Introduction: These notes are mainly meant for the philosophy students of Chittagong University. It is unfortunate that we have to abide by some very stifling rules and practices here. We have to follow a syllabus that defined us a course as "Mathematical Logic", but in terms of the content the course actually indicates an amalgam of some very rudimentary set theory and Boolean Algebras. So rudimentary is the set theory portion – at least in terms of its sketchy details and in terms of the questions set for it in the past few years – that it should be actually good for a grade 7 student; you might have already learned it at that stage. The syllabus, it seems, was set very carelessly. There is also another worry. Our question setting and marking procedures have to go through so called "moderation board" and a number of examiners:"first examiner, second examiner and third examiner". We are already a very slowpaced university (due to campus violence the university gets often closed, and we – both the students and the teachers – spend very little time for learning and teaching – let alone for research). We become slower due to these procedures. But worse are the grotesque results out of these procedures. The question which you have to answer at last will be far from consistent – you may find repetitive and even wrong questions. With regard to the marking of your answer you might be a little lucky if your course teacher – or one who has directly taught you the subject – is the first examiner. But the second examiner happens to be, in practice, less an expert on the field; moreover he/she happens to be not aware of what you were taught – at least he/she lacks the firsthand experience of teaching you. This situation may become much worse with the third examiner. Nevertheless, despite all these adversities we have to proceed; we have to learn. Here is the plan. We will not exactly follow it but keep it, the book SET THEORY AND THE RELATED TOPICS (Second edition) by Seymour Lipschutz, as a checkpoint and content setter. (I wish I could start with a different book, but deviating that much from the syllabus seems to be a little risky). Lecture 1. (July 3, 2012 with 4 th yr; July 4 with 3rd yr) Why philosophers are interested in sets? [Plan and
hints: Platonic forms. Nominalism: A class/set of individuals as a substitute
for a form] You may feel a little discomfort to buy this argument. Would the Quran itself
exist if we human beings didn’t exist? “Surely, it would, after all it is a
holy book”, you may murmur this if you happen to be a little religious. But,
how about the song? Would it exist if we were not here? It is less likely that you will agree to confer
such an “existential” (or “ontological”)
status to the song. Instead you may claim that the song is actually a part of our
culture and history. In fact this is how many philosophers tend to react to
Platonic entities. They won’t accept such entities. They would propose that in lieu of such an entity we actually find a set of items which are concrete
– not abstract – and these items are similar with
each other in some respect. So the Quran, these philosophers would claim, is
actually a set of items – like various hard copies, electronic copies, audio instances,
and so on – which are similar, for at least we call all these items (instances
of) "the Quran". These philosophers are nastik – atheists (or, more accurately
they are called “nominalists”). So using the notion of sets the nastik philosophers can do away with Platonic entities. They do this because they adopt an economic policy (known as “Occam’s Razor”): less is better. Intuitively a set means a collection of some items. But there are other words with similar meaning like – “collection”, “class”, “category”, “group”, “aggregate”, “bundle” and so on. Why do the philosophers prefer sets? One reason is often acknowledged, that the notion of sets is quite crispy – for the identity condition of a set is very clear. Historically, this notion has been sharpened and developed mainly by the mathematicians; the philosophers seem to be happy getting such a ready made tool from the mathematicians. Sets are, however, not immune from becoming Platonic entities. A set itself is abstract, though it is a set of some concrete items; moreover, sets eventually pave a way to infinite number of infinities (to Cantor’s heaven). Those nastik philosophers are not unaware of these troubling issues pertaining to sets. Nevertheless, their solace seems to be this: the trouble pertaining to sets is much less than the trouble with embracing the other Platonic entities. Lecture 2. (July 6, 2012 with 4 th yr) Sets : at first glance A set is a collection
of some given items. In fact it can be any
collection of any “definite” items. The
adjective “definite” is said to be used by Georg Cantor, who invented set
theory. An item in a set is known as an element or member of the set. That the
item is definite means that it is certain
– not probable – that the item is a member of the set. Moreover a set can be
any kind of collection – in what so ever manner its members have been
collected. True, there are sets whose members are similar or homogenous in some
respect. But, there can be sets having very dissimilar or inhomogeneous members
– which might have been collected very randomly.
The set of all natural numbers have homogenous members – for all the members are natural
numbers. But we can have a set containing my nose, the Milky Way, and the
theory of relativity; its members are disparately different from each other. In order to mean the former set, the set of
all natural numbers, we write {x  x
is a natural number} or just {x  N(x)} – where N shortens the predicate “ ... is a natural number”. The
latter set will be written as follows: {my nose, the Milky Way, the theory of
relativity}. You can see now that we use a pair of braces in order to mean a
set; if the members of the set have a common property P we will write {x  P (x)} – the set of all Ps. If there is no such common property we will just list corresponding names between the braces (provided that the members have to be denumerable; we will see later what "denumerable" means). Two more things. The elements of a set have to be not "ordered"  whatever "being ordered" means. So there will be no difference between {a,b,c} and {b,a,c}. Moreover, there is no scope of repeating a member in a set: so {a,b,c} and {a, b, c, a} will be counted as same set. [If order
matters and repetition is allowed too we get the notion of a “multiset”, but
our focus is on “set” not on “multiset”. ]
If two sets A and B are such that all elements of A are also elements of B (but not necessarily vice versa) then A is called a subset of B and B is called a superset of A. In symbol we write A ⊂ B.  plan Cantor’s def: any collection of any definite items ⇒ *no vague objects ** any collection – it can be random #Sets defined with respect to certain condition {xP(x)} Elements (no order is needed in listing the elements/ multiset) Subset / superset Empty set {unique empty set in standard set theory} Power set A magic/twist
x ≠{x}
contain vs include ⊆ ⊂ ≠ ∊ Identity condition of a set / extensionality Philosophy Lecture 3. Equivalent Relation and Infinite sets 44444444444444444444444444444444444444444444444444444 (10 july, 2012) arity of a relation binary relation (binary function λ x,y,z. x+y=z) function vs relation equivalent relation, 44444444444444444444444444444444444444444444444444444 (15 july, 2012) equinumerosity infinite sets  equinumerous with its own proper subsets (Dedekind's definition) various types of number Cantor's Theorem 44444444444444444444444444444444444444444444444444444
ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ cardinality, cardinal numbers embedding, injection ℕ = ℤ = ℚ < ℝ diagonal technique Cantor's theorem Numbers as equivalent sets Foundations Philosophy sssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
