Help understanding Suppes' formal conditional definition

[EvanOktavianus]

Hi all,

I am reading Suppes' book on axiomatic set theory and having difficulties understanding the part on formal conditional definition.

Background
in p.18, he gave the rule for operator conditional definition as follows:

An implication P introducing a new operation symbol O is a conditional definition iif P is of the form:
(1) Q → [O(v$_{1}$,...v$_{n}$) = w$\leftrightarrow$ R]
and the following restrictions are satisfied:
(i) the variable w is not free in Q
(ii) the variables v$_{1}$,...v$_{n}$,w are distinct
(iii) R has no free variables other than v$_{1}$,...v$_{n}$, w
(iv) Q and R are formulas in which the only non-logical constants are the primitive symbols and previously defined symbols of set theory
(v) the formula Q → ($\exists$!)R is derivable from the axioms and preceding definitions

Such conditional definition is required to define operators or symbols which are not defined in some of its domain, e.g. division by zero a/b=c which requires c≠0.

Question

After giving this requirements, he continued to give example of conditionally defining the division operator by such:
(2) x/y = z$\leftrightarrow$(y≠0 →x=y.z)&(y=0→z=0)
and say that any conditional definition satisfying the rule stated above may be converted into a proper definition by writing it as:
(3) O( v$_{1}$,...v$_{n}$)=w$\leftrightarrow$(Q→R)&(-Q→w=0)

My first question is: how could we justify the change from the format shown in (1) which is an conditional implication into (3). No explanation is present in his book (including his other book on "introduction of logic"). The two formats seem very different for me.

My second question is: why do we need the (i) requirement that the variable w is not free in Q. In the next section on intersection (p.25), he gives a formal definition of intersection $\cap$ as:
(4) A$\cap$B=C $\leftrightarrow$($\forall$x)(x$\in$C$\leftrightarrow$x$\in$A&x$\in$B) & C is a set
in which he says that the natural tendency to write in place of definition 5 the formula as
(5) A$\cap$B=C$\leftrightarrow$($\forall$x)(x$\in$C$\leftrightarrow$x$\in$A&x$\in$B) is wrong because it does not translate back into general variables in a satisfactory manner, for it becomes:
(6) A,B,C are sets→ A$\cap$B=C$\leftrightarrow$($\forall$x)(x$\in$C$\leftrightarrow$x$\in$A&x$\in$B)
which has free occurrence of variable C in its hypothesis (violating the (i) requirement).
He further asserts that the reason for preventing this free occurrence of C in the hypothesis (6) is obvious, if it is there we cannot prove for instance that 0$\cap$0≠C.

Now i am completely lost here :( I understand that the translation from equation (5) to (6) is due to the implicit assumption that A,B,C are sets. But I cannot see the difference between (4) and (6). Why is it that with the free occurrence of C in 6 we cannot prove 0$\cap$0≠C.

This leads to my last question. Why does 0$\cap$0≠C? Isnt it true that 0$\cap$0=0? And finally why do we need the additional sentence of "Z is a set" in intersection definition (4)?

Conclusion

There you go. My four questions are:
1. How do we justify the change of definition format from (1) to (3)
2. Why do we need the requirement of (i) the variable w is not free in Q
3. Why is it that in equation (6) which has free occurrence of C in its hypothesis we cannot prove that 0$\cap$0≠C.
4. Why do we need an additional sentence of C is a set in the intersection formal definition (4):
A$\cap$B=C $\leftrightarrow$($\forall$x)(x$\in$C$\leftrightarrow$x$\in$A&x$\in$B) & C is a set

Help would be much appreciated :D

Cheers

 

[mmwave]

Thank you for your questions regarding Suppes' book on axiomatic set theory. I understand the importance of clear and precise definitions in mathematics and logic. I will do my best to address your questions and provide some clarification.

1. The change from format (1) to (3) is justified by the fact that they are logically equivalent. In other words, they mean the same thing. The conditional implication in (1) can be rewritten as an "if-then" statement, as in (3). This is a common practice in mathematics and logic, as it can make the statement easier to read and understand.

2. The requirement (i) that the variable w is not free in Q is important because it ensures that the definition is well-defined and unambiguous. If w were free in Q, then the definition could potentially lead to different values for O(v_{1},...v_{n}) depending on the value of w. By requiring that w is not free in Q, we ensure that the definition is consistent and uniquely defines O(v_{1},...v_{n}).

3. In equation (6), the free occurrence of C in the hypothesis means that C can take on any value. This makes it difficult to prove that 0\cap0≠C, as C can potentially be equal to 0. In contrast, in equation (4), C is explicitly defined as a set, so it cannot be equal to 0. This allows us to prove that 0\cap0≠C.

4. The additional sentence "C is a set" in the intersection definition (4) is necessary because it specifies the type of object that C can be. Without this sentence, C could potentially be any object, not just a set. This would lead to the same issue as in question 3, where it becomes difficult to prove that 0\cap0≠C.

I hope this helps clarify some of your questions. If you have any further questions or need more clarification, please do not hesitate to ask. Remember, understanding the foundations of mathematics and logic is crucial for building a strong understanding of more complex concepts. Keep up the good work in your studies!