Math Forum Discussions

Shmuel (Seymour J.) Metz

Posts: 3,473
Registered: 12/4/04

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

Plain Text

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[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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