Cabbage
Banned
- Joined
- Nov 24, 2002
- Messages
- 2,598
I believe you missed my point.
It is in this part of the discussion that I believe we are in agreement--I tried to make that clear in my previous post, but apparently I wasn't successful.
I'm not denying the existence of infinites, infinitesimals, or the surreals in general. I think it's great to study extensions of the reals in which we have infinites and infinitesimals, I'm certainly not so blind as to limit my thoughts to the real numbers alone.
All I've been saying is that we need to be clear on what our "universe" of numbers is in a given discussion, and I see no benefit in thinking, "Well, we have the surreals, and so that's all we should consider, since they encompass all of the integers, rationals, reals, ordinals,...." Sometimes is useful to "pretend" our "universe" consists only of a certain set of numbers alone--the integers alone, for example, or the rationals, or anything along those lines.
A basic example: Someone has a bag containing whole apples, and nothing but whole apples. They say that if they had twice as many apples as they have now, they would have exactly five whole apples. In our "universe" of natural numbers, I know this is impossible--there is no natural number whose double is five. I may wish to remark on the fact that if we allow half-eaten apples, that the above situation is possible, but that doesn't change the fact that within this specific context, it is impossible.
Taking it a step further, I assume you're familiar with the idea of geometric constructions using a compass and straight edge. Some classic questions were:
1. Can you square an arbitrary circle?
2. Can you duplicate an arbitrary cube?
3. Can you trisect an arbitrary angle?
To answer these questions, we need to consider what our universe is--what numbers (represented as lengths of line segments) can we construct with a compass and straight edge. So here we have an interesting (in my opinion, anyway) question where it is useful to restrict our attention to a specific set of numbers much smaller than even the real numbers. It doesn't mean those other numbers don't exist in some broader sense, it just means that they are of no interest for the problem at hand.
And, of course, we learn that with these geometric constructions, we add, subtract, multiply, and divide numbers, and take square roots as well. The cube root of 2, e, and pi are not in our universe--these numbers do not exist in the problem at hand. Being able to do #2 and #3 would imply that we can construct a root of an irreducible (over the rationals) polynomial of degree three, which we know we can't do (those roots live outside our universe), hence they're impossible. #1 is even worse--it would imply that we can construct pi, which is transcendental (not the root of any polynomial with rational coefficients), so it's impossible as well.
And then there are fifth degree (and greater) polynomials over the rationals. A big question was whether or not there is a general formula (using radicals along with the basic four operations) to find the roots of an arbitrary polynomial of degree five or better. And, of course, it turns out there is not. Once again, that doesn't mean that those roots do not exist, just that they can't be expressed using the given operations, and I find that fascinating. We can find the roots, but we need to introduce additional functions in order to do so.
Or take integral calculus--what functions have "nice" antiderivatives (constructible from finite combinations of basic functions, such as the four basic operations, radicals, trig functions, exponential and logarithmic functions)? e<sup>-x<sup>2</sup></sup> is a typical example of a function without one. It most certainly has an antiderivative, but it most certainly also falls outside of our universe of constructible (as defined above) antiderivatives, and once again, I think that's interesting.
All I'm trying to say is that there are various sets or classes of numbers--integers, rationals, geometrically constructible, algebraic, reals, and surreals, to name just a few. Depending on the situation under consideration, it may be useful to restrict ourselves to the rationals only, for example, or it may be useful to extend our viewpoint to the surreals. It's all about context, in some sense.
Last question first. The cardinality of the reals (c) is certainly not aleph-naught. It's certainly consitent (within ZFC set theory) for c to be aleph-one (the continuum hypothesis), but most modern set theorists tend to think the continuum hypothesis is false--c may be much larger than aleph-one. In fact, all we actually know about c is that it has cofinality greater than omega--and, in fact, it has been proven that it is consistent with ZFC set theory to assume that c is any given cardinal with cofinality greater than omega.
First question now. I get what you're asking, but I'm afraid I don't know enough of the history to give you a specific year (or even a general time period). Let's just say that when I refer to "the real numbers", I am referring to the complete ordered field (there is only one (up to isomorphisms), so this sufficiently clarifies it).
As for your middle question, I must admit, I don't quite get what you are asking. Could you elaborate on it?
Again, I'm not exactly following you here, but hopefully I've already clarified myself on this point earlier in this post.
Summing it all up with reference to the OP, it really does make a difference in what you mean by .999... (*) as to whether or not it equals one.
If by * you simply mean a string of decimal digits (as the guy in xouper's link seems to be doing, at least in part), then no, of course * does not equal one--by inspection, they are obviously different strings of digits, and so are obviously not equal.
If by * you mean the limit of the (countable) sequence .9, .99, .999, .9999,... in the surreal numbers, then once again, no, of course * is not equal to one. There are surreal numbers infinitesimally smaller than one, so if this sequence does converge, it must certainly converge to a number such as that (though I'm not convinced that it actually converges to any surreal at all).
Of course, there is a limitless supply of interpretations in which * is not equal to one; I won't bother with listing any more.
However, if by * you are referring to the standard decimal representation of a real number (there are no surreals in our universe now, simply because I am choosing to restrict my attention to the reals), then * is undeniably equal to one.
This is just as true as the facts that there is no rational whose square is two, and there is no real number whose square is -1. The discovery/invention (whichever you want to call it, I don't care) of the irrational and imaginary numbers, respectively, did nothing to change these facts--there is a square root of two, but it's not rational; there is a square root of -1, but it's not a real number. Similarly, introducing infinitesimals does not change the fact that, in the real numbers, .999... is equal to one. It's perfectly fine to talk about infinitesimals, but if you do, you're talking about something other than the real numbers (the complete ordered field). It's simply a matter of definition, and that definition exists only to give clarity on what the topic of discussion is--the definition does not exist to claim that these are the only types of numbers that exist, only to clarify exactly which numbers are under consideration:
1. There are no negative natural numbers. There are negative numbers, but they aren't natural numbers.
2. There are no rational square roots of two. There are square roots of two, but they are not rational.
3. There are no geometrically constuctible irrational cube roots. There are irrational cube roots, but they aren't geometrically constructible.
4. There are no nice antiderivatives of e<sup>-x<sup>2</sup></sup>. There are antiderivatives of e<sup>-x<sup>2</sup></sup>, but they aren't "nice" (in the sense I defined above).
To name a few. Similary,
5. There are no infinitesimal real numbers. There are infinitesimal numbers, but they aren't real numbers.
It really is quite the same thing as, for example, the difference between the definitions of "dog" "cat" and "mammal", for example.
If you point at a beagle and say "Dog", I'll say, "Yep, that's a dog, all right".
The same if you point at a collie, a golden retriever, dachsund, or chihuahua.
But if you point at a cat and say "Dog", I gotta jump in and say, "No, I'm afraid that's not a dog".
It's not like I'm suddenly denying the existence of cats, but you seem to be taking it that way over and over again, and I really can't understand why.
To say that a cat is not a dog does not imply that I think cats do not exist.
In fact, I'm quite content in calling a cat a "mammal". I do believe in cats.
And in fact, that's the extent of all I'm saying when I say there are no real infinitesimals. I do know there are infinitesimals, it's just that they are not a part of what is defined as "real numbers".
Do we agree on this?
I do not think I can state my position any more clearly than that.
You should see the big deal he is making about finding me over here, like he learned how to tie his shoes or something. I posted what my username on this forum was about 3 1/2 weeks ago or so on that forum. Nice detective work.
