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Re: distinguishability  in context, according to definitions
Posted:
Feb 21, 2013 3:33 AM


In <PO6dnbZX0sYOZr7MnZ2dnUVZ_jGdnZ2d@giganews.com>, on 02/19/2013 at 04:24 PM, fom <fomJUNK@nyms.net> said:
>It may not be a great book, but it is what I have.
I don't believe that the problem is in the book. The problem is that there is no rule of inference allowing you to get from a=a and b=b to a=b.
>Granted, my illustration was without any rules,
How is it an illustration when it isn't a valid inference?
>Yes. But, in what way?
The constants appear in the axioms of the system; the notational conventions do not.
>If we sit in a roomful of dogs. And, if I continue >to refer to "the dog." You will ask, "Which dog?"
Unless you have defined "the dog".
>I say, "THE dog. Of course!" Because you see many dogs in the >room, it is clear that I am not describing the situation sensibly.
That, however, is not a mathematical issue.
>By the received paradigm on these matters, "objects" are >"selfidentical." Constants and singular terms purport to refer to >individuals.
That's philosophy. A mathematical theory applies to anything that satisfies the axioms.
>When Dedekind devised his sequence of ordinal numbers, >he did not "purport" invariance. He formulated a theory based on >successive involutions. That is, he *modeled* it: >Definition: A system N is said to be simply >infinite when there exists as similar transformation >F of N in itself such that N appears as chain of >an element not contained in F(N). We call this >element, which we shall denote in what follows by >the symbol '1',
Are you sure that you quoted accurately? In particular, are you sure that his definition didn't include singling out a specific F?
>Correct. But is that the kind of set theory which had >originally been envisioned?
Most[1] set theories that came out in the wake of Russell's paradox either had no universal class or didn't allow the universal class to be an element.
[1] NF used a different mechanism to avoid the paradox.
 Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
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