Are those statements first order or second order: $$\{X\in 2^{A}:|X|=2\} \\ \{X \subset 2^{A}:|X|=2\} \\ \{X \subset A:|X|=2\}$$ why?
Are $\{X \in 2^{A}:|X|=2\}$, $\{X \subset 2^{A}:|X|=2\}$, $\{X \subset A:|X|=2\}$ first order or second order?
2
$\begingroup$
logic
set-theory
higher-order-logic
-
3Neither is a statement. In the usual formalized set theory, the statement that $W$ is equal to either the first or the second is first-order. In many other settings, the statement that $W$ is equal to the first is first-order, and the statement that $W$ is equal to the second is second order. For the first, all we need to say is that $x_1$ and $x_2$ satisfy the predicate $A$, and $x_1\ne x_2$. – 2012-02-24
-
0@AndréNicolas I don't fully understand what you mean by setting. For this example we may assume FOL with ZFC axioms, I think it is all that is needed to define syncatical setting. – 2012-02-24
-
0If you are working within ZFC, everything is first-order. If you have been asked this question in a course, things have to be interpreted in the (unknown to me) context of that course. My guess is that you are supposed to say the second is second-order, because you are considering subsets rather than elements. – 2012-02-24
-
0@AndréNicolas I am not taking any course this question emerged when I was trying to answer [this question](http://math.stackexchange.com/questions/112935/notation-for-all-subsets-of-size-2) about how to state all set that have two elements(second notation is obvious mistake but third is valid). I do not understand why you claims that everything is first order it is obvious that I can not quantify over predicate so not everything can be first order. – 2012-02-24
-
0Predicates can be realized as sets in ZFC. – 2012-02-24
-
0What is A^{|2|} ? Is it a standard notation? – 2012-02-25
-
0This is mistake it should be $2^{A}$ which is notation for powerset. I have corrected this mistake. – 2012-02-25
-
0I have found a [good lecture](http://videolectures.net/ssll09_tiu_intlo) which briefly explains those king of things. – 2012-03-18