Someone asked for an explanation of 
Here goes. Begin with the concept of counting a set of objects. We can count a set of objects if we can arrange the objects in a list: first object, second object, third object, and so on, so that each object in the set appears on the list. Listing the objects in a set is the same as assigning a positive integer to each member of the set. Some countable sets are infinite. For example, the set of positive integers itself, the set of positive even integers, the set of positive odd integers as well as the set of primes.
To see that the positive even integers are countable just line up the members of that set and match each member of the positive even integers to a member of the positive integers. The function 
Having the property of being able to be put into such a correspondence is called a set’s cardinality. The cardinality of finite sets is just the number of elements they have; two sets have the same cardinality if, and only if, they have the same number of members/elements. For example, the cardinality of a set of a dozen eggs and a set of a dozen pencils is 12.
Any infinite set that can be put into a one-to-one correspondence with the positive integers is countably infinite. We can also speak of the cardinality of countably infinite sets (like the set of positive integers and the set of positive even integers). We can speak of the number of elements in these sets, which is an infinite, or transfinite number, denoted by
So the set of positive integers 
Finally, suppose we’re really crazy and want to add 
So we have shown that 
While
is not lawlike coextensive with a physical predicate 
, where 
and 
are appropriate vocabularies, the attribute expressed by 
is mental, but not physical. Nevertheless, by the principle of physical exhaustion (PE), the attribute is physical, but not mental (ontologically), as it is possessed solely by physical objects.
where 
is a rank in the hierarchy of the physical), which allows us to say that everything is exhausted by the physical, every attribute is mathematical-physical (i.e., existing somewhere (possibly high-up) on the set-theoretic hierarchy; see 
, and for any attribute that expressed by a predicate that makes essential use of members of 
of predicates of a language 
to the universal they give expression to 
, where 
is the set of universals of 
and 
are mapped to the same universal 
just in case it is a scientific necessity that these two predicates apply to the same things. In other words, 
, 
is a law of science.
in 
-truth and 
-structures, 
-reference, but within this same class of structures, 
does not inductively define 
of code numbers of sentences true in 
is not definable. This has negative consequences for the prospects of aritmetically defining a truth predicate for arithmetic. Nevertheless, Tarski showed how to define truth in terms of satisfaction and by giving an inductive definition of satisfaction beginning with atomic sentences and on up with sentences of higher and higher complexity in terms of the satisfaction of their parts. While a stronger set theory or higher-order logic is still required to convert the inductive definition into an 
, no formula 
is true in 
of numbers such that 
-models of arithmetic. Now extend each model 
by adding the class 
of all sets 
of natural numbers from the domain of 
such that 
of 
) as a subsystem of the axiom system 
in Hellman’s example. Just because it’s so much fun, let’s talk about 
indicates the successor function that returns the successor of 
.
![(\textup{L2}) \ \forall x \forall y [x \dot{<} \dot{\mathsf{S}}y \leftrightarrow (x \dot{<} y \vee x \dot{=} y)]](http://s0.wp.com/latex.php?latex=%28%5Ctextup%7BL2%7D%29+%5C+%5Cforall+x+%5Cforall+y+%5Bx+%5Cdot%7B%3C%7D+%5Cdot%7B%5Cmathsf%7BS%7D%7Dy+%5Cleftrightarrow+%28x+%5Cdot%7B%3C%7D+y+%5Cvee+x+%5Cdot%7B%3D%7D+y%29%5D&bg=ffffff&fg=333333&s=0 "(\textup{L2}) \ \forall x \forall y [x \dot{<} \dot{\mathsf{S}}y \leftrightarrow (x \dot{<} y \vee x \dot{=} y)]")
![(\textup{L3}) \ \forall x \forall y [(x \dot{<} y) \vee (x \dot{=} y) \vee (y \dot{<} x])](http://s0.wp.com/latex.php?latex=%28%5Ctextup%7BL3%7D%29+%5C+%5Cforall+x+%5Cforall+y+%5B%28x+%5Cdot%7B%3C%7D+y%29+%5Cvee+%28x+%5Cdot%7B%3D%7D+y%29+%5Cvee+%28y+%5Cdot%7B%3C%7D+x%5D%29&bg=ffffff&fg=333333&s=0 "(\textup{L3}) \ \forall x \forall y [(x \dot{<} y) \vee (x \dot{=} y) \vee (y \dot{<} x])")
![(\textup{A2}) \ \forall x \forall y [x \dot{+} \dot{\mathsf{S}}y \dot{=} \dot{\mathsf{S}}(x \dot{+} y)]](http://s0.wp.com/latex.php?latex=%28%5Ctextup%7BA2%7D%29+%5C+%5Cforall+x+%5Cforall+y+%5Bx+%5Cdot%7B%2B%7D+%5Cdot%7B%5Cmathsf%7BS%7D%7Dy+%5Cdot%7B%3D%7D+%5Cdot%7B%5Cmathsf%7BS%7D%7D%28x+%5Cdot%7B%2B%7D+y%29%5D&bg=ffffff&fg=333333&s=0 "(\textup{A2}) \ \forall x \forall y [x \dot{+} \dot{\mathsf{S}}y \dot{=} \dot{\mathsf{S}}(x \dot{+} y)]")
![(\textup{M2}) \ \forall x \forall y [(x \dot{\times} \dot{\mathsf{S}}y) \dot{=} (x \dot{\times} y) \dot{+} x]](http://s0.wp.com/latex.php?latex=%28%5Ctextup%7BM2%7D%29+%5C+%5Cforall+x+%5Cforall+y+%5B%28x+%5Cdot%7B%5Ctimes%7D+%5Cdot%7B%5Cmathsf%7BS%7D%7Dy%29+%5Cdot%7B%3D%7D+%28x+%5Cdot%7B%5Ctimes%7D+y%29+%5Cdot%7B%2B%7D+x%5D&bg=ffffff&fg=333333&s=0 "(\textup{M2}) \ \forall x \forall y [(x \dot{\times} \dot{\mathsf{S}}y) \dot{=} (x \dot{\times} y) \dot{+} x]")
, which has historically been called “Robinson Arithmetic”. ![(\textup{Q0}) \ \forall x [x \dot{=} \dot{0} \vee (\exists y) (y \dot{=} \dot{\mathsf{S}}y)]](http://s0.wp.com/latex.php?latex=%28%5Ctextup%7BQ0%7D%29+%5C+%5Cforall+x+%5Bx+%5Cdot%7B%3D%7D+%5Cdot%7B0%7D+%5Cvee+%28%5Cexists+y%29+%28y+%5Cdot%7B%3D%7D+%5Cdot%7B%5Cmathsf%7BS%7D%7Dy%29%5D&bg=ffffff&fg=333333&s=0 "(\textup{Q0}) \ \forall x [x \dot{=} \dot{0} \vee (\exists y) (y \dot{=} \dot{\mathsf{S}}y)]")
-
with,
) fail to hold, in addition to 
and 
. At the same time, howerver, 
and 
and natural interpretations of 
, 
, 
, 
fail to hold.
) of sentences true in the interpretation. The extension of any proposed truth predicate in such a system escapes the definitional machinery of the system. Of course, the truth predicate for first-order arithmetic can be defined with appeal to a stronger system, like second-order arithmetic, in the case of the Peano Axioms, etc.
(where 
contains no explicit definition of 
containing all models of 
such that for each 
, 
. 
if, and only if, every member of the class 
is a 
.
, there is a formula 
such that for all 
, 
. But since the function 
is effectively computable, it is definable and so, 
, for some 
. But this means that,
and ![\psi(\dot{m}, \dot{p}) \in \mathsf{Th}(\Omega)]](http://s0.wp.com/latex.php?latex=%5Cpsi%28%5Cdot%7Bm%7D%2C+%5Cdot%7Bp%7D%29+%5Cin+%5Cmathsf%7BTh%7D%28%5COmega%29%5D&bg=ffffff&fg=333333&s=0 "\psi(\dot{m}, \dot{p}) \in \mathsf{Th}(\Omega)]")
![m \in \textup{D}_{\textup{U}} \Longleftrightarrow \exists y [\phi(y) \wedge \psi(\dot{m}, y)] \in \mathsf{Th}(\Omega)](http://s0.wp.com/latex.php?latex=m+%5Cin+%5Ctextup%7BD%7D_%7B%5Ctextup%7BU%7D%7D+%5CLongleftrightarrow+%5Cexists+y+%5B%5Cphi%28y%29+%5Cwedge+%5Cpsi%28%5Cdot%7Bm%7D%2C+y%29%5D+%5Cin+%5Cmathsf%7BTh%7D%28%5COmega%29&bg=ffffff&fg=333333&s=0 "m \in \textup{D}_{\textup{U}} \Longleftrightarrow \exists y [\phi(y) \wedge \psi(\dot{m}, y)] \in \mathsf{Th}(\Omega)")
,
is definable. This is a contradiction. So, 
effectively countable.
