Originally posted by doublezSmaller than what?
from the quote you gave (from my source) it implies that when one infinity is a subset of another it is smaller.
When you deal with finite sets it is clear that a proper subset is smaller because it has less elements than the set itself.
With infinite sets that is not so. All infinite countable sets are equivalent e.g. even numbers are as many as all natural numbers, all rational numbers are as many as all natural, etc.
I recommend you read some of Cantor's Theories on the subject just in case you realy want to understand why operating with infinite sest is so counter-intuitive.
Originally posted by THUDandBLUNDERUM... "there are even more individual numbers than there are squares".
You haven't a clue what you are talking about.
In future you should stick to finite sets, sonny.
Originally posted by doublez
[b]from the quote you gave (from my source) it implies that when one infinity is a subset of another it is smaller.
It said 'smaller', not smaller. Notice the difference? Duh.
[i]Originally posted by doub ...[text shortened]... nfinity + Infinity = Infinity
or, more precisely,
Aleph(zero) + Aleph(zero) = Aleph(zero)
[/b]
Originally posted by doublezSorry doublez, that isn't the way it works. A common error people make when people compare these kinds of sets goes something like this:
UM... "there are even more individual numbers than there are squares".
"There are more individual numbers than there are squares. If we count from 1 to 100, we've got 100 numbers and only 10 squares. And if we count up to 1,000,000 we've got 1,000,000 numbers and only 1000 squares. Clearly, there are more individual numbers than squares."
For any finite set, that is true. But in an infinite set, we're not bounded by some upper counting value. All we have to count out more squares. So the counting would go something like this:
"If we count from 1 to 100, we have 100 numbers. To find an equal that number of squares, we look just look at the first 100 squares (up to the number 10,000)."
"But wait a minute!" you might say. "If we count up to 10,000 then we would have 10,000 numbers, and that's clearly more than the number of squares!" But all we'd have to do there is count the first 10,000 squares (up to the number 100,000,000).
"But wait a minute!" you might say again. "That's not fair! You're counting much higher to get the same number of squares!" which is true. However, you must remember that these are all finite slices of two infinite sets. When you're dealing with infinity, there is no ceiling to hit, no finish line to cross. This is a progressive embodiment of what Galileo called the "potential" of infinity in the link you provided. So it's not cheating to count ahead with one set (squares), because the other set (numbers) will catch up. There is no "head start" we can give to either set that the other set can't surmount.
That's why there are the "same" number of squares and individual numbers (meaning the sets have the same degree of infinity). Of course, real numbers (1, -2, 1/3, 0.5832298...) are a different kettle of fish... =)
I hope that helps.
3 edits
Originally posted by doublezI tought you wouldn't reply until a taunt came out, and I guess that was the case. Honestly, I don't care who's right as long as the truth gets spelled out for all to hear. I've been wrong many times, and although it's hard to swallow sometimes, it makes me feel better to learn a new truth. Does what I say make sense?
so... the source is wrong?
EDIT: I'm drunk right now...
Originally posted by doublezFrom "your" source:
yes but the the source clearly states that "there are more individual numbers than there are squares. " without any brackets etc in the text.
Galileo has spotted something very special about infinity. The normal rules of arithmetic don't really apply to it. You can effectively have 'smaller' and 'bigger' infinities, one a subset of the other, that are nonetheless the same size.
Now can we drop that issue already?
And just so yo get your definitions straight read this:
http://mathworld.wolfram.com/InfiniteSet.html