[FOM] Re: Arithmetic-free theory of formal systems?

[Timothy Y. Chow]

On Fri, 21 May 2004, Vladimir Sazonov wrote:
> What do you really want?

I am not sure.  However...

> In my previous posting to which you have replied
> "I find that I disagree with you much less than I originally thought"
> I already suggested *in which informal sense* syntactic entities *may
> be considered* as really simpler. It looks that you did not notice that.
[...]
> A theory of feasible numbers (based on an appropriate weaker version
> of classical logic) which I already mentioned in posting to you gives the
> necessary example of (imaginary) N which is closed under +, but
> does not contain 2^1000.  Did you read this place in my posting?

I did see this, but I didn't understand it.  You gave a sketch, but in 
particular I did not understand what you meant by:

>a formal *theory of feasible numbers*
>based on some modification of classical first order logic

Is this formal theory of feasible numbers published somewhere
(in English)?  It sounds interesting.

> Of course, these two versions of N cannot be represented as sets in
> "the" universe for ZFC. But corresponding axiomatizations are
> consistent.

In other words, you sacrifice the completeness theorem?

> Is it true the sentence forall n log log n < 10 according to
> your platonism (if you accept at all my, as you say,
> "clarification")? Do you have a unique meaning of "and so on"
> according to your platonism? In the context of our exchange
> I do not understand what is your platonism?

It was not my primary purpose to defend platonism, only to argue that the 
inconsistency of PA would not present any *new* challenges to platonism
that aren't already provided by known mathematical results.  You yourself
have argued that formal set theory and formal number theory are parallel
in many ways; the inconsistency of PA would not, as far as I can see now, 
present new philosophical challenges that, say, Russell's paradox did not 
already present.

Tim