[FOM] FOM: Set theory and ordinary language

[Jeremy Clark]

Dean,

The answer to your question is 'no'. You could take, for example, the
reverse ordering of
any successor ordinal: this has a first and last element, and since in a
well ordering
any element has a successor, in the specified ordering, every element will
have a predecessor.
Even if you add condition (e), that every element has a successor, you will
still have examples
like \omega + \omega* (a copy of omega, followed by a copy of omega
reversed). In fact,
no collection of conditions specifiable in first-order sentences will
capture finite linear-orderings,
as a result of the compactness theorem. You need some sort of second-order
condition
(like the usual set-theoretical ones which capture finiteness).

Regards,

Jeremy Clark


> Consider ordinal descriptions, such as "the first man", "the second man",
> "the third man" and so on, as they are used in ordinary language.
>
> Suppose we can show that any sequence of such expressions (a) has a linear
> ordering (b) has a first member (c) has a "last" member (d) contains no
> "limit ordinal", ie. contains no ordinal without a direct predecessor.
>
> Does this then guarantee that the sequence determines a finite set, and
thus
> a
> set which is well-ordered, and thus a set to which for which finite
> induction is automatically valid)?
>
>
> Dean
>
>
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