[FOM] FOM: Set theory and ordinary language

[Richard G Heck]

Dean Buckner wrote:

>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)?
>
No. An ordering of the form omega + -omega will satisfy these 
conditions. So, for example, the set:
    {1,2,3,...n,...,...-n...-3,-2,-1}
satisfies all of your conditions. There is a general reason to be 
suspicious: The four conditions can be specified as a single first-order 
sentence (which will, of course, be a conjunction, but nevertheless).

Richard