Transcendentals and definability

[daithi]

Not all transadental numbers exist

Does a tree falling in the woods make a sound if no one hears it fall?
For sake of argument let's assume the answer is - No. The tree does not make a sound.

Using the above reasoning I would argue that transadental numbers that can not be described or constructed do not exist either. If you can't describe the number or represent the number in any finite way then it is no different than a tree falling in the woods that nobody can hear.

Transadental numbers such as Pi and e certainly exists. This is because they can either be described or represented. We can also describe an infinite number of transadental numbers. For example, we could say something like "replace every 5th digit in Pi with the number 3." There is an infinite number of ways of modifiying Pi in this manner, so we could easily come up with an infinite number of transadental numbers.

However, here is the cool part. We can count every single one of these transadental numbers. All we have to do is order their English descriptions by length and alphabetical order and start counting.

Therefore, I have intuitively proved that transadental numbers are either countable, or I have proved that even if no one hears a tree fall that it still makes a sound.

 

[EnumaElish]

By daithi's argument either real numbers do not exist, or we can count every one of them.

Edit: But there may be more to this argument. There is an interpretation of the late-19th century real analysis (invention of the real numbers, the continuum, the infinity, &c.) as a response to the Archimedean/Zenoean paradoxes of the infinitesimals (converging series). But, as we all probably know, the converging series have very little (if at all) to do with the real numbers; they are all expressed in rational numbers. So in that interpretation, Cantor's "inventions" were a little like inventing the-Hubble-telescope-and-the-electron-microscope-all-in-one just in order to see the rabbit taking over the turtle. (Comic effect intended.)

 

[chronon]

However, here is the cool part. We can count every single one of these transadental numbers. All we have to do is order their English descriptions by length and alphabetical order and start counting.

And eventually you get to the 'description':

The unterminating decimal that is formed by applying Cantor's diagonal method to this list.

Do you count that as describing a number or not?

 

[NateTG]

However, here is the cool part. We can count every single one of these transadental numbers. All we have to do is order their English descriptions by length and alphabetical order and start counting.

That notion leads to some Goedelesque problems. Clearly, we have some ordering, so I could, for example, describe a number as:

The 'smallest' number that cannot be described within ten words.

 

[NateTG]

And eventually you get to the 'description':

The unterminating decimal that is formed by applying Cantor's diagonal method to this list.

Do you count that as describing a number or not?

You'd need to be a bit more formal. I would guess that daithi might like to investigate the computable numbers...

http://en.wikipedia.org/wiki/Computable_number

 

[HallsofIvy]

Because they didn't understand it!

 

[Office_Shredder]

However, here is the cool part. We can count every single one of these transadental numbers. All we have to do is order their English descriptions by length and alphabetical order and start counting.

This isn't the definition of countable, so all you've done is, well... nothing.

You might as well have just written down every single number instead of giving it an english description (you can write the number down next to the list if you want). Then we can apply Cantor's diagonalization argument all over again

 

[daithi]

Office Shredder, NateG, and chronon pointed out a couple of objections to my post.

Office, I probably could have been a little clearer in my original post, but in the scenario I described you can't write down every number that has an unending and non-repeating stream of decimals. Each number needs to be represented in some finite way in order for it to exist. The number itself can be transfinite, such as pi, but in order for it to exist it needs to be represented in some finite way. Under this limitation you can count all the transfinite numbers that meet my definition of existence.

chronon, since in the scenario I described not all numbers that have an unending and non-repeating stream of decimals (i.e. not all real numbers) can be represented it would mean that Cantor's diagonal method would have no meaning. Actually, I suppose it would have the same meaning as adding 1 to the natural numbers.

NateG, I think you make a good point. I'm not sure what the Godel ramifications would be concerning this view of numbers, but then again the real numbers have their own Godel ramifications. The wiki article on constructable numbers is facinating. That article made two great objections for which I don't have an answer. The first, is that all real numbers are required in order for a closed set of numbers to exist, and without a closed set of numbers Analysis (i.e calculus) wouldn't work. The second objection is that constructivist mathematics does not use the law of excluded middle, and I kind of like this law. So I think sticking with the real numbers is probably the best idea, but this is an interesting concept.

 

[Hurkyl]

There are other issues with constructivism -- IMHO the most severe is that Cantor's argument rears its head in the form of the undecidability of the Halting Problem. This has some nasty consequences: for example, equality of real numbers is not a well-defined relation!

(On the other hand, equality of algebraic numbers is decidable)


IMHO, it is much more straightforward, and achieves most of the same goals, to simply take the traditional formulation of mathematics, and study things like constructive analysis in that framework... it's a lot clearer than trying to study constructive analysis in a constructivist framework.

In particular, it makes it easier to avoid the accidental level slips and other little errors that lead to paradox[/url] and other issues.

 

[honestrosewater]

However, here is the cool part. We can count every single one of these transadental numbers. All we have to do is order their English descriptions by length and alphabetical order and start counting.

Perhaps I missed the point, but can you define "an English description"? If you mean simply a finite-length string on some finite alphabet, then the set of all English descriptions is countable, being the union of countably-many at-most-countable sets, so you've limited yourself right there and of course an injection from it won't have an uncountable range, right?

Now, if you give English a bit of respect and let a description be an actual English sentence that can describe sets of numbers, that might be interesting. But working with natural languages isn't a day at the beach. And they can be just as precise as any formal language. It's only the speakers you have to contend with.