[FOM] First-order arithmetical truth

V.Sazonov@csc.liv.ac.uk V.Sazonov at csc.liv.ac.uk
Tue Oct 24 18:30:13 EDT 2006

Quoting "Timothy Y. Chow" <tchow at alum.mit.edu> Sun, 22 Oct 2006:

> It is an attempt to keep the discussion on the original track---which was
> my attempt to get Francis Davey to see Arnon Avron's point.  Notice that
> *you* are the one interrupting the discussion in order to inject your own
> agenda, so I do not feel obligated to answer *your* questions, as opposed
> to any questions Francis Davey might have about what I am trying to say.

If asking question "what does it mean" (the real goal of that my post) 
is interruption of a well-going discussion with good mutual 
understanding and all concepts well defined then I am really guilty. 
But I think my question was just complementing or even sharpening the 
question of Francis Davey which also goes, as I understand, in somewhat 
"orthogonal" direction to your way of thought likewise my question.

> But in any case, since you agree with the parallelism between (e.g.)
> "formal string of symbols" and "natural number," perhaps this will be
> helpful to Francis Davey too.  Any skepticism about the naive (resp.
> formal) concept of the integers carries over directly into skepticism
> about the naive (resp. formal) concept of a formal system.

Yes, exactly.

Anyone who
> thinks that formal systems are crystal clear while integers are vague and
> suspect

You could recall my reply where I wrote that both are non crystal 
clear, or, more precisely, all FOUR versions (two more arising due to 
adding the adjective "naive") are not crystal clear, but in different 
ways. Two are too naive and "feasible", and the other two are too 
abstract, although obeying some formal rules, and imaginary. Is not our 
imagination something inherently vague?

Then you could notice that my suggestion was to concentrate on naive 
formal systems (based also on the naive idea of feasibility) as the 
base for formalising mathematics? Naive formal systems are something 
from our real life and (for example, computer) practice. They are quite 
rigid and reliable and even can be represented physically by computer 
systems. The main point is that they are REAL unlike ABSTRACT 
mathematical numbers and abstract (meta)mathematical formal systems 
(quite similar by the nature to mathematical numbers).

Again, naive formal systems are REAL and RELIABLE. THIS IS THE POINT. 
This makes them a REAL and RELIABLE foundation of mathematical rigour 
(despite they are naive, not crystal clear, and involve the vague idea 
of feasibility).

---and who tries to argue that the existence of nonstandard models
> lends support to that idea---is simply suffering from a blindspot that
> prevents him from seeing that his skeptical arguments apply equally to
> formal systems.

I definitely do not suffer from such a blindspot. Moreover, I make 
clear distinction between naive and abstract (meta)mathematical 
concepts discussed and easily follow your line of thought, whereas you 
seemingly do not notice my points.

You are right. Abstract (meta)mathematical formal systems can have 
nonstandard formulas and derivations. Exactly the same as for abstract 
mathematical numbers.

> Here's another way to put it, taken from an email reply I just sent to
> another FOM subscriber (off-list).  The other subscriber wrote:
>> It's easy to formulate a formal system as string rules, like one does
>> in Theory of Formal Languages courses in c.s.
> [...]
>> there seems no "standard" way to write a standard formal system on a
>> sheet of paper that isolates exactly the natural numbers w/o getting
>> non-standard numbers as well.

In place of that person I would replace here "standard formal system" 
by "NAIVE, CONCRETE formal system" because it is continued with "on a 
sheet of paper". This is not about an abstract (meta)mathematical 
concept. This is from real human activity of writing symbols, 
symbolically presented rules and practical ability to follow these 
rules. No theory explaining this activity is needed. People just are 
able to do this in practice. That is why this activity is both NAIVE 

But why do you ignore the words "sheet of paper"? They are crucial here!

> To this I replied:
>> Let N be the standard natural numbers,

I guess you would say that you know what it means. If you assume what 
is called \omega in ZFC then you are out of the topic of the 
discussion. The point is that some people (and I guess you too) think 
that the concept of standard natural numbers is meaningful in an 
absolute way, independently on any axiomatic formal system like ZFC 
what I do not accept. So, this your sentence has for me an unclear 

and let PA be the standard
>> first-order system of Peano arithmetic.

Same problem as above. What does it mean? Something absolute or only 
relative to ZFC? Or may be what I call NAIVE (i.e. from our real life)?

>> If there is no formal system in the language of arithmetic that isolates
>> exactly N, then there is no formal system in the language of syntax that
>> isolates exactly PA.

Exactly so, provided some clarification should be made. First, I would 
add 'abstract (meta)mathematical' between your 'exactly' and 'N' (resp. 
'PA'). Second, "formal system in the language of syntax" should be 
understood as "formal theory of (abstract) finite strings of symbols in 
fixed alphabet". Otherwise it sounds a bit ambiguous.

If the former state of affairs leads us to worry
>> about the determinacy of N, then the latter state of affairs should lead
>> us to worry about the determinacy of PA.

Exactly so, but with the above comments.

Given any text that
>> purportedly describes PA in terms of "string rules," as you suggest,

That person suggested something on a sheet of paper, that is something 
from our real life (a system of rules of PA, even not the text 
describing/explaining this system of rules; there is no reason to 
overcomplicate things in the simple and naive context of a sheet of 
paper). But you miss this crucial point by identifying the real life 
with the abstract world of imaginary mathematical objects and making 
the following conclusion:

>> can perversely interpret that text as referring to *nonstandard* string
>> rules, yielding something very different from the "intended" PA.

NO!!! Just symbolic rules written on a sheet of paper are assumed - 
quite real physical objects, nothing abstract or nonstandard.

> Tim

I hope you see that I can follow your line of thought in all essential 
details (with making some necessary clarifications and distinctions), 
but I do not see that you can follow the lines of thought either by me 
or Francis Davey or the other person you cite above. I mean just 
"follow". Of course you could disagree with some steps.


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