These notes cover in some more detail what the textbook discusses in Chapter 5. Some details of my presentation will be different in irrelevant ways from the details in the book. I will be somewhat terse in terms of examples. Let me know if you feel that they are sorely needed, and I’ll amend the notes accordingly.
Predicate logic extends propositional logic in that we now allow the presence of quantifiers, and therefore we also allow the presence of statements that depend on variables. This complicates the definition of semantics, and also introduces additional complications at the syntactic level. On the other hand, this new flexibility allows us to code in straightforward ways many theories actually used in practice. Set theory is but on example of a theory that can be formalized in this context, and we will see others.
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is a formula, a generalization of 
are different or that they are present in 
)
be a term, let 
be a variable, and let 
or 
with 
a variable occurring in 
units from its natural length is given by 
where 
is a constant that only depends on the spring.
where units of distance are measured in meters, and forces in Newtons. This means that 
or 
The work required to stretch the spring 10 m beyond its natural length is then 
and asked to rotate about the 
-axis the segment where 
To find the area of the resulting surface, we can consider splitting the surface into little shells. A typical `thin’ shell, at height 
where 
is the radius of the shell, in this case just the value of 
i.e., 
Also, 
is here the arc length element, given by 
so 
and the lines 
and 
We rotate it about the 
Its volume is 
Here, 
is the exterior radius, which in this case is always 2, and 
is the interior radius, given by 
This means the washer contributes 
of the full volume, and adding all of them together we obtain 
The upper limit of 4 comes from observing that the curves 
and height 
Its volume is 
where 
and 
is the length of the shell, in this case 
We thus proceed to express 
Since 
then 
Hence the thin shell contributes 
of the full volume, and the volume is obtained by adding all these contributions, obtaining 
This is expression (c).
for 
is given by the integral 
In this case, 
so 
and 
so the integral reduces to 
which is expression (b).
]
can be rewritten as 
which is clearly separable. (T)
for 
is four times the area swept by 
Intuitively, we are making lengths twice as long, and areas carry two dimensions of length. Think of the area of a square of side 2, for example. Formally, we can check that this is the case: The area swept by the first curve is given by 
while the area swept by the second curve is given by 
(T)
the given statement is clearly true.
if originally the amount is 
is given by 
where 
so 
Then 
and 
(T)
whenever 
is a nonprincipal ultrafilter on the natural numbers, and
for all 
or
is a nonprincipal filter on a set 
then there is a nonprincipal ultrafilter on 
that extends 
Here, each 
is a constant symbol, 
is another constant symbol, and 
is a symbol for a binary relation (which we will interpret below as membership).
A model 
of this theory 
would look a lot like 
except that the natural interpretation of 
namely, 
is no longer nonprincipal in 
is a common element of all these sets.
thanks to the compactness theorem.
let 
and note that 
the thin slice of water in the tank at depth 
and weighs 
This is a constant force, so the work required to remove it to ground level is just 
where 
is the depth at which the slice is located, i.e., 
The total work is obtained by adding all these contributions, i.e., 
ft-lb.
Since 
the graph is symmetric about the 
is in the graph, then so is 
).
the graph is symmetric about the origin (because whenever 
).
Here 
so 
which is impossible, so there is nothing to graph here. Consider now what happens when 
As 
increases, 
increases, from 
to 
So the same occurs with 
This means that 
, and 
decreases from 
The part with 
is undefined. This corresponds to the fact that at the origin the tangent to the curve is the 
Show:
is definable (with parameters) and that 
Show that 
is finite.
Show that there is some infinite 
Here, 
is treated as a relation, and in 
we may have placed whatever functions and relations we may have need to reference in what follows; moreover, we assume that in our language we have a constant symbol for each real number. (Of course, this means that we are lifting the restriction that languages are countable.) To ease notation, let’s write 
for 
The convention is that we identify actual reals in 
with their copies in 
so we write 
rather than 
etc.
is a nonstandard model of the theory of problem 1. (In particular, check that the indicated restrictions of 
and 
have range contained in 
)
such that 
Otherwise, it is infinite. A (nonstandard) real 
but for all positive (finite) natural numbers 
We write 
to mean that either 
Show that infinite and infinitesimal numbers exist. The monad of a real 
such that 
which we may also write as 
and say that 
is an equivalence relation. Show that if a monad contains an actual real number, then this number is unique. Show that this is the case precisely if it is the monad of a finite number. In this case, write 
to indicate that the (actual) real 
We also say that 
is continuous at a real 
iff 
for all infinitesimal numbers 
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Argue as follows to show that 
with 
such that 
Conclude that the same holds if 
Let 
and argue that the maximum of 
then also 
Combined with the soundness Theorem 
and any formula 
if 
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