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 >"self-identical." 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 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.
 NF used a different mechanism to avoid the paradox.
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