# [FOM] Follow-up to Tennant

Ross A. Finlayson apex at apexinternetsoftware.com
Mon Feb 17 22:25:22 EST 2003

```On Sunday, February 16, 2003, at 09:17 AM, Dean Buckner wrote:

>
> Neil Tennant's entertaining parable about the crocodiles neatly
> illustrates
> the difficulty of reconciling a theory of plural reference with set
> theory.
>
> Given the pair axiom, and the existence of the empty set, then
> ("Kaboom")
> "the hierarchy of hereditarily pure sets mushrooms up in the Platonic
> portion" of the crocodiles mind.  {} exists, and {{}} exists, and {{{}}}
> exists, and ...
>

Eventually what you can get there is a powerset of the finite ordinals
which contains via an element-of operator/relation an infinite ordinal.

> Everything in Omega exists.  Yet Omega need not exist.  So what exactly
> _are_ we asserting to exist when we say that Omega also exists?

The powerset of N, or omega, is constructed thus that for any element E
of P(N) that N >= E.

> ... [content omitted]
> As should  be plain, in any case from
>
>         The set S contains a large number of objects
>
> The expression "a large number of objects" applies to a large number of
> objects.  The expression "The set S" does not, otherwise the whole
> sentence
> would assert that S contains itself.  Similarly, "the set containing
> all the
> numbers between 1 and 100" refers to something different from "all the
> numbers between 1 and 100".  Yes?
>
>

I would think about this in the computer science or linguistic way where
there are certain operations that can be applied to both of those two
things with the same results.  Then, depending upon what type those two
nearly equal things are said to subsume, they may be almost equivalent.
For calling the two nouns A and B, then the element with the highest
value of both A and B is 100.  A and B each have an element of operator
with the for-any-element, for-each-element, and for-all-elements.  They
almost exactly refer to the same thing, and may be considered to be the
same thing.  Each describes the collection of any and all numbers where
the value of the number is between one and a hundred.

I approach the logic in a similar way as Dean has professed in
preferring mechanisms that are easily explicable in plain language, in
this case, English.

I think there is a lot of utility in Chu spaces, collections, ramified
and stratified type theories, and that those things are part of a set of
all sets, where any thing has a unique identifier, and its complement
and context in the set of all sets helps identify the item as unique.

Ross Finlayson
```

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