Beyond Turing: question for the math and computer whiz kids

You can crank out numbers to the tiniest precesions for a million years on the fastest computers, and there will still be numbers you haven't defined... because there's no such thing as infinite technology. In fact, the word "definition" implies some notion of being finite... which goes against the concept of the real number system.

Is this what you're trying to get at?

As DB said, what do you specifically want to know?

this would be a very big (or small) number.

The computable numbers include all specific real numbers which appear in practice, including all algebraic numbers, e, pi;, et cetera. Indeed they must since, as explained above, no uncomputable element can be expressed using a finite number of symbols. In some sense the computable numbers include all numbers which are individually "within our grasp". So the question naturally arises of whether we can dispose of the reals entirely and use computable numbers for all of mathematics. This idea is appealing from a constructivist point of view since it would allow us to work without uncountable sets. It has been hypothesized that most of analysis could be reconstructed using computable numbers. A great deal of traditional analysis has been done in a constructive framework. Nevertheless, it is necessarily more complicated than classical analysis would be. In any case, most mathematicians see no need to restrict themselves to computable numbers, even if this can be done.

and, Alan Turing proved the existence of definable but uncomputable numbers, some of which can be approximated, some of which can't (Chaitin's Constant is apparently an example of the former).

Using all possible algorithms, no computable number wouldn't eventually, given infinite time, be educed. My question, restated, is: Would an analogous list of all possible definitions of uncomputable numbers include all the rest, or are there numbers that are neither computable nor definable?

I mean, you all know about how pi goes on forever in a non-repeating decimal value. Heck, some of you may even have created album tracks based on it, Kate and Ned.

But how do they know? What makes the numbers? Where from?

But short of measuring a large perfect circle and it's diameter, where do they get that level of precision from?

(1/1) - (1/3) + (1/5) - (1/7) + (1/9) - (1/13) + (1/15) - ....

If you want a more precise value, you add on a few more terms in the series.

So to what extent do we want ineffability? It strikes me the more we seek it, the more we will probably be able to discribe it in a meta langauge. Remember 2 is not two (nor is that) it is just a way of describing the concept. The question assumes a degree of platonism, that numbers and mathematics exist without a mind to do them in. I am not a platonist, so my answer to the question is NO!

xpost What's red and invisible? No tomatoes.

You can make numbers, or at least prove they exist through Set Theory (if you so desire) and it's fairly simple once you get hang of the terminology.

Godel raises a problem (Which I suppose is why DFW calls him the Dark Prince of 20th Century Mathematics), that problem being that any formal system (in the case of his Incompleteness Theorem this is Arithmetic, but it could be anything) will always have well-formed statements that can be made using its rules that make no sense. (they're neither true nor false - simply undecidable) and even if you extend the system's rules to include the statement you've just dissallowed there will always be statements that are beyond it. Ineffability is inevitable.

As for computability - even given the Turing Halting Theorem stuff (ie. you can't know if your problem is even soluble - which is, sort of, equivalent to Godel's Incompleteness for Information Theory), there's also a limit on computablity in the real world. It's called Bremmerman's Limit and it basically shows that there are some problems which, although they may be theoretically doable, can't actually be done because there isn't enough matter in the universe to convert to computing resources.

Here's what Wiki has to say:
Bremermann's Limit is the maximum computational speed of a self-contained system in the material universe. It is derived from Einstein's mass-energy equivalency and the Heisenberg Uncertainty Principle, and is approximately 2 x 1047 bits per second per gram. This value is important when designing cryptographic algorithms, as it can be used to determine the minimum size of encryption keys or hash values required to create an algorithm that could never be cracked by a brute-force search.

For example, a computer the size of the entire Earth, operating at the Bremermann's limit could perform approximately 1075 mathematical computations per second. If we assume that a cryptographic key can be tested with only one operation, then a typical 128 bit key could be cracked in 1E-37 seconds. However, a 256 bit key (which is already in use in some systems) would take about a minute to crack. Using a 512 bit key would increase the cracking time to 1071 years, but only halve the speed of cryptography.

This also runs into the P-NP (which refers to polynomial versus non-polynomial time) thing. Which (until some clever bugger solves it - but see above as to whether it's soluble or even decidable) means you can't tell how long a given problem is going to take you to solve - which implies the size of the computng resources you'd have to devote to solving it. Obviously you can tell in some special cases, but there's no way of telling for all cases.

If all the above seems overly complex...Well you did ask a difficult question.

Your second sentence was basically what I was getting at. Numbers exist when we create them, and the whole Russellian/Fregan ste theory construction is the best way of doing it. Thus these ineffable numbers by the very nature of not being definable (and hence non-computable) do not exist. But then this is clearly drifting into philosophy: and since I am on the whole a constructivist (with quai-empiricist sympathies) I would say all that.

There's a very good short story by Greg Egan called "Luminous" that plays with this point.

In the beauty of mathematics we see the face of God? Not me.

I'm actually surprised the IDiots haven't used this argument...Which probably shows how they have evolution induced tunnel vision.

No. The square root operation cannot be performed on negative numbers. i is the number whose square is -1.

You could have a number so large (or small) that it couldn't be expressed, vocally or by a supercomputer or whatnot, within the remaining lifespan of the universe.

This assumes the universe will "die".

that problem being that any formal system (in the case of his Incompleteness Theorem this is Arithmetic, but it could be anything) will always have well-formed statements that can be made using its rules that make no sense.

It has to be a sufficiently powerful system, like one that allows for recursiveness.