FOM: Categorical Foundations
Colin Mclarty
cxm7 at po.cwru.edu
Sun Jan 18 10:59:37 EST 1998
Reply to message from friedman at math.ohio-state.edu of Sat, 17 Jan
>Remember, you are talking to
>elementary undergraduates who may eventually become statisticians,
>physicists, business leaders, philosophers, lawyers, etcetera.
Absolutely. And so they will not raise objections based on
any mismatch between what I say and what they expect from prior
exposure to ZF.
As at OSU, we categorists have to choose one of many
possible syllabi. Here I have chosen a conservative one, closely
approximating yours so as to give the clearest proof that the
categorical approach is logically and pedagogically possible.
I have in fact published an article with a related sketch in
in "Numbers can be just what they have to" in NOUS 27 (1993) 487-498.
>
>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?
Subsets of a given set A are equal if and only if they have
the same elements. There is a set of natural numbers. Only
sets have elements. Every object is either a set or a function.
An element of a set A is a function 1-->A.
>2. The set N of all natural numbers exists. Analog?
There is a set of natural numbers--I included this above.
>3. The successor of any natural number is a natural number. Only natural
>numbers have a successors. Analog?
There is a successor function from natural numbers to themselves.
>3. 0 is a natural number which is not the successor of any natural number.
>Analog?
The same, verbatim
>4. Any two natural numbers with the same successor are equal. Analog?
The same, verbatim
>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?
The same, verbatim
>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?
The same, verbatim
>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?
For any condition expressible as above, and for any set A, there
is the subset of all elements of A that obey that condition.
>8. For any set A, there is the set of all subsets of A. Analog?
The same, verbatim
>9. For any two objects x,y, there is the set consisting of exactly x and y,
>written {x,y}. Analog?
The same verbatim
>10. For any set x, there is the set consisting of the elements of the
>elements of x. Analog?
The same, verbatim
>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?
No analogue, we have no use for it.
>12. We prove that the Cartesian product AxB of any two sets A,B, exists.
>Analog?
We stipulate that the Cartesian product AxB of any two sets
A,B exists.
>13. We prove that there are unique functions on NxN into N obeying the
>usual conditions for addition and multiplication and exponentiation. Analog?
The same, verbatim
All the rest are the same, verbatim. I include them for easy
reference:
>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.
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