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Math 4000: Foundations, Fall 2007
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Homework
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Assignment
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Assigned
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Due
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Problems
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| HW1 |
08/29/07
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09/05/07
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1.
Show that the following 3 statements are true for all natural numbers m and n:
0+m=m, m++n=(m+n)+, m+=m+1.
2. Show that m+n=n+m
for all natural numbers m, n.
3. Define multiplication recursively by
(i) m x 0 = 0;
(ii) m x n+ = (m x n) + m.
Show that multiplication is associative.
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HW2
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09/05/07
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09/12/07
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1.
An ordinal is a successor ordinal
if it has an immediate predecessor, otherwise it is a limit ordinal. Show that an
infinite cardinal is a limit ordinal, but some limit ordinals are not
cardinals.
2. It is possible to extend addition to all ordinals with the following
definition: for ordinals m
and n,
(i) m + 0 = m;
(ii) m + n+ = (m + n)+;
(iii) if n
is a limit ordinal, then m+n
is the union of all sets m+p
with p < n.
Using this definition, show that there are ordinals m and n such that m+n and n+m are different.
3. Use Theorem 0B to prove that the set of 1-variable polynomials with
rational coefficients is countable. |
HW3
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09/12/07
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09/19/07
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Section
1.1: 4, 5
Section 1.2: 4
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09/19/07
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Midterm
1, no HW
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HW4
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09/26/07
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10/03/07
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Section
1.3: 7
Section 1.5: 2, 3
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HW5
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10/03/07
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10/10/07
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Section
1.5: 4
Section 2.1: 1, 5
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HW6
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10/10/07
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10/17/07
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Section
2.2: 3, 4, 6
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HW7
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10/17/07
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10/24/07
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1.
Let Q = <{rational numbers}; +, x > be the rational numbers
under addition and multiplication. Show that the set ([1/2,
3/4] union [5, 6]) is definable in this structure. (It
helps to define the order relation first. For this you may want to use
the theorem of Lagrange that states that every nonnegative integer is a
sum of squares of four integers.)
Section
2.2: 18, 28
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10/24/07
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Midterm
2, no HW |
HW8
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10/31/07
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11/7/07
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Section
2.4: 2(a)(c), 5, 6 |
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11/7/07
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No
HW
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HW9
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11/14/07
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11/28/07
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1.
Modus Tollens (MT) is the rule of inference that states ``From
(p->q) and (not q), derive (not p).'' Consider a formal system that
is exactly like the one we are studying except that Modus Ponens is
replaced by MT. Show that this formal system is not complete. (Hint:
Show that any theorem of the new system is either an axiom or starts
with a quantifier or a negation symbol. Use this to exhibit a valid
formula that is not a theorem.)
2. Let Gamma be the
set of all L-sentences true in an L-structure A. Show that Gamma proves a
sentence phi if and only if phi is true in A.
Section
2.4: 4
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HW10
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11/28/07
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12/5/07
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Recall
that a theory is a set of sentences closed under entailment.
1. Show that a theory is also closed under provability.
2. Suppose that T1 and T2 are theories whose union is
inconsistent. Show that there is a sentence in T1 whose negation
is in T2.
Recall that the theory of an L-structure
A, written Th(A), is the
set of L-sentences true in A.
3. Let L be the language whose nonlogical symbols are {+,*} -- plus and
times. Show that in this language the theory of the rational numbers is
different from the theory of the real numbers.
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HW11
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12/5/07
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No due date
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These
problems will not be collected. Please try them. If you get stuck,
please ask about them in class or in my office hours.
Section 2.5: 2 (We did this in class, but make sure you understand the
argument.)
Section 2.5: 5, 8
Section 2.6: 7, 8
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