"The poet only asks to get his head into the heavens. It is the logician who seeks to get the heavens into his head. And it is his head that splits." G.K. Chesterton

Saturday, August 24, 2013

Tangibility of the transcendental 2 - updated.

I'm re-posting this blog entry with a crucial correction to the λ function, which previously wasn't properly defined. 

"I was doing a bit of reading about transcendental numbers today, and had an idea of constructing my own well defined transcendental. I was inspired by the Liouville constant. Anyway, my idea was to construct a number which is a base 10 concatonation of the digits of consecutive prime numbers:
mp =  0.2357911131719232931...
or the natural numbers:
mN = 0.12345678910111213...
The construction proceeded as follows. First we define the function λ:\mathbb{N}\mathbb{N}. Intuituively λ(n) gives the number of digits in n, i.e. the "length" of n in base 10.

Now we can express the numbers as infinite sums:

Where pn is the n'th prime.

Initially I thought this to be an exercise in creative procrastination, since firstly the idea is dead simple, and secondly who would be silly enough in going to the trouble of formulating such numbers. Surely only a fresh graduate with too much time on his hands.

To my surprise the idea behind those numbers has been already formulated by not one, but a few mathematicians. My mN is in fact a constant previously formulated by D. G. Champernowne. And it turns out that mp is already known as the Copeland-Erdos constant:

Notice the striking similarity in the formulation - I have formulated mp completly oblivious to the Copeland-Edros formulation. The similarity lies in the fact that my λ function acts a lot like the floor of base 10 logarithm of y. In fact the relation is the following:

Which makes the sums in the exponents of the definitions of mp and the Copeland-Edros constant equal, since for each i we subtract 1 i.e. we subtract 1, k times, so we need to add k back to the sum. And consequently,

Well, I guess I feel better knowing that I'm in good company deeming quite obscure and prima facie pointless ideas valuable. I knew that both mN and mp are irrational, but now I also know that mN is in fact transcendental (mission accomplished!), and hence don't have to provide a proof, which I suspect would be a herculean task."

Friday, August 16, 2013

A simple idea.

Physicists disagree about what exists, but they tend to agree that nothing is just the lack of that which exists. Note that this is physical nothing, or physical impossibility. Nothing "contains", as it were, all the physically impossible objects, but there are none, so nothing is "empty", as it were. There is one more interesting point to add, which will play a crucial role later on - sure physicists will agree that nothing is the lack of any physical thing, but since what counts as a physically possible object needn't be congruent across physical theories (it's not), physicists will disagree about the list of stuff that nothing is meant to denote the lack of.

Is it really that difficult to extend this idea to logical impossibility? I think not. Analogously, we extend the idea of physical nothing to logical nothing as the lack of what can exist. Here, nothing "contains", as it were, all the logically impossible objects, but there can't be such things, so nothing is "empty", as it were. Furthermore logicians will tend to agree that nothing possible is the lack of any possible thing, but since what counts as a logically possible object needn't be congruent across logical theories (it isn't), logicians will disagree about the list of stuff that nothing possible is meant to denote the lack of.

Clearly, in some set theoretic ontological model nothing possible can be identified with the empty set. That is we think of the empty set ØK(L) as the object associated with all the impossible objects in the L theory, K. We can thus speak of the order of the empty set, here called the rarity, defined as the cardinality of all the K(L) impossible objects.