[FOM] Solution to Buckner?

[Harvey Friedman]

Dear FOM,

I will post a clarification of my posting #167 in light of Solovay's 
posting shortly, where I make it clear what versions of predicate 
calculus are used, etc.

**************************************

Reply to Buckner 6:05PM  5/8/03.

>
>Thanks for the note.  But I am a little confused about what constitutes PA,
>PA' and ACA0.
>
>...

>I always understood that PA not only excluded the Axiom of Infinity, but
>included its negation.  This suggests it is a natural analogue of "NL" - the
>system embedded in natural language.
>
>But do PA' and ACA0 therefore include the Axiom of Infinity?  If they do, I
>can't see this from the definitions you give.  If not, a cursory inspection
>of Prof. Simpson's book (Chapter 1 on website) suggests that ACA0 supports
>the concept of real numbers, how is this consistent with not including the
>Axiom?

PA does not have the Axiom of Infinity, since PA doesn't directly 
talk about sets at all. PA talks only about natural numbers, 0, 
successor, addition, and multiplication. Any kind of "negation of the 
axiom of infinity" that is implicit in PA comes immediately from the 
fact that in PA one has induction.

PA' also has no axiom of infinity. However, some predicates on the 
natural numbers are introduced by definition as I indicated. But 
these predicates are NOT used as objects.

So PA and PA' are almost certainly entirely acceptable to you. I 
assume that you accept induction, even if the induction hypothesis 
mentions all natural numbers, but does not use any infinite sets.

On the other hand, the main point is that ACA0 does include the Axiom 
of Infinity, blatantly. So the main point is that ACA0 is 
UNACCEPTABLE to you.

Now here is the CRUCIAL point.

Although ACA0 is unacceptable, PA' is acceptable.

BUT, FORTUNATELY, ACA0 IS

$$$$$$$$$$$$$$$$$INTERPRETABLE$$$$$$$$$$$$$$$$

IN PA'.

(I use dollar signs to make sure that I have your attention).

I did NOT say that ACA0 is

$$$$$$$$$$$$$$$$PROVABLE$$$$$$$$$$$$$$$$

in PA'.

The latter is obviously false.

But the former claim about

$$$$$$$$INTERPRETABILITY$$$$$$$$

is enough to give a kind of justification, indirectly, of ACA0.

To make this point clear - i.e., the power of interpretability in 
this discussion, let me tell a little story.

George is a Euclidean geometer who revels in thinking about points in 
the plane, lines in the plane, Euclidean diagrams, etc.

Fred is a hard core mathematician who thinks only about real numbers. 
Fred doesn't think that the Euclidean plane is meaningful, Fred 
doesn't think that points in the Euclidean plane are meaningful, Fred 
doesn't think that lines are meaningful, etc.

Now Fred wishes to make some sort of good sense out of what George is doing.

Fred is never ever going to accept that what George is doing is 
directly meaningful.

However, Fred has a way of making some sort of sense out of what 
George is talking about, and is interested in doing so for several 
reasons. Among these reasons are that Fred wants to make sure that 
George isn't going to run into a contradiction. Fred wants to make 
sure that George, even if he is misguided, is at least not going to 
be trapped into a logical disaster. (Why? Because Fred cares about 
George - George is Fred's best friend. They went to school together, 
belong to the same political party, like the same TV shows, etc.)

So Fred has figured out that he can "interpret" or "translate" 
George's thinking within Fred's thinking.

E.g., whenever George thinks "point" Fred thinks "pair of real 
numbers". Whenever George thinks "x,y,z are colinear points" Fred 
thinks "the appropriate algebraic equations".

Fred finds that every assertion that George makes becomes something 
Fred knows how to prove AFTER Fred translates it into Fred's terms.

One day Fred asked George to list all of his axioms that he uses 
about points, lines, etc.

George gave Fred the list. Then Fred translated every one of George's 
axioms into Fred's terms, And then Fred proceeded, successfully, to 
prove every one of these translated axioms.

Now Fred was satisfied that George is thinking straight and won't 
ever come up with a contradiction. Why? Because if George came up 
with a contradiction then Fred could translate that argument into his 
own personal contradiction in his own (Fred's) terms. But Fred 
believes in what Fred is talking about.

So now Fred can rest easy, knowing that his best friend makes 
perfectly good sense under Fred's translation or interpretation, even 
if George cannot see that the way he is talking about the world is 
misguided.

There is even more interesting parts to this story, but I hope this 
is enough for you to work with.

Now how does this story connect up with the situation at hand?

GEORGE uses ACA0, that misguided thing with infinite sets, etc.

FRED uses PA', something even Buckner accepts.

We know that ACA0 has an interpretation, or translation, into PA'. 
This is a proved result from f.o.m.

So everybody is happy. Or at least, this is probably as happy as 
f.o.m. is going to make Buckner. Can we make Buckner any happier?

>...
>
>(i)  some sets are infinite
>(ii) without infinite sets, some real numbers would not exist
>
>Common sense, and the evidence of popular works on the subject suggests that
>(ii) is correct.

There are extremely interesting axiomatizations of the system of real 
numbers that have no sets at all in them. This is a fairly long story.

>For example it's commonly said that the Axiom of Infinity
>or some version thereof is required before we can get to work on real
>numbers.

This is true only if we are doing what is called the set theoretic 
interpretation of the real numbers. Of course, this is the standard 
way of doing things in f.o.m. But there are other approaches in 
f.o.m. that are very interesting. One can take real numbers as 
primitive and a lot of important things result from this 
investigation. However, these investigations are not aimed at 
satisfying your concerns. They are aimed at other kinds of goals.

>...Anyway, common sense suggests that, if there are
>infinitely many objects of a certain kind, there must be a set of them.

You do NOT want to say this! There are infinitely many natural 
numbers, as you have accepted. But you have not accepted the idea 
that there is a set of all natural numbers.

>
>So, the evidence suggests that real analysis requires the Axiom of Infinity,
>or some variant, and that therefore NL (the set of asusmptions embedded in
>ordinary language) does not support real analysis.
>
>A further point: in your posting you say an option would be to "Rip apart
>and reassemble real analysis in order to do it in PA".  But if the common
>sense view is true, and if PA does not embed the Axiom of Infinity, this
>wold surely be impossible.  Are you seriously saying that it is not
>conclusively established that PA cannot support real analysis?

There is a sense that, in many situations in real analysis, one can 
systematically approximate the real numbers in question close enough 
by rationals to avoid the real numbers in favor of rational numbers. 
This will not work all of the time, and when it works, it can be very 
messy.

The interpretation of ACA0 into PA' avoids this kind of complication. 
But if one wants to get very refined information about estimates and 
the like, then there may be no choice but to roll up your sleeves and 
"rip real analysis apart", etc.