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SURREAL ORDERED EXPONENTIAL FIELDS – ERRATUM
- PHILIP EHRLICH, ELLIOT KAPLAN
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- Journal:
- The Journal of Symbolic Logic / Volume 87 / Issue 2 / June 2022
- Published online by Cambridge University Press:
- 09 March 2022, p. 871
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- June 2022
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Are Points (Necessarily) Unextended?
- Philip Ehrlich
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- Philosophy of Science / Volume 89 / Issue 4 / October 2022
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- 12 January 2022, pp. 784-801
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- October 2022
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Since Euclid defined a point as “that which has no part” it has been widely assumed that points are necessarily unextended. It has also been assumed that this is equivalent to saying that points or, more properly speaking, degenerate segments, have length zero. We challenge these assumptions by providing models of Euclidean geometry where the points are extended despite the fact that the degenerate segments have null lengths, and observe that whereas the extended natures of the points are not recognizable in the given models, they can be recognized and characterized by structures that are suitable expansions of the models.
SURREAL ORDERED EXPONENTIAL FIELDS
- Part of
- PHILIP EHRLICH, ELLIOT KAPLAN
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- Journal:
- The Journal of Symbolic Logic / Volume 86 / Issue 3 / September 2021
- Published online by Cambridge University Press:
- 13 August 2021, pp. 1066-1115
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- September 2021
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In 2001, the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway’s ordered field ${\mathbf {No}}$ of surreal numbers was brought to the fore by the first author and employed to provide necessary and sufficient conditions for an ordered field (ordered $K$ -vector space) to be isomorphic to an initial subfield ( $K$ -subspace) of ${\mathbf {No}}$ , i.e. a subfield ( $K$ -subspace) of ${\mathbf {No}}$ that is an initial subtree of ${\mathbf {No}}$ . In this sequel, analogous results are established for ordered exponential fields, making use of a slight generalization of Schmeling’s conception of a transseries field. It is further shown that a wide range of ordered exponential fields are isomorphic to initial exponential subfields of $({\mathbf {No}}, \exp )$ . These include all models of $T({\mathbb R}_W, e^x)$ , where ${\mathbb R}_W$ is the reals expanded by a convergent Weierstrass system W. Of these, those we call trigonometric-exponential fields are given particular attention. It is shown that the exponential functions on the initial trigonometric-exponential subfields of ${\mathbf {No}}$ , which includes ${\mathbf {No}}$ itself, extend to canonical exponential functions on their surcomplex counterparts. The image of the canonical map of the ordered exponential field ${\mathbb T}^{LE}$ of logarithmic-exponential transseries into ${\mathbf {No}}$ is shown to be initial, as are the ordered exponential fields ${\mathbb R}((\omega ))^{EL}$ and ${\mathbb R}\langle \langle \omega \rangle \rangle $ .
NUMBER SYSTEMS WITH SIMPLICITY HIERARCHIES: A GENERALIZATION OF CONWAY’S THEORY OF SURREAL NUMBERS II
- PHILIP EHRLICH, ELLIOT KAPLAN
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- Journal:
- The Journal of Symbolic Logic / Volume 83 / Issue 2 / June 2018
- Published online by Cambridge University Press:
- 05 February 2018, pp. 617-633
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- June 2018
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In [16], the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway’s ordered field ${\bf{No}}$ of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field to be isomorphic to an initial subfield of ${\bf{No}}$, i.e., a subfield of ${\bf{No}}$ that is an initial subtree of ${\bf{No}}$. In this sequel to [16], analogous results for ordered abelian groups and ordered domains are established which in turn are employed to characterize the convex subgroups and convex subdomains of initial subfields of ${\bf{No}}$ that are themselves initial. It is further shown that an initial subdomain of ${\bf{No}}$ is discrete if and only if it is a subdomain of ${\bf{No}}$’s canonical integer part ${\bf{Oz}}$ of omnific integers. Finally, making use of class models the results of [16] are extended by showing that the theories of nontrivial divisible ordered abelian groups and real-closed ordered fields are the sole theories of nontrivial densely ordered abelian groups and ordered fields all of whose models are isomorphic to initial subgroups and initial subfields of ${\bf{No}}$.
An Essay in Honor of Adolf Grünbaum’s Ninetieth Birthday: A Reexamination of Zeno’s Paradox of Extension
- Philip Ehrlich
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- Journal:
- Philosophy of Science / Volume 81 / Issue 4 / October 2014
- Published online by Cambridge University Press:
- 01 January 2022, pp. 654-675
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- October 2014
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We suggest that, far from establishing an inconsistency in the standard theory of the geometrical linear continuum, Zeno’s Paradox of Extension merely establishes an inconsistency between the standard theory of geometrical magnitude and a misguided system of length measurement. We further suggest that our resolution of Zeno’s paradox is superior to Adolf Grünbaum’s now standard resolution based on Lebesgue measure theory.
The Absolute Arithmetic Continuum and the Unification Of all Numbers Great and Small
- Philip Ehrlich
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- Journal:
- Bulletin of Symbolic Logic / Volume 18 / Issue 1 / March 2012
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- 15 January 2014, pp. 1-45
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- March 2012
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In his monograph On Numbers and Games, J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many less familiar numbers including −ω, ω/2, 1/ω, and ω − π to name only a few. Indeed, this particular real-closed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered fields—be individually definable in terms of sets of NBG (von Neumann–Bernays–Gödel set theory with global choice), it may be said to contain “All Numbers Great and Small.” In this respect, No bears much the same relation to ordered fields that the system ℝ of real numbers bears to Archimedean ordered fields.
In Part I of the present paper, we suggest that whereas ℝ should merely be regarded as constituting an arithmetic continuum (modulo the Archimedean axiom), No may be regarded as a sort of absolute arithmetic continuum (modulo NBG), and in Part II we draw attention to the unifying framework No provides not only for the reals and the ordinals but also for an array of non-Archimedean ordered number systems that have arisen in connection with the theories of non-Archimedean ordered algebraic and geometric systems, the theory of the rate of growth of real functions and nonstandard analysis.
In addition to its inclusive structure as an ordered field, the system No of surreal numbers has a rich algebraico-tree-theoretic structure—a simplicity hierarchical structure—that emerges from the recursive clauses in terms of which it is defined. In the development of No outlined in the present paper, in which the surreals emerge vis-à-vis a generalization of the von Neumann ordinal construction, the simplicity hierarchical features of No are brought to the fore and play central roles in the aforementioned unification of systems of numbers great and small and in some of the more revealing characterizations of No as an absolute continuum.
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- Edited by Daniel Patte, Vanderbilt University, Tennessee
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- The Cambridge Dictionary of Christianity
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- 05 August 2012
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- 20 September 2010, pp xi-xliv
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John L. Bell. The continuous and the infinitesimal in mathematics and philosophy. Polimetrica, International Scientific Publisher, Monza-Milano, 2005, 349 pp.
- Philip Ehrlich
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- Journal:
- Bulletin of Symbolic Logic / Volume 13 / Issue 3 / September 2007
- Published online by Cambridge University Press:
- 15 January 2014, pp. 361-363
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- September 2007
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Corrigendum to “Number Systems with Simplicity Hierarchies: A Generalization of Conway's Theory of Surreal Numbers”
- Philip Ehrlich
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- Journal:
- The Journal of Symbolic Logic / Volume 70 / Issue 3 / September 2005
- Published online by Cambridge University Press:
- 12 March 2014, p. 1022
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- September 2005
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Number systems with simplicity hierarchies: a generalization of Conway's theory of surreal numbers
- Philip Ehrlich
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- The Journal of Symbolic Logic / Volume 66 / Issue 3 / September 2001
- Published online by Cambridge University Press:
- 12 March 2014, pp. 1231-1258
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- September 2001
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Introduction. In his monograph On Numbers and Games [7], J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many other numbers including ω, ω, /2, 1/ω, and ω − π to name only a few. Indeed, this particular real-closed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered “number” fields—be individually definable in terms of sets of von Neumann-Bernays-Gödel set theory with Global Choice, henceforth NBG [cf. 21, Ch. 4], it may be said to contain “All Numbers Great and Small.” In this respect, No bears much the same relation to ordered fields that the system of real numbers bears to Archimedean ordered fields. This can be made precise by saying that whereas the ordered field of reals is (up to isomorphism) the unique homogeneous universal Archimedean ordered field, No is (up to isomorphism) the unique homogeneous universal orderedfield [14]; also see [10], [12], [13].
However, in addition to its distinguished structure as an ordered field, No has a rich hierarchical structure that (implicitly) emerges from the recursive clauses in terms of which it is defined. This algebraico-tree-theoretic structure, or simplicity hierarchy, as we have called it [15], depends upon No's (implicit) structure as a lexicographically ordered binary tree and arises from the fact that the sums and products of any two members of the tree are the simplest possible elements of the tree consistent with No's structure as an ordered group and an ordered field, respectively, it being understood that x is simpler than y just in case x is a predecessor of y in the tree.
The Absolute Arithmetic and Geometric Continua
- Philip Ehrlich
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- Journal:
- PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association / Volume 1986 / Issue 2 / 1986
- Published online by Cambridge University Press:
- 28 February 2022, pp. 237-246
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- 1986
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With the appearance of J.H. Conway's On Numbers and Games (1976), the mathematical and philosophical communities have much to celebrate. It is Conway's important discovery that the familiar Dedekind cut and von Neumann ordinal constructions are part of a more general construction which leads to a proper class of numbers embracing the reals and the ordinals as well as many less familiar numbers including -ω, ω/2, l/ω, √ω and ω-π, where ω is the least infinite ordinal. Conway further shows that the arithmetic of the reals may be extended to the entire class yielding a real-closed ordered field (1976, pp. 40-42); that is, an ordered field where every positive element is a square, and every polynomial of odd degree with coefficients in the field has a solution in the field.