[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.

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