FOM: Categorical Foundations

[Harvey Friedman]

In an attempt to productively refocus the discussion on what I call
categorical foundations, which includes topos theory as foundations, I
would like to review the beginnings of how one develops real analysis in
the ordinary set theoretic way at the undergraduate level at U.S.
Universities. Normally, the material is covered in two or more courses -
where each course covers other material as well. At least that is the way
it is done here. Actually, here at OSU some of the material is in "discrete
mathematics," some in "foundations of mathematics," and some in "real
analysis." These are all at an elementary undergraduate level.

There is a major issue as to how to handle the natural numbers (nonnegative
integers). One way is to do this purely set theoretically, with epsilon
connected transitive sets. However, the current trend is not to do this
because natural numbers are regarded as being so intellectually primitive.
So it has been customary to use natural numbers as urelements, with the
constant 0 and the operation S for sucessor. However, I don't want to get
too formal in this presentation today, but that is the approach I am going
to take. And in this presentation today, I don't want to exactly delimit
the axioms, as it will be clear what the axioms needed are in this
treatment. Let me just say that we use the induction scheme for all
formulas on the natural numbers.

Now at each stage of the development, I am going to ask for the categorical
analog. By this I don't mean some analog for professionals, together with
some justification that it is completely analogous. By analog here I mean:
what **replaces** what I have written from set theoretic foundations that's
in categorical foundations when lecturing TO THE ACTUAL UNDERGRADUATE
CLASS? In order words, what is it in your "categorical foundations"
lectures that replaces what I have in my "set theoretic foundations"
lectures? We want to compare the two different approaches microscopically
in terms of simplicity, teachability, intellectual coherence, philosophical
clarity, ease of motivation, etcetera. Remember, you are talking to
elementary undergraduates who may eventually become statisticians,
physicists, business leaders, philosophers, lawyers, etcetera.

1. Sets are equal if and only if they have the same elements. Natural
numbers are not sets. Only sets have elements. Every object is either a set
or a natural number. Analog?
2. The set N of all natural numbers exists. Analog?
3. The successor of any natural number is a natural number. Only natural
numbers have a successors. Analog?
3. 0 is a natural number which is not the successor of any natural number.
Analog?
4. Any two natural numbers with the same successor are equal. Analog?
5. For any objects x,y,z: x = x. if x = y then y = x. if x = y and y = z
then x = z. Analog?
6. For any condition expressible with for all, there exists, and, or, not,
if then, iff, membership, being a natural number, 0, and successor, if it
holds of 0, and if whenever it holds at a natural number x, it holds of its
successor S(x), then it holds of all natural numbers. Analog?
7. For any condition expressible as above, and for any set A, there is the
set of all elements of A that obey that condition. Analog?
8. For any set A, there is the set of all subsets of A. Analog?
9. For any two objects x,y, there is the set consisting of exactly x and y,
written {x,y}. Analog?
10. For any set x, there is the set consisting of the elements of the
elements of x. Analog?
11. The ordered pair <x,y> of two objects x,y, is {{x},{x,y}}. We prove
that <x,y> = <z,w> iff x = z & y = w. Analog?
12. We prove that the Cartesian product AxB of any two sets A,B, exists.
Analog?
13. We prove that there are unique functions on NxN into N obeying the
usual conditions for addition and multiplication and exponentiation. Analog?
14. We prove the usual collection of basic equalities of arithmetic
involving addition, multiplication, and exponentiation, on N. Analog?
15. We define the ordering on N in terms of addition. Analog?
16. We prove the usual collection of basic inequalities of arithmetic
involving the usual ordering on N, addition, multiplication, and
exponentiation, on N. Analog?
17. We write 1 for S(0). The concrete integers consist of 0, and the
ordered pairs <n,0>, and the ordered pairs <n,1>. The set of all concrete
integers exists. Analog?
14. We explicitly extend the usual ordering on N, and addition,
multiplication, to the concrete integers. We prove the usual collection of
basic equalities and inequalities of arithmetic involving these notions.
Analog?
15. We define the concept of ordered ring and discrete ordered ring. Analog?
16. We define the concept of Archidmedian ordered ring. We define the
general concept of structure and isomorphism between structures. We prove
that any two Archimedean ordered rings are isomorphic by a unique
isomorphism. Analog?
17. We define the concrete rationals as certain ordered pairs of concrete
integers (reduced form with strictly positive denominators). We define
order, addition, multiplication. We prove the usual equalities and
inequalities. Analog?
18. We define the concept of ordered field and prove that the concrete
rationals form the least ordered field up to isomorphism in an appropriate
sense.
19. We define the concept of Dedekind cut in the concrete rationals and
prove basic facts about these cuts. Analog?
20. We prove the existence of the set of all Dedekind cuts in the concrete
rationals. We call these Dedekind cuts the concrete real numbers. We define
the basic ordering, and addition and multiplication on the set of all real
numbers. Analog?
21. We prove the basic equalities and inequalities involving the basic
ordering, and addition and multiplication on the set of all real numbers.
In particular, we prove the least upper bound principle. Analog?
22. We define the complete ordered fields. We prove that any two complete
ordered fields are uniquely isomorphic. Analog?
23. We define the (finite and infinite) sequences of real numbers as
functions from initial segments of N into R = the set of all real numbers.
We define Cauchy sequences, monotone sequences, bounded sequences, and
prove the fundamental facts such as Cauchy completeness and the existence
of limits of bounded infinite sequences. Analog?
24. We define the continuous functions from R to R. We prove the
intermediate value theorem. We prove the attainment of maxima and minima.
Analog?

REMEMBER: Clarity, simplicity, coherence, teachability, etcetera.