Re: What is the 1st order formal system known as PA?



David C. Ullrich wrote:



The parts of this that make sense, and are not simple misunderstandings on your part, have not been neglected,
they've very well known. To be more specific:

You evidently think that the function symbols in FOL
actually refer to specific n-ary relations. They do not.

They do not _until_ we start talking about an _interpretation_
of the language (a "model"). If we're talking about interpretations of first-order PA then yes, "S" can have
many different interpretations. Nobody has every said
otherwise - it's awesomely obvious that "S" can have
many different interpretations.


Yes, it's "awesomely" obvious that S can have many different [model-]
interpretations. But it seems much less obvious that, up-and-down
the ladder of mathematical _introspection_, model-interpretation
is *not* the only kind of legitimate interpretation, albeit the fact
it's the only kind we *normally* have to deal with. For example,
what about the semantic-interpretation of the language (or of a
portion of it)? Are you sure any 2 reasoning beings can _always_
agree what, say, a particular symbol of the language _mean_?
And in this case, how can any 2 beings [human or not] guarantee
that they semantically mean the same existence of a successor
function? [And while for most of the familiar formulae such as
Axy (x+y = y+x), that they might semantically mean 2 different
successor functions - by the single name "S" - won't matter at all,
how can we be so sure it won't matter to _all_ formulae of L(PA)?
For instance, how can we be so sure it won't matter to GC(PA)?]


---Nam 



************************

David C. Ullrich 

--
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