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AMATH/PMATH 331

Assignment 1

Due Wednesday January 25 at 11:59 pm (end of day).

(1) In the lecture we have defined an inductive set to be a set S for which the following is satisfied: ∅ e S  and  Ax e S  x ∪ {x} e S.

Now, let N be the intersection of all inductive sets. Verify the following:

(a) The set N, defined that way, is itself inductive.

(b) Denote ∅ by 0, and accordingly set 1 = {0} = {∅}, 2 = {0, 1} = {∅, {∅}}, and 3 = {0, 1, 2},‘and so on’.  Actually, we still can’t say  ‘and so  on’, because the definition of a sequence requires finishing the work of defining and studying N, which is still ongoing to the end of this question! As a correct rephrasing, for an element n e N, we denote n ∪ {n} by the symbol n + 1.  Since N is inductive, the elements (sets) 0, 1, 2, and 3 lie within N.  On N, define order by writing n < m if and only if n e m. Show that N is transitive, every element is transitive, and that for each n e N,n = {m e N ∶ m < n}.  (Hint:  These are exercises 1.3-1.4 in T. Jech’s Set Theory book. It is also recommended to think about exercises 1.5-1.9)

(2) When two sets A and B have the same cardinality, they are often called equinumerous, and we write A ∼ B . Prove that R ∼ [0, 1], and use that to show R is uncountable.

Remark 0.1. The cardinality ∣R∣ is conventionally denoted by c, whereas that of N is denoted by ℵ0 . Thus, this question is about showing that ℵ0 < c as cardinals (or ordinals). A great research was done to discover if there are any cardinalities in between.  The statement that we can’t have such middle cardinalities is called the Continuum Hypothesis (CH). It was shown by Kurt G¨odel and Paul Cohen that the continuum hypothesis is independent from the ZFC axioms. Cohen used what is called the method of forcing.  In my opinion, Cohen’s proof using forcing is a great amusement to read, and I recommend it for interested students. You can read about this in many textbooks, including T. Jech’s book.

(3) Prove that a countable union of countable sets is countable.

(4) Prove that the interval (0, 1) is equinumerous to the set of functions f ∶ N → {0, 1} (these are the binary sequences in other words).

(5)  (+7/100  Bonus  marks  for  the  simplest  proof) Prove that N contains uncountably many infinite subsets (Nα )α∈R such that Nα ∩Nβ  is finite whenever α ≠ β . Please note that, unfortunately, none of the proofs available online is simple enough for this contest Q.

(6) Complete reading the construction of R, try to solve the inquiries you have about the operations. (This is for you, not to be submitted).