502 – First order (or predicate) logic

September 28, 2009

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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502 – Predicate logic

September 19, 2009

These exercises (due September 28) are mostly meant to test your understanding of compactness.

  1. Let M be a nonstandard model of {\rm Th}({\mathbb N},+,\times,0,1,<). Show:
    1. (Overspill) Suppose that A\subset M is definable (with parameters) and that A\subseteq{\mathbb N}. Show that A is finite.
    2. (Underspill) Suppose that A\subset M is definable and that A\cap {\mathbb N}=\emptyset. Show that there is some infinite c such that all the elements of A are larger than c.
  2. Let M be a nonstandard model of {\rm Th}({\mathbb R},{\mathbb N},+,\times,<,\dots). Here, {\mathbb N} is treated as a relation, and in \dots 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 ({}^*{\mathbb R},{}^*{\mathbb N},{}^*+,{}^*\times,{}^*<,\dots) for M. The convention is that we identify actual reals in {\mathbb R} with their copies in {}^*{\mathbb R}, so we write \pi rather than {}^*\pi, etc.
    1. Show that ({}^*{\mathbb N},{}^*+\upharpoonright {}^*{\mathbb N}\times {}^*{\mathbb N},{}^*\times\upharpoonright {}^*{\mathbb N}\times {}^*{\mathbb N},0,1,{}^*<\upharpoonright {}^*{\mathbb N}\times {}^*{\mathbb N}) is a nonstandard model of the theory of problem 1. (In particular, check that the indicated restrictions of {}^*+ and {}^*\times have range contained in {}^*{\mathbb N}.)
    2. A (nonstandard) real r is finite iff there is some (finite) natural number n such that -n<r<n. Otherwise, it is infinite. A (nonstandard) real r is infinitesimal iff r\ne 0 but for all positive (finite) natural numbers n, one has that -1/n<r<1/n. We write r\approx 0 to mean that either r is infinitesimal, or else it is 0. Show that infinite and infinitesimal numbers exist. The monad of a real r is the set of all s such that r-s\approx 0, which we may also write as r\approx s; and say that r and s are infinitesimally close. Show that the relation \approx 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 s={\rm st}(r) to indicate that the (actual) real s is in the monad of r. We also say that s is the standard part of r.
    3. Show that a function f is continuous at a real a iff {\rm st}({}^*f(a+h))=f(a) for all infinitesimal numbers h.
    4. Suppose that f is continuous on the closed interval {}[0,1]. Argue as follows to show that f attains its maximum: For each positive integer n, there is some integer i with 0\le i\le n such that f(i/n)={\rm max}\{f(j/n)\mid 0\le j\le n\}. Conclude that the same holds if n is some infinite natural number, i.e., there is some (perhaps infinite) “natural number” i with 0\le i\le n such that {}^*f(i/n)={\rm max}\{{}^*f(j/n)\mid 0\le j\le n\}. Let r={\rm st}(i/n), and argue that the maximum of f is attained at r.