[Predendum (post facto prologue!) - For all you peoples' info, all of the following argument is wrong... This is not the way to determine that the set of Reals is greater than the set of Rationals, or that the Reals are uncountable. I fracked it up, merely ended with a collection of the same infinities. It's his trick (actually Dedekind's suggestion) of making ALL rational numbers end in a string of 9s that make Cantor diagonal proof work. 1.5 = 1.499999,,, for e.g. Still it was 2am when I was thinking it, and it had never been successfully worked out before until 1870 or whatever, what can you expect? Also I didn't do much math after year 10.]
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I was about to go to bed - it's 1:30am and I have a flight first thing in the morning, but I have to get this down.
I think I have had an epiphany about German mathematician George Cantor's uncountable numbers. Surely David Foster Wallace did not explain it at all well in "everything and more"… Too many words and symbols mixed together, not enough diagrams. I am going to try it with just words and no symbols - I am dylfectiv on math symbols.
Anyway he had just tried to explain how to make an irrational number from a rational one, and I JUST DIDN'T GET IT. Then somehow, I understood, not the way he explained it, but my way, with words not equations and symbols.
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Now, it is the irrational numbers that make the Number Line a continuum, the rational numbers (integers and fractions) even though there are an infinite number of them, they count for zero on the number line, as they are points and points are dimensionless, therefore they cannot contribute to the actuality of the number line. Get it? No? Move on. This is what I am going to "prove" slash "explain". (When I should be asleep. Maybe I am asleep and this is just a dream.)
Here are my words. Sorry, no diagrams either.
OK, take a breath.
Firstly. Again, a point is a DIMENSIONLESS site, say the place where a number is. A line is a solid continuum made up of points, between two points. Huh? How can a line be solid if it is made up of dimensionless points? I'm glad you asked that question.
Let's say I want to count all the numbers on the Number Line between 1 and 2. That is to say, all the points (numbers) on the line of points (numbers) between 1 and 2.
Go.
1. The first rational number in this line is the integer 1, rational because it can be written as (an infinite number of) fraction(s): 1 = n/n, e.g. 2/2, 3/3, etc…, and, cue the drum roll, it can also be considered to be equal to the infinitely long number 0.99999...recurring. How? See footnote. (My God, I am turning into DFW!)
2. 0.99999…recurring, is therefore, like I said, a rational number (equal to 1).
3. I can make a new number, one that is not a rational from this, just by changing the second 9 to an 8. Like this...
4. 0.98999…recurring. However, it is NOT a rational number, I just made it up. It can’t be written as a fraction.
5. Trust me. My son knows someone who is good at math. Next.
6. 0.99899…recurring is NOT a rational number either. Next.
7. 0.99989…recurring is also NOT a rational number. Next.
8. Etc., etc, literally and mathematically, ad infinitum
9. From the first rational, 1, I have created an infinite set of irrational numbers, each just slightly different from the last.
10. If I now tried to COUNT all those irrational numbers based on 1 (0.99999recurring), I could show that there is a 1 to 1 correspondence with the integers, meaning that it is a COUNTABLE set.
11. By counting, I mean that I could go: 1 is to 0.99999…, 2 is to 0.9899999…, 3 is to 0.998999…, 4, 5, 6, etc…
12. However I would never be able to STOP COUNTING those new irrational numbers because there are an infinite number of them. They are countable, yes, look, I am counting them, but they are infinite.
13. However there is this NEXT rational number that I want to continue the counting procedure on… 1.1, (or whatever the next rational number is…) I have to get to 2, remember.
14. Hang on, I can't start to COUNT this number, or the irrational variations I can make from it, because I am STILL COUNTING the FIRST INFINITY of the irrational variations of the first rational number. That counting will never end. It is infinite. Duh!
15. The second irrational and the rest, are UNCOUNTABLE is the sense that I am still waiting to get to count them...
16. In other words, infinity is busy, taken up with the first of the rational number, I can’t use it anymore to count the second. Or the third. Or the rest.
17. I'd have to have ANOTHER infinity, one infintity in fact for each rational number, of which there are already an infinite number, remember, between 1 and 2.
18. In fact, therefore (don't you wish I'd not say that word?), I'd need an INFINITY OF INFINITIES to get from (to count from) 1 to 2 on the Number Line, because I need to include all those uncountable irrationals.
19. If we removed all the rational numbers (an infinite number), the number line would be just as solid, because it is so jam-packed solid that it wouldn't missed just ONE infinity. It is made up of an infinity to the power of infinity of all the irrational numbers.
20. This is a new class of infinity. An uncountable infinity. Aleph, is I think what Cantor called it.
21. This is the continuum of the number line, by my interpretation of the clues I got from DFW explaining George Cantor's definition. I think. (If not, whatever. I'm using it - it means I can keep reading the book!)
22. This sort of reasoning about infinity is what got Galileo in trouble.
23. Only God can be Infinite because infinity is Perfect.
24. Galileo was threatened by the Inquisition.
25. George Cantor went mad. And religious.
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I may be wrong, but I am proud of myself for at least doing a little bit of abstract thinking for once in my life, nonetheless.
I am going to bed. It's now AFTER 2:30am. I have a finite time to sleep.
Damn. After I do the footnote...
E@L
Footnote:
1. Let x = 0.9999999recurring.
2. Multiple x by 10.
3. 10x = 9.999999recurring.
4. Subtract x
5. 9x = 9.0
6. Divide by 9
7. x = 1
8. Therefore 1 = 0.999999recurring.
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6 comments:
Close, but no cigar. Your argument would also work for the set of integers (order them positive ones first, ascending, then zero and below, descending: you never get to start counting the negative ones - whereas you can construct a different countable set {0, 1, -1, 2, -2, ...) that has every integer in it, and can be "counted". Similarly, you can construct an ordering for rational numbers {0, 1/1, 2/1, 2/2, 1/2, 3/1, ...} and show every rational number occurs in it somewhere.
Cantor's diagonal proof starts by assuming such a set exists for rationals, and then constructs a number that isn't inside it - and therefore such a set doesn't exist.
OK, that was what I later on was afraid I hd done. Obviously I am not as smart as those guys, or you.
But DFW explains what you said really badly in my opinion. Obviously, again, because I didn't get it. Things that are not all that contentious he gets ultra-worried about, but then skips over the crucial steps with a superficial blase attitude IMHO. A lot of teachers do that, at least mine must have. They place the emphasis on the wrong points.
Actually, showing this new number is what he says Cantor does, but his explanation doesn't I thought, and diagram doesn't make any sense to me, with this X1.a1a2a3a4a5... X2.b1b2b3b4b5b... etc... Does this proof only work if the real numbers only have the same post decimal point number? Like does 'a' ALWAYS have to 9? Like 0.9999999 as in my attempt? He never *says* they have to be. This is why I went off on my tangent, trying to explain it to myself.
As I said I am really dyslexic when I come to math notation, so his equations just do not help. It's yet another language I can't speak.
But hang on, isn't Cantor's proof then sort a LEM? I thought DFW had earlier said they were not strong enough proofs for this level of abstraction.
We need beers to sort this out together... Or not. We just need beers.
Cantor's proof works whatever the numbers after the decimal point are: if we assume we have a list X1.a1a2a3a4a5..., X2.b1b2b3b4b5b like you said, then you construct a new number from the diagonal a1b2c3d4... by adding one to each digit modulo 10 (1->2, 2->3, ..., 9->0, 0->1) and then your new number differs from each one in your, supposedly complete, list, in at least one digit, and hence it's not on the list. Ergo, the list isn't complete, so our assumption that such a list could be constructed is false.
What's a LEM? More than happy to talk maths over some beers some time - do you know what's non-orientable and swims? MΓΆbius Dick! hahaha
Yeah, well DFW didn't say that, and that's why I was confused and drifted off into a fantasy world of my own.
LEM - law of excluded middle. Proving the null hypothesis to be true. Or false. Or not. I don't know, I need sleep... Brain hurts. Yes, God, I'm coming to thee!!!!!
well, ok, i'll just sit in the corner or how about i just bring the beer over? xoxox
We sorted this out by SMS, don't worry Sav. Suffice to say he was right and I was wrong. He did "set theory" to university level. New Math was coming in for the year below at primary I think. We just memorized the times tables until high school.
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