The nature of infinite things 

For 60 years I've believed the argument that there are an uncountable infinity of real numbers. The argument goes roughly like this:

Suppose that there are a countable number of reals in [0,1), i.e. at least 0 and less than 1. This means that there is a 1-1 correspondence between those reals and the natural numbers from 1 up. We create this list of reals showing the sequence of decimal digits for each. But we can now create a new real number in [0,1) by setting its nth digit to be 1 more modulo 10 (i.e. 9->0) than the nth digit of the nth entry. Now we've created a new real that differs from every real in the list, contradicting our assumption that the list is complete.

 I now believe that this argument is wrong because it assumes that one can carry out operations that can't actually be carried out. To see this let's consider a related argument: that there exist non-computable reals.

Our definition of a computable real in [0,1) is a procedure that takes a natural number n, and returns that number's nth digit after the decimal place. A procedure in any turing-complete language is just a sequence of characters from some finite alphabet. So we can represent each character by a sequence of bits and concatenate those bits to form a number that corresponds to our procedure. So all possible procedures in our language are a subset of the natural numbers in this representation. Now make a list of all the procedures that create computable reals in [0,1) -- it doesn't matter if some numbers occur multiple times. We can now create a new computable real by a procedure that, given a position n, asks the nth procedure for its nth digit, adds 1 to that modulo 10, and returns that for its nth digit. This is different from every element in our supposed list of computable reals contradicting the assumption that all reals are computable.

But what we actually proved is that there is no computable way to put the computable reals into a list, because if we can compute that list then we can, using our diagonal argument, create a new procedure that is a computable real that is not in the list.

And it seems that a lot of mathematics is like this. We assume that some non-computable thing exists, then we find that this implies that all sorts of other non-computable things must also exist.

 So, to be perfectly clear:

Non-computable things don't exist.

You don't agree: show me one.

However it is a lot easier to think about the real numbers rather than the computable real numbers, and the sets of natural numbers rather than the computable sets of natural numbers. So my 2nd point is:

It's ok, in a programming language, to define types which ostensibly include non-computable values, even though the type's constructors only cover computable values.

Bob Simpson and helmets in sport

The death of Bob Simpson reminds me of when I used to watch him in Sydney grade cricket. I particularly remember a century in a little over an hour, without balls hit in the air and with fielders around the fence. Incidentally the Western Suburbs team at the time had a better bowling line up than Australia, since it included the recently retired Alan Davidson at the height of his powers, Graeme Corling, and Johnny Watkins and Simpson to bowl spin, and a guy who bowled medium pace or spin who Tony Grieg said would have been a sensation on English conditions.

Simpson never wore a helmet. For fast short bowling he would watch the ball closely as it whistled past his nose. The problem with helmets is that they give the batters a false sense of security. Yes, they will stop you from getting a broken jaw and other external damage. However they don't offer enough protection against concussion which is more likely to have long term consequences. Helmets should give the batters confidence to keep their eye on the ball and evade it. Instead we see the dangerous option of turning their back. Players facing fast bowling need to be trained to do the right thing without thinking.

Update on who to eliminate in STV (preferential) voting

 As I said after Australia's 2022 election (https://grampsgrumps.blogspot.com/2022/05/who-to-eliminate-in-stv-preferential.html), we should eliminate candidates who can't win against any remaining candidate. The 2025 election reinforces this. There have been several cases where one candidate has a good lead but can't win against either of the next 2 candidates. The result is then determined by who finishes 2nd. This causes 2 problems:

• The voters who backed that leading candidate are deprived unnecessarily of their right to influence the winner with their preferences.
• Voters may feel that they have to vote strategically to get their preference between expected 2nd and 3rd candidates counted. This means that they don't get to indicate the policies they really prefer, and might even prevent their preferred candidate winning if she has more support than expected. Avoiding strategic voting is one of the key advantages of STV over 1st past the post.

Eliminating the candidates who can't win is easy by computer if the votes are digitised, 

If forced to do it by hand you can make a lot of progress this way. If there are N candidates then draw up an NxN table. For each vote, for each pair of candidates X and Y: if X has a higher preference then add 1 to his square against Y and subtract 1 from Y's square against X. We can have lots of people doing this at once on different lots of votes, then add the tables at the end. When all votes completed, we eliminate every candidate that only has negative numbers. We  can also eliminate groups of candidates at the bottom who can't combine to get ahead of any other candidate. Then repeat.  No redistributing preferences needed. Hopefully at the end there is one candidate who wins.  If not then we do have to see who can't win. And if we end up with X beats Y and Y beats Z and Z beats X, or worse, then we have to do something more arbitrary, like eliminating the one with the lowest top preferences, as now, or the most bottom preferences, as I somewhat prefer.

Impressive Adelaide scholars

Speaking of AI and quantum and English: I just saw this insanely impressive talk:

https://youtu.be/rGAynRTUMVI?si=h4jpSsZC2D5RZcBG. May I say how nice it is to hear an Aussie accent. However Phiala Shanahan talks at high speed, and I wonder if the rest of the world can hear it all. [I once gave a talk in the US and the South accent people had trouble understanding me.]

I notice that Dr Shanahan is from Adelaide. Another impressive Adelaidian is Terry Tao. I think we can assume that genetics are not involved. Maybe it's the water. They get their water from the Murray River. I once visited Adelaide during a drought in the Murray-Darling catchment. The water was horrible. But maybe there's a magic ingredient.