FOM: Another attempt at defining Math
Yoav
yoavy at math.huji.ac.il
Thu Aug 31 13:39:20 EDT 2000
There are several plausible definitions of math which enjoy the property of
being precise, or at least seem to. Examples are:
Math = formal systems,
Math = first order logic FS,
Math = ZFC.
I think these examples can be shown to be equivalent (though in a way which
shows they aren't so precise), implying we really have a good definition (this
we know every time a few seemingly different definitions turn out to be the
same, e.g. the definitions of a compact metric space, or the definitions of a
stable theory).
I'll try to show that the examples above are part of a full circle:
0. Natural numbers (N) - meaning naively the ability to verify facts like
"5+7=12", or even "there exists a prime n" by exhibiting that 7 is, but NOT
facts like "for all n,m n+m=m+n" (no axioms/proofs/theorems yet).
1. Formal systems (FS) - here we take a finite collection of symbols
(=language), define a finite collection of finite strings of these symbols
(=axioms), and a finite collection of decidable ways to make new strings from
old ones (=inference rules). If we want a recursive collection of axioms we can
use another FS to make them. Then we consider the collection of strings we can
make (=theorems) by applying inference rules to axioms a finite number of times
(=proof).
The concept 'any natural number' is needed here in at least one place: a string
is a theorem iff we can make it in any natural number of steps. When we pass to
Go:del codes this means the code of the proof can be any natural number.
Remember that, for now, the answer to a question like "is there any natural
number with property P?" can only be "yes - here it is: n."
2. First order logic formal systems (FOL-FS): this is a special case of FS,
allowing us to talk about an inconsistent FOL-FS (where you can make "\phi" and
"~\phi").
3. Standard set theory (ZFC): a special case of 2, powerful enough for most
mathematicians.
4. Mathematical 'objects' (or sets): In ZFC one can define an 'object' to be a
formula \phi(x) for which "there exists one and only one x s.t. \phi(x)" is a
ZFC-theorem, and say that the 'objects' \phi(x) and \psi(x) are actually the
same 'object' if "for all x \phi(x) iff \psi(x)" is a ZFC-theorem. Note that,
for now, questions like "is \phi(x) an object?" or "are \phi(x) and \psi(x)
equal?" can only be answered "Yes - here is the proof: ..."
5. A consistency assumption: we assume that ZFC is a consistent FOL-FS,
otherwise all objects are equal.
6. \omega (the set of 'natural numbers'): one defines \omega as usual to be the
'object' "x is a subset of every inductive set" (inductive= includes the empty
set and closed under the successor operation). One also has in ZFC the
operations on 'naturals', and using Go:del codes one can find in ZFC 'objects'
for all of the above: 'FS', 'axioms','proofs', 'ZFC', '\omega', etc.
In particular one can 'prove' in ZFC facts like "for any 'natural' number...",
so we are back home to N (are we?!), only our power of expression is better.
This seems to have brought us full circle, so one is tempted to define
Mathematics as "anything lying in the above circle of concepts". Since, like
Oscar Wild, I can resist anything but temptation, this is the definition I
propose. (If you think this is long check out the definition of Scheme... :)
However, Go:del's Incompleteness Theorem makes this definition fuzzier than one
had hoped. Go:del showed the above 'circle' is not really closed, but looks
more like a spiral. This also strengthens my feeling that there is no natural
standing point from which we can build up our definitions, but rather that all
the undefined concepts above are actually needed in defining one another (like
'point', 'line' and 'lies on' define one another through Euclid's postulates).
So, why can't the circle be closed? It's easy to embed N in \omega - look at
'objects' of the form \phi(x) = "x=SSS...S(0)". But is it the case that
\omega=N ? And if so, are all the properties of N provable in ZFC ?
Let's take a Go:del sentence G = "if n is in \omega then n does NOT code
a ZFC-proof of G". By our consistency assumption there's no n in N
which codes a ZFC-proof of G, since such an n can be used to give a
ZFC-proof of ~G.
If Omega=N then G codes a property of the natural numbers not provable
in ZFC, so the circle has some 'gap'. We can use this gap to open the circle
into a spiral rising in consistency strength, by adding axioms on each bend.
On the other hand, ~G might be a ZFC-theorem. In this case we can
define an object M as the minimal n in \omega which does code a ZFC-proof
of G, and show that M is not in N, therefore \omega ~= N. If this phenomenon
continues, i.e. the Go:del sentences can be refuted, we get a spiral that rises
more naturally - each bend the concept 'natural number' will get an (elementary)
extention.
Since I'm no expert in any of the relevant fields I would really like to get
comments (of any kind) on this.
--------------------------------------------------------------------------------
Name: Yoav Yaffe
Occupation: Ph.D. student in mathematics
Institution: Hebrew University in Jerusalem, Israel
Interests: Model theory of differential fields
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