u/LorenzoGB

Can Aristotle’s notions of potency and act be applied to mathematics? I ask because I think it can for consider the following. Suppose we had a circle. Then that circle is potentially a line because I can cut the circle at some point on the circumference and deform it into a line. Suppose we had an infinite line and that infinite line had negative infinity and positive infinity as end points. Then that infinite line is potentially a circle because I can glue the end points together to form a circle.

According to the Philosopher (Aristotle): Again, to be or being signifies that some of the things mentioned are potentially and others actually. For in the case of the terms mentioned we predicate being both of what is said to be potentially and of what is said to be actually. And similarly we say both of one who is capable of using scientific knowledge and of one who is actually using it, that he knows. And we say that that is at rest which is already so or capable of being so. And this also applies in the case of substances. For we say that Mercury is in the stone and half of the line is in the line. And we call that grain which is not yet ripe (Metaphysics 5:7).

With this being said let us define has as in X has Y as follows:

1. For all X, X has X

2. For all X and Y, if X has Y and Y has X then Y equals X

3. For all X, Y, and Z, if X has Y and Y has Z, then X has Z.

Let us also add the following too:

1. For all X and Y, X has Y potentially if and only if X has Y and Y is not in act.

2. For all X and Y, X has Y actually if and only if X has Y and Y is in act.

3. For all X and Y, if X has Y then Y is in act or Y is not in act.

Now let us consider the following: The extremities of a line are points. This can be written as follows: For all X, if X is a line then X has the property of having points as extremities. For all X, if X has the property of having points as extremities, then the property of having points as extremities is either in act or not in act. Therefore, if X is a line, then the property of having points as extremities is either in act or not in act.

The difference between a well-ordered set and a well-orderable set is the following:

1. For all X1, X1 is a well-orderable set if and only if X1 is a set and there exists X2 such that X2 well-orders X1.

2. For all X1, X1 is a well-ordered set if and only if X1 is a set and there exists X2 such that X1 has X2 and X2 well-orders X1.

What is the definition of a well-ordered set? I ask because I thought the definition of a well-ordered set is the following: For all X1, X1 is a well-ordered set if and only if X1 is a set and there exists X2 such that X2 well-orders X1.

1. For all X1, if X1 is a set then there exists X2 such that X2 arranges X1 in such a way that every subset of X1 has a first and a last member.

2. For all X1 and X2, if X2 arranges X1 in such a way that every subset of X1 has a first and a last member, then all subsets of X1 have a first and a last member.

3. For all X1, if all subsets of X1 have a first and a last member then X1 is a complete lattice.

4. For all X1, if X1 is a set then X1 is a complete lattice.

Is the following valid and sound: For all X1, if X1 is a causal series then X1 is a set. For all X1, if X1 is a set then there exists X2 such that X2 well-orders X1. For all X1 and X2, if X2 well-orders X1, then X1 is well-ordered. For all X1, if X1 is well-ordered then X1 satisfies the greatest lower bound property. For all X1, if X1 satisfies the greatest lower bound property, then X1 satisfies the least upper bound property. Therefore, for all X1, if X1 is a causal series, then X1 satisfies the least upper bound property.

The Well-Ordering Theorem can be written as follows: For all X1, if X1 is a set then there exists X2 such that X2 arranges X1 in such a way that every non-empty subset of X1 has a first member.

With this being said, would this be equivalent to the Well-Ordering Theorem: If X1 is a set then there exists X2 such that X2 arranges X1 in such a way that every non-empty subset of X1 has a last member.

Is the Compactness Theorem equivalent to Proposition 2? I ask because of the following:

1. For all X1, if X1 is a theory in FOL and for all X2, if X2 is a finite subset of X1 then X2 is consistent, then X1 is consistent.

2. Therefore, for all X1 there exists X2 such that if X1 is a theory in FOL and X2 is not finite or X1 is a theory in FOL and X2 is not a subset of X1 or X1 is a theory in FOL and X2 is consistent, then X1 is consistent.

3. Therefore, for all X1 there exists X2 such that if X1 is a theory in FOL and X2 is not finite then X1 is consistent.

4. Therefore, for all X1 there exists X2 such that if X1 is a theory in FOL and X2 is not a subset of X1 then X1 is consistent.

5. Therefore, for all X1 there exists X2 such that if X1 is a theory in FOL and X2 is consistent, then X1 is consistent.

With regard to generalized quantifiers, the following deduction is invalid: There are exactly three cats. All cats are mammal. Therefore, there are exactly three mammals. This is because the generalized quantifier exactly three is non-monotonic. Yet suppose I interpret the quantifier exactly three as the following: There exist X1, X2, and X3 such that X1, X2, and X3 are cats and none of them are identical to each other. Then the reasoning is valid because exactly three mammals can be interpreted as there exist X1, X2, and X3 such that X1, X2, and X3 are mammals and none of them are identical to each other. So can non-monotonic generalized quantifiers be interpreted as strings of existential quantifiers of whatever cardinality?