Subsets of the continuum

Following on from the discussion of the set of all the integers, with order $\aleph_0$, and the set of all its subsets– the continuum, of order $c$ with $c = 2^{\aleph_0}$ – what can we say about the set of all subsets of the continuum?

Like any other set of order $N$ we can say it has order $2^N$, in this case $2^c$ or $2^{2^{\aleph_0}}$. But what do its members represent in the physical world?

By interleaving digits each point on the continuum can also be thought of as a representation of cartesian co-ordinates: for instance 0.55011200…. can be thought of as (0.5010…, 0.5120…) using interleaving, so we can think of a subset of the continuum as a subset of points on the cartesian plane and all the subsets as all the possible curves in the cartesian plane (including disjoint curves).

According to Wheels, Life and Other Mathematical Amusements– the book is a good thirty years old so I am hedging it here a bit – there are no known physical representations of yet higher transfinite orders – they exist in a purely mathematical world: discussion of the philosophical implications of which is fascinating but beyond me.

Looks like a great book

My copy of C. J. Date‘s SQL and Relational Theory: How to Write Accurate SQL Code turned up today – I have only read one chapter so far but already I have a feeling that it is going to have been worth every penny.

Date is concerned not to write another SQL primer but to give those with some SQL experience a good grounding in the relational model and the set theory on which it is based. It just feels more like a proper scientific/mathematical text book and it also seems to be well written.

Very pleased with the purchase and will keep you all posted on how it goes: at the moment I am just wishing I had bought it months ago.

Bought on a whim but seems like a good one

I bought this book this evening on the way home from work – The Annotated Turing – and have already got through fifty pages.

(Actually I wish I had bought it from Amazon because the price there is less than half I paid for it).

Those pages covered much of the same ground as the early chapters of  Godel, Escher, Bach: An Eternal Golden Braid – a much more famous book (and not a bad read) –  and in a more formal, mathematical way, but it also seems to do it in a much clearer fashion.

So, while I haven’t yet got to the chapters that dissect Turing’s On Computable Numbers, with an Application to the Entscheidungsproblem, so far I have no hesitation in recommending it as a good introduction to the issues of computability.