# [FOM] First-order arithmetical truth

V.Sazonov@csc.liv.ac.uk V.Sazonov at csc.liv.ac.uk
Sat Oct 21 16:42:07 EDT 2006

```Quoting Arnon Avron <aa at tau.ac.il> Sat, 21 Oct 2006:

>> Does anybody here have this ability?
[to distinguish the intended model of PA from another model - VS]

Arnon:

> I certainly do.

Dear Arnon,

In fact I asked about clear criteria how to check whether anybody has
this ability. This could be a precise definition of the standard model,
if possible at all. But I see that you rather present your subjective
beliefs and appeals to beliefs of others.

So does everybody else here (well, almost everybody).
> What is more: so does almost every person from some age onwards.
> Thus everybody who uses the notions of "ancestor" and "descendant"
> distinguishes the intended model of PA from all the others.

Does "almost every person" know and understand what is induction axiom
or what are non-standard numbers? Does this person "from the street"
have the same idea and "picture" of natural numbers as professional
mathematicians? Even the majority of professionals have no idea on
nonstandard numbers and understand induction axiom not in the same way
as logicians, knowing nothing about subtle distinctions. Definitely,
all of us have somewhat different pictures and intuitions on N in our
minds.

Take
> my word: when God promises Avraham in the old testiomony
> to give this or that to Avraham's descendants, he meant only
> his standard ones, and no reader has ever expected him to keep
> the promise to any nonstandard descendant (especially that God
> seems to find difficulties in keeping his promises even to
> the standard descendants...).

In fact, this context assumes no mathematics. All of this is rather on
feasible numbers understood naively (see my last posting to FOM). Even
more than that, it is nowhere assumed here that there will not be the
last descendant. The idea of infinity is sufficiently non-trivial and
without mathematisation (formalisation) it can be discussed only on
non-serious, speculative level, something analogous to discussions on
"how many devils can fit at the end of the needle". That is, the naive
concept of N and mathematical one are quite different.

>
> Do you pretend not to understand the notion of a descendant,

On the naive "feasible" level I understand, as well as on the formal
mathematical level. But you seemingly intend to mix these two levels.

and
> can you imagine yourself having nonstandard descendants?
> (what can be the possible meaning of this??)

The concept of nonstandard number is highly theoretical one based on
the concept of formal theory and using model theory, typically in the
framework of ZFC. Not everything in mathematics has a direct
application in the real world. So, I do not consider the above question
as meaningful as well. And no contradiction in my views, as you
seemingly wanted to demonstrate.

>
>  But the whole question is wrong. You emphasize in your question
> "models of PA". But what is so special about PA? The only
> interest in PA is due to the fact that it codifiersd a part of what
> we know to be true about the natural numbers ("the intended model").

The only non-speculative and healthy way to deal with mathematical
concepts such as natural numbers is via a formalisation. PA is one of
the standard contemporary formalisations of numbers. On a preliminary
level we can work in a semiformal way which is by no means
self-sufficient. Even in the older times when PA was not explicitly
formulated the mathematical proofs presented on numbers were
sufficiently formal (and, a posteriori, formalisable in PA or ZFC).
Anything mystical and non-formalisable (this or other way) was excluded
from mathematical proofs by the future generations of mathematicians.
So, implicitly, mathematics was always based on the formal approach.
The current discussion on the "intended" model of PA is also outside of
mathematics.

> Conceptually and historically, the structure of the natural
> numbers precedes PA,

There was no such a fixed structure. People had only SOME APPROXIMATE
idea on N, but sufficiently clear rules (studied just by training with
the help of already well-trained mathematicians) how to operate with
this imaginary N. It is these rigid rules which leads to the illusion
of something absolutely solid and unique.

and nobody would have dreamt to introduce
> PA otherwise

See above. The process of formalisation can take ages. This is the
lesson we learnt from the previous century essentially after Hilbert
who urged to formalise mathematics. N existed only as fragmentorary
formalised what was considered as highly non-satisfactory. Despite
being old, the question on what is N remained OPEN (not only for people
"from the street" but for the most advanced mathematicians), say, in
the form "Was sind und was sollen die Zahlen?" (Let me stress on
"sollen" = "must" or "should" - something what we can understand as our
will - axiomatisation and rules imposed by us on N.) The solution we
have now is an appropriate, although non-unique (first-order)
formalisation, like PA or like the (relatively) unique standard model
in the framework of ZFC (following to the idea of Dedeking). But,
nevertheless, both PA and ZFC are first order theories and cannot fix N
uniquely so that it is in fact unclear what exactly is this "standard"
N. And this is absolutely unnecessary for doing mathematics. We have
intuition, imagination, axioms and proof rules - that is more than
enough. No "super-fictions" more than simple and modest intuition which
does not pretend on anything absolute and eternal are necessary.

(I wonder whether even you really take PA+CON(PA)
> and PA+not-CON(PA) to be of the same significance and interest,
> even though I dont see what possible reason you might have to prefer
> one over the other).

As I wrote recently CON(PA) does not express adequately the informal
consistency statement (which assumes considering only proofs of
feasible length), nevertheless it is a natural extrapolation. So there
is strong temptation to accept it according to the general
methodological approach to try to accept extrapolations, likewise there
is a temptation to accept the induction axiom which is also an
extrapolation from some finite pictures. (Let me recall some postings
of Harvey Friedman on ZFC as an extrapolation of simple considerations
on finite sets.)

Now PA, being a first-order theory, has more
> than one model: the intended one,

You have given no satisfactory explanation what it is. You like to
think that it exists. This is your right. But your beliefs are outside
of "objective" mathematics.

and many others. So what?
> So do many other theories, like Q (which is weaker) and PA+CON(PA)
> (which is stronger). Why should nonstandard models of PA
> be anymore interesting or better candidates to be "alternatives"
> to the real natural numbers than nonstandard models of Q
> or of PA+CON(PA)?

Again, the point is that objectively there is no such thing as the
standard model. Moreover, ALL (imaginary) models of PA are non-standard
in a sense. (See my last posting to FOM.)

Well, perhaps you answer that it is the
> most natural first-order approximation of Dedkind-Peano
> second-order  categorical axiomatization.

The latter (only RELATIVE!) categoricity argument is formalised only in
first order ZFC (or the like). In this sense there is nothing second
order here. I am puzzled why do you ignore this crucial point.

Even so, attaching
> great significance to nonstandard models of PA

Just because there are no others at all!

just means
> attaching unjustified philosophical importance to first-order
> axiomatizations, despite the fact that first-order languages
> are simply too weak for characterizing interesting structures.

Unjustifiable??? There is no known *mathematical* way at all to
absolutely characterise these (imaginary and already by this reason
vague, non-standard) structures. Formal approach like PA (i.e. first
order or even second order theory which is a hidden first order) is the
only one which makes mathematics to be mathematics. Any informal
speculations, whichever fruitful they are, MUST be formalised.
(Standard model in the ABSOLUTE sense you mean is not formalisable!)
Until this moment it is not TRUE mathematics. Recall, for example, the
history of Analysis. Why epsilon-delta approach, Dedekind cuts, etc.
were invented at all? Even before that happened Analysis was quite
successful and even applicable. Physicists using Analysis probably
would not bother themselves too much. Why mathematicians were
non-satisfied? Only needed to "codify" their work, as you say, or
because this is vital for mathematics, because it is not a "true" and
"healthy" mathematics otherwise, because the question was in fact on
the very nature of mathematics?

Needless to say, the Natural numbers
> can be categorically characterized in this language.
[of "ancestral logic" - VS]
Of course,
> unlike first-order logic, there is no r.e. axiomatization
> of "ancestral logic". This is indeed unfortunate, but nobody promises
> us that everything should be according to our wishes...

This is indeed unfortunate that there is no standard model, but nobody
promises us that everything should be according to our wishes... -:)

That is, you witness that this "ancestral logic" is not a full-fledged
formalisation. I guess it is interesting and instructive logical
language, but it should be put again in the framework of ZFC to make
the categoricity argument mathematical. So, the same story.

It is clear for me that it is almost useless trying to convince
believers. But, from the point of view of methodology of science, I
still consider important to distinguish beliefs from real mathematics.
Believers can have their own impulses and ways to arrive to rigorous
and valuable mathematical considerations, but, Arnon, unfortunately,
the usual side effect of subjective beliefs is dogmatism. And this is
harmful for mathematics and science in general. Recall, for example,
beliefs

widespread belief which stopped Gauss even from publishing(!) his work
on non-Euclidean Geometry),

or on absolute time (despite it was already known that the speed of the
light is finite; so there was a real but unused chance for Special
Relativity to appear much early before Einstein, let even in the status
of imaginary space-time geometry awaiting experimental confirmation; it
is this belief which Einstein was needed to overcome by asking the
crucial scientific question "WHAT DOES IT MEAN OBJECTIVELY?").

Now we have subjective beliefs in absolute mathematical truth and
uniquely existing (absolute) standard model of PA...

Arnon, I understand your argumentation quite well, however I cannot
accept your beliefs as having no real grounds.