistency Strength of Reflection Forms

This theory is inconsistent.

We note that by 1 and 2 that if set(x) and y⊆x, then set(y).

(a) There is a v such that ∀x(set(x)-->x∈v).

Proof:Suppose not. Then ∀v∃s∃t(s∈t∧s∉v). By 3 there is transitive x such that

  set(x) and ∀v(v⊆x-->∃s∃t(s⊆x∧t⊆x∧s∈t∧s∉v). In particular 

  (x⊆x-->∃s∃t(s⊆x∧t⊆x∧s∈t∧s∉x).  But this is impossible.

Suppose that ∀x(set(x)-->x∈V). Then ∃w∀t(t∈V-->t∈w). By 3 there is transitive x

such that set(x) and ∃w(w⊆x∧∀t(t⊆x∧t∈V-->t∈w)). Since t⊆x implies set(t),

t⊆x implies t∈x. By 2, there is a c such that t∈c<-->(t∈x∧t∉t). Since set(c),

c∈c<-->c∉c.

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Related Solutions

[Math] Who discovered the surreals

Set Theory – Consistency Strength of Reflection Forms

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