Foundations of arithmetic

I've messed with this idea before (I found some examples by searching my blog for 'arithmetic', and was a little bit surprised at how many times ...).

I think I have a clearer picture of it, now, though:

The incompleteness and inconsistency proofs show that any attempt to formalise arithmetic will generate open question paradoxes.

As I've tried to suggest (clumsily) in earlier posts, a recursive approach to the OQ trap is also possible for arithmetic. This is what needs a clearer statement.

It cannot be a consequence of Gödel's paradox that arithmetic is not possible (that its inconsistency is catastrophic per Duns Scotus) because the construction of the paradox itself depends upon the reliability of arithmetical computations (e.g. factorisation) and the truth of arithmetical theorems (e.g. about the unique outcomes of factorisation). So Gödel's argument is as much a statement of the possibility of arithmetic as it is a demonstration of its inconsistency.

In other words, because all formal proofs can be represented within arithmetic (using an equivalent of

Gödel's coding method), the statement "There is some true theorem in arithmetic" (the equivalent of "It is possible to do arithmetic") is an isomorph of "It is possible to formally prove something". Since this is a consequence of "It is possible to say something", we have the possibility of valid arithmetic recursively from the possibility of being able to talk.

We can only talk if we can agree about the consequences of what we say. We can only explicitly show some behaviour to be linguistic if we can give an account of this agreement - in the form of 'A implies B' types of statement. In order to explicitly pursue certain very general (unbounded) enquiries, we need to have a language which can make these kinds of statements about its own components.

The consequences of 'A implies B' statements must themselves include much of first order logic if they are not to be inconsistent (and they would not be meaningful otherwise because their meaning consequences would be catastrophically inclusive).

In this kind of language, then, we have the possibility of logic; and so, the possibility of arithmetic. And we also, necessarily, have open question paradoxes, since we cannot constructively prove that the language itself makes sense - although we cannot deny this within the language.

Since Gödel's own coding method depends upon the 'fundamental theorem of arithmetic' - that all numbers have unique prime factors - his particular generation counts as a kind of meta-proof of that theorem:

First order logic is consistent because it is possible for us to talk to one another;

Arithmetic is possible because first order logic is consistent;

Arithmetic would only be possible if the fundamental theorem were true.

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It may also have another consequence which I've mentioned before, I think:

If factoring large numbers was 'tractable', in a way which brought Gödelised logical proofs within the scope of practical arithmetic, then practical arithmetic would include the computational inconsistencies that he shows must be a consequence of arithmetic being able to demonstrate its own completeness.

There may be a relationship - it seems likely to me that there is a relationship? - between a language being 'open', in the sense of not placing explicit constraints on meta-linguistic statements, and it's being usable for the pursuit of certain (philosophical? fundamental?) enquiries. There must be no stipulated boundary on 'why?' questions. We tacitly explore kinds of boundaries as we explore the limits of intelligibility ("why do 'why' questions make sense?" (!)), but these boundaries cannot be explicitly delimited within the language.

Arithmetical constructions permit, in principle, a kind of indefinite complexity. We can point to numbers - e.g. Borel's number - which we say must 'exist', in the sense that their composition does not breach any arithmetical rules (however practically inconvenient they might be).

Our practical experience of the tacit limits of meta-linguistic enquiry has an ambiguous quality - it is hard to discriminate between complexity and inconsistency. We don't know whether we are looking at a flat contradiction, or whether we have failed to see a (possibly convoluted) set of meaning commitments which would render a puzzling statement intelligible. We cannot (Kripke) rule out the possibility that a speaker is following a set of rules that we do not yet understand.

The potential complexity of arithmetic (its 'open-ness') may depend on the intractability of factorisation. If we could factor indefinitely large numbers in some straightforward way, we could extract all their 'rules' and demonstrate that we had done so.  Questions of the form 'this number has this property' would always be tractable.

Gödel's method may have the corollary that this will never be possible.