[FOM] First-order arithmetical truth

[V.Sazonov@csc.liv.ac.uk]

Dear Arnon,

Unfortunately, I should note that there is almost complete 
misunderstanding from your part of what I am trying to say.

At least about one thing we could agree that no (first-order) formal 
system can define uniquely the concept of natural numbers.

I assert that the only way to demonstrate that what we are doing is 
indeed a mathematics is to present this formally (rigorously) either as 
a derivation of a theorem or as an explicit definition of a concept in 
the framework of a formal system or as an implicit definition of a 
concept (model) by a formal axiomatic system and desirably to prove a 
metatheorem (in some other formal system) that this axiomatic system is 
consistent. Sometimes we just postulate a new axiom like in the case of 
CON(PA) (just CON(PA) itself or, for example, epsilon_0-induction 
allowing to derive CON(PA)). Whichever way will you argument that some 
non-first-order non-r.e. logic (language + semantics only) is "nice", 
you eventually will present this in a context of a formal system (such 
as ZFC) and deduce formally some theorem on its "niceness" (say, that 
it characterises natural numbers uniquely). Whatever will you do will 
be eventually presented formally (rigorously) once you pose yourself as 
a mathematician. And I am sure that you behave exactly in this way, 
whatever you assert in your informal comments.

IN THIS SENSE, any your considerations on uniqueness of N are done 
eventually RELATIVE to a (first order) formal system (typically, ZFC) 
and therefore are NOT ABSOLUTE. IN THIS SENSE you DO NOT and CAN NOT 
DEFINE an ABSOLUTELY UNIQUE/INTENDED/STANDARD MODEL OF NATURAL NUMBERS.

I repeat, IN THIS SENSE!

I hope you should confirm the last paragraph and we can find the point 
of agreement thereby PUTTING ASIDE (or ISOLATING) the point of 
disagreement which still remains existing and essential.

Best wishes,

Vladimir


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