As of last class, "infinite" is no longer enough to describe any set fully. While we had ℝ, ℕ, and many other infinite sets, these only comprised two distinct sizes of infinity. In order to find a new size of infinity, we had to prove that there
existed an infinite set from which no 1:1 correspondence could be formed
with either ℝ or ℕ.
Instead of just testing sets though, which wouldn't be very practical,
we tried finding a function that could take any infinite set and, from
it, produce an infinite set of higher cardinality. To do this, we
considered the power set of a given set, which is a set consisting of
every possible subset of the original set. And lo and behold, it worked!
We proved that with any set, finite or infinite, its power set is
larger than it is. We can then repeat this as many times as we want on
the resulting power sets, yielding a new size of infinity each time,
leaving us with many, many of sizes of infinity. However, we still cannot classify the number of sizes of infinity properly yet. As far as we know right now, it is countable, but we may yet prove otherwise.
I am still processing the "infinity grenade" that Professor Hamman lobbed at us on Monday. So the infinity that we know and love really isn't as big as we thought it was? There are even bigger infinities that exist? So there are countless larger infinities within our smaller infinity? WHAT?! I'm totally amazed and still shaking my head...... Oh, and WELL DONE Julian!
ReplyDeletei really like this summary. Don't really have any comments to add at the moment, but just thought i'd toss that out there.
ReplyDeleteI feel so excited when we we taught that we can create one bigger than a given infinite set. More than that, we can create as infinite infinite set that are bigger as we want. THerefore, i think the opportunity for us to do that is unlimit. I don't really agree (or did i miss something in class?) with what you're saying at the last sentence: "it is countable". Anyway, your summary is so understandble. Like button for it.
ReplyDeleteHow will we classify the number of sizes of infinity? big, bigger. more bigger? AHHH the paradox!
ReplyDeleteJulian, I love your title..."how far can we go?" It's a good question, and when it comes to humans, there is no limit. It reminds me of a characteristic of human - never satisfy with what we have....find something big, want to go bigger...find the bigger one, want to go beyond and find something that even bigger. Sorry, this probably not even related to anything we talked about. Anyway, after the previous class we had, I'm still a little blur on the power set and the sub set thing.
ReplyDeleteHi Julian,
ReplyDeleteI don’t know if I’m the only one wondering about this power set. I have gone through the reading but all I’m seeing is a finite set and I’m wondering why the sets are close ones without anything showing that there is more subset. If a power set has a subset of 16 then how will I compare that to infinity? The null set which is a subset to the power set at least makes me wonder that there might be more numbers or whatever it contains that we don’t know. I find it very fascinating even though it can be larger set or smaller I don’t see it relation to infinity. It looks like we are the ones trying to generate infinity by putting individual set together to get a subset. I’m still not convinced on Cantor’s power set theorem but I will read over again and may be someone can explain it more to me.
Yaa
Great summary Julian, and thanks again for using the word "countable" :)
ReplyDeleteYaa: A link that might help clarify this:
http://www.mathacademy.com/pr/minitext/infinity/
I think it gives a good summary of what we have learned thus far.
I think if we can create a Power Set in which the size is always going to be bigger than it's original/or previous set; then those words lends itself to sound like a sequence. Not just any sequence but an infinite sequence. I think the argument over the "n+1"th term is upon us...
In class power sets made sense, but now i'm lost. If there are bigger and smaller infinite sets then my mind dismisses it as no longer infinite.
ReplyDeleteNot my post, but all too curious as to what the very LAST link in google would give me: this was the second to last.....Apologies to prof. Hamman but I had to do it: "An infinite number of mathematicians walk into a bar.
ReplyDeleteThe first asks the bartender for a beer.
The second asks for half a beer. The third one says he wants a fourth of a beer.
The bartender interrupts, puts two beers on the bar and says:
“You guys need to learn your limits.” "