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I'm currently reading John H. Conway's On Numbers and Games, but without a good understanding of transfinite induction and/or recursion, progress is very slow.

What's a good resource for learning this topics? I'm looking for something that emphasizes transfinite induction and/or recursion with respect to arbitrary well-founded relations, as opposed to focusing exclusively on well-orderings.

Asaf Karagila
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goblin GONE
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  • You might take a look at Claus Tøndering’s PDF Surreal Numbers — An Introduction; it’s a very gentle introduction, and at least the first few inductions are done in considerable detail. This student paper introducing them also offers quite a bit of detail. – Brian M. Scott Jun 27 '13 at 06:26
  • @BrianM.Scott, neither of those resources is currently comprehensible to me. For instance, consider the definitions of the Surreal numbers given in those articles. The former defines that: 'A surreal number is a pair of sets of previously created surreal numbers. The sets are known as the “left set” and the “right set”. No member of the right set may be less than or equal to any member of the left set.' The second defines that, 'Every number corresponds to two sets of previously created numbers, such that no member of the left set is greater than or equal to any member of the right set [1].' – goblin GONE Jun 27 '13 at 09:15
  • I don't currently have the knowledge to make these definitions rigorous. For instance, lets just simplify the whole discussion by restricting ourselves to games. My question is: does the following definition of 'game' work? 'Define that $x$ is a game iff there exist sets $L$ and $R$ such that $(L,R)=x$ and every element of $L \cup R$ is itself a game.' I don't know whether this uniquely defines the concept 'game' or not; and even if it does, how do I define relations and/or functions on games recursively? – goblin GONE Jun 27 '13 at 09:16
  • And how do I prove things about games by induction? What am I allowed to assume - what's the inductive step? I asked essentially those questions here, and one of the comments mentioned $\in$-recursion. Hence my interest in understanding transfinite induction and recursion. – goblin GONE Jun 27 '13 at 09:17

2 Answers2

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Would it be enough a couple of examples? It seems to me there is no much difference in how to use it from the usual induction on the naturals.

Induction on a transfinite but still well-ordered set $X$

You are proving the proposition $P$ for every $\alpha\in X$. You need to check $P(\alpha_0)$ for the first elements $\alpha_0$ of $X$. And then the induction is completed by showing that if you assume $P(\alpha)$ for every $\alpha<\beta$ then $P(\beta)$.

Exercise (if I am not forgetting something): Show that if the (totally) ordered set $X$ (any cardinality) satisfies induction then every non-empty subset has a first elements. Show the converse too.

Induction on a well-founded set $X$

Let us take an example in which $X$ is still countable, but it doesn't matter too much. Suppose $X$ is indexed by $\mathbb{N}^2$. $X:=\{a_{n,m}\}$, where you order $\mathbb{N}^2$ by the Cartesian order, i.e. $(a,b)\leq(c,b)$ iff $a\leq c$ and $b\leq d$.

You could prove $P$ for all elements of $X$ if you check: 1. $P(a_{0,0})$ 2. $P(a_{n,m})$ implies $P(a_{n+1,m})$ and $P(a_{n,m})$ implies $P(a_{n,m+1})$.

Does this clarifies it? Do you need more concrete examples?

Answer to the question in the comments:

Question 1:

The point $2$ would be better written (and should/could had been written that way too above) "If $P(\alpha,\beta)$ for every $(\alpha,\beta)<(\alpha_0,\beta_0)$ then $P(\alpha_0,\beta_0)$." Remember the how we ordered the pairs. the assumption is, in this case, that is true for $\alpha_<\alpha_0$ and $\beta<\beta_0$.

Question 2:

Example: Suppose we want to define the sequences $a_{n,m}$by:

  1. $a_{0,0}=1$
  2. $a_{n+1,m}=F(a_{n,m})$
  3. $a_{n,m+1}=G(a_{n,m})$. for some functions $F$ and $G$.

It is induction what tells you this sequences are defined for all $\mathbb{N}^2$.

Exercise: Take the proposition $P(n,m)$ to be "a_{n,m} is defined" and prove it by induction as above. (notice that "well defined" here is a different problem since $a_{1,1}$ can be defined using $F(a_{1,0})$ and $G(a_{0,1})$. We are only proving that it can be defined. To get well defined we need to impose more condition on the $F$ and $G$).

Bonus info: Induction over the reals

If $X=[0,\infty)$ and

  1. $P(0)$ is true and
  2. whenever $P(\alpha)$ is true for every $\alpha<r$ then $P(r)$,

then $P$ is true on all of $X$.

This is equivalent to the nested closed intervals property, Lagrange theorem, Rolle theorem, ...

So, in the same way that almost all probable statements about the naturals can be proven by induction, also almost all (those involving continuity of the reals) probable statements on the reals can be proved by this induction as well.

OR.
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  • This is helping. Okay, suppose instead of $X = \mathbb{N}^2$ we had $X = \omega 2 \times \omega 2$, how would that change Part 2? – goblin GONE Jun 26 '13 at 13:54
  • Yep, great. There's three more concepts I really need to understand. Firstly, how do we define a class of entities inductively? Secondly, how do we define functions/relations on those entities recursively? Thirdly, how do we use transfinite induction to prove things about entities/functions/relations defined in this way? Btw, the 'bonus info' is super cool!! – goblin GONE Jun 26 '13 at 14:08
  • Don't worry if you don't have time to answer some/any of the above questions. This has been helpful anyway. – goblin GONE Jun 26 '13 at 14:10
  • Franklin, I don't understand the exercise... what is the meaning of defined, as opposed to well-defined? – goblin GONE Jun 28 '13 at 01:39
  • That was just because I was lazy so make the exercise better. Assume defined is that you can use the formulas to get one that will give you the value of each element of the sequence. Take well defined to be that any way to get to the value of an element of the sequence using those formulas will give you the same result. E.g. you won't get $a_{1,1}=F(0,1)=0$ and $a_{1,1}=G(1,0)=1$ at the same time. Here the formulas are defining $a_{1,1}$ but not consistently. So, not well defined. – OR. Jun 28 '13 at 18:22
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I haven't seen Conway's book so I don't know if this will be useful. But Ciesielski's "Set theory for the working mathematician" has numerous interesting applications of transfinite induction to set theoretic real analysis. When combined with Hajnal and Hamburger's book, I like to think that this could make a really interesting first course in set theory for undergraduates.

hot_queen
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