[FOM] First-order arithmetical truth
Francis Davey
fjmd1 at yahoo.co.uk
Fri Oct 13 14:54:01 EDT 2006
Stephen Pollard wrote:
>
> The first-order number theoretic truths are exactly the first-order
> sentences in the language of arithmetic that follow from the axioms
> of Peano Arithmetic supplemented by the following version of the
> least number principle: "Among any numbers there is always a
> least." (This principle is not firstorderizable; but that doesn't
> make it unintelligible.)
>
I'm not sure that really answers my problem (though I would be
interested to know if I am right about that). The LNP sounds like
something you would need to formalise using a concept of "set", which is
even harder to understand than that of "natural number".
I can convince myself (just about) that I know what a natural number is
-- but I know its easy to convince oneself of things that are not true.
Godel's incompleteness theorem seems to tell me that I can't get a
handle on the set of number theoretic truths.
Now, I know just enough about set theory to know that it is far less
clear what a "set" is than what a "number" is. The large cardinal axioms
of ZF set theory (for example) suggest I don't really know what a set is
at all and the fact that a number of quite plausible formalisations
exist which are not equivalent (NF seems just as plausible really as ZF
for example).
So, I have no idea how I would tell the intended model (of PA) from
another model.
I'm worried by the idea that one can "see" that the Godel sentence is
true. One obviously can't inside the model, but to do so outside looking
in requires a more elaborate logical apparatus, which has its own problems.
Francis
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