Dennis E. Hamilton
dennis.hamilton at acm.org
Wed Jan 14 16:43:08 EST 2004
My immediate impression was that the question is not well-stated enough for there to be any meaningful discussion.
It seems to me that one must give explicit statement to a large collection of tacit assumptions behind "the same mathematical objects" and the system in which ordinals and cardinals are identified/represented/expressed/exist (or something). Then the question can be addressed in the context of those givens/premises. Maybe mathematicians have that be settled. I am not at that point.
After nosing around, I came to wonder is this a Cheshire Cat question? Or is the subject "ontology" because the first question must be what does it mean to say "*the* (finite) ordinals" and "*the* (finite) cardinals?" (Your placement of emphasis may vary.)
My unsophisticated appreciation of the question is best expressed by Quine's straightforward statements:
"Numbers as measures of multiplicity are called *cardinal**numbers* [Emphasis his]. Thus the natural numbers are adequate as finite cardinals." [STL: section 30].
I find this next a little deeper, so I am on thinner ice: "Because of the general comparability of well-orderings, we can adopt measures for them -- numbers of a sort, called *ordinal**numbers* [ditto]. ... For finite well-orderings they will be simply the natural numbers, or might as well be; for, as remarked, the lengths of finite-orderings compare as the quantity of the things ordered." [STL: section 22]. I like the tie-in with the finite-well-ordering presumption and length-to-quantity (-to-multiplicity -to-cardinality).
I reckon that the question asked is deeper than this. What is the context?
On the likely chance that I am taking this far more seriously than is intended, I have just performed an act of penance for lack of humor that will further your becoming rich, famous and universally loved. [;<).
[STL] Quine, Willard Van Orman. Set Theory and Its Logic. Revised edition. Harvard University Press (Cambridge, MA: 1963, 1969). ISBN 0-674-80207-1 pbk.
From: Thomas Forster on Wednesday, January 14, 2004 10:14 (pst)
An essay question:
Are the finite ordinals the same mathematical objects as the finite
cardinals? Give reasons... [\aleph_0 marks]
[ ... ]
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