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Set Theory under the guise of "Mathematical Logic" (Notes for CU students, 2012)

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 slow-paced 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]

            I shall start with Platonism. Think of the Quran, the holy book. There are various copies of the Quran; in various sizes, shapes and styles. All these copies are the Quran; they manifest or express the Quran. But they themselves are not the Quran itself. The Quran itself is rather abstract; it will not be affected if you destroy those copies.  And, a copy of Quran can be not perfect – it may express only a part of the Quran; but the Quran itself is perfect. Thus argues a Platonist. Take another example, a musical piece or a song. The musical piece or the song can be played or sung by many people at various moments; there can be various performances even by a single (set of) performer(s). And a performance is usually – in fact, always is imperfect; it will fall short of the ideal, the real music or the real song. The real music/song is actually above all those performances; it is abstract.  The Quran itself is an abstract entity; and it is real and perfect. So is the case with the music or the song. Let us call these – the abstract Quran, or the abstract piece of music or the song – Platonic entities. Though abstract the Platonic entities are real and perfect; therefore they exist in some sense of “existence” or “being”.

            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 Occams 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 Cantors 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.

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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  {x|P(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

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arity of a relation binary relation (binary function λ x,y,z. x+y=z)

function vs relation

equivalent relation,

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equinumerosity

infinite sets -- equinumerous with its own proper subsets (Dedekind's definition)

various types of number

Cantor's Theorem

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cardinality,  cardinal numbers

embedding, injection      |ℕ| = |ℤ| = |ℚ| < |ℝ| 

diagonal technique          Cantor's theorem

Numbers as equivalent sets

Foundations

Philosophy

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