Interesting question.
I take it to be something like (a).
Consider: (Ay)(Fyc -> Fyy): For any y, if y likes John, then y likes himself. 'likes' is 'F', John is a constant -- c. Informally, this statement says: 'anyone who likes John, also likes himself.' There's no implication that ALL people...
But there is a question about such 'infinite' verifications. We can't complete an infinite process and so there's a question whether we can talk about verification here.
For instance, would you argue that the continuum hypothesis is either true or false because you can just run through all the...
I'm not sure about some of the assertions that have been made here.
The Godel formula is not non-constructive. It's mathematically artificial, but it can be written down.
There are arithmetically 'natural' statements which are not provable in PA. Such as the Paris-Harrington theorem...
He argues that 'u is a member of u' is not a proposition and then concludes that u is not a set. In those cases where 'u is a member of v' is not a proposition, u and v are not sets. Here is a statement whose variables do not include sets. And he says that his first axiom tells us when u is not...
But almost any theory -- physical or mathematical -- could be said to be composed of propositions in this sense. So, while I think I see what you mean, this boils down to a fact about the nature of theories.
Yes, but I don't think it's sneaky or hidden or restricted to set theory. It's a...
I think i see where OP is coming from but, to be honest, I'm having a hard time of making sense of Schuller's discussion.
I do agree with others that the questions decidability is better understood as something to be totally distinguished from the questions of truth-value. We have no way, at...
I think that a commitment to first order logic is fairly small. I agree that the *model theory* of first order logic can be very complicated, and the model theory of certain axiom systems of first order logic can require some heavy-duty set-theory.
Second order logic's arguably *is* connected...
This is a good question. However, it's been found that you can get the effect of defining binary relations in systems that go just a little beyond first order logic, but which do not require sets.
More precisely (though from memory), in a system with plural quantification, and which contains...
If A and B have different interpretations of 'x is a member of y' then A and B will (usually) disagree over which entity is the empty set.
However, if all you want from a mathematical definition is to characterise a role in a structure -- so that {} is just that entity which contain no...
There are no constructive definitions of 'and', 'all', '=' either. But this doesn't mean that these are to be thought of as merely formal marks on paper. They are meant to be meaningful.
The empty set is defined as that set which contains no members. I'm not sure where you're going with your...
The empty set is defined as the set which has no members. So 'x is a member of y' is used to define the empty set. So without a definition of 'x is a member of y' there is no definition of the empty set.
ZFC at face value contains the predicate 'x is a member of y'. And this is a basic...
I can think of two things that may be worrying you.
1.
'Different concrete theories seem to prove different subsets.'
Yes -- but why is this a problem? Different theories and different subsets go hand in hand.
Begin with a first order axiomatisation of arithmetic and let A be the smallest...
A theory in mathematical logic is often understood to be a set of sentences closed under logical implication.
So understood, it doesn't matter whether the sentence was proved from axioms or using rules of inference. You can formalise the same theory either way. So the details about whether...
They are different concepts -- there are equivalence relations that aren't identity -- and you need to tell me which one you want to talk about. I'm now simply unsure whether you want to talk about identity or whether you want to talk about equivalence relations. I believe: (a) it's trivial that...
Subset, Union, intersection, complement are usually defined notions in set theory. A U B is defined as the set which contains precisely the objects which are members of A or (inclusive or!) members of B. 'is a member of' is usually a primitive of the theory.
One way or another, a reasonable...