How far can we go?

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.

Infinite Hotel- Extended Discussion

Hi everyone,
When the class started, we elaborated more on some perspective on infinity and we jumped on infinity hotel which gave us an idea of infinite and how it works. After reading my notes I realize some of the attribute come into place when you compare the hotel room situation. Some of them were limitless, irreducible, endless and boundless. I chose this 4 because no matter how the hotel room was filled, it could still take infinite number of people in the bus.
It is limitless because it has beginning but you can never know the end and there is no last person at the hotel room. No matter how you break it down there is more room available to be occupied, small or big doesn't define it so it's irreducible. It is endless or boundless because we added as many infinite buses as we like but they all had a room at the hotel.
We used different method like n-2, 2n-2, 6n-2 etc to make it easy to move people from room one and then it went on and on but at the end of the class I felt like they all come down to some of the attribute or perspective that the class listed. In my point of view infinity is undefined no matter how we think about it; there is always more to it. I was asking myself that if everything in this World is infinite how will the World be like to live in? If no one dies and everyone keeps on living will the World be a better place to live after all? Thanks a lot
Yaa

Infinity+Infinity+Infinity

It seems Infinity is a bit slippery and just out of reach at ALL TIMES. It is bounded, unbounded, AND order matters? "The Rule of Infinity" is a concept that became a little less foggy(However, not at first!) for me when Professor Hamman proposed the infinite bus/hotel room scenario. The idea that infinity plus infinity (or even infinity + infinity + inifinity) really can be partially compartmentalized in a predictable way, can be accounted for and has value is astounding to me. I feel like I have started to grasp what the bounded aspects of infinity represent and how order plays a role and little about the unbounded possiblities it holds.

Extended discussion on Infinity- Class 1

As we were ending our debate on infinity, I had mentioned that perhaps we assign things as infinite when we've come to our limit of being able to comprehend it. In other words, when we realize we're limited with what we can see or, where logic can no longer be attainable. I think classifying something as infinite almost saves you from going insane! Jk. Even if you believe that everything has to have finite attributes or will eventually come to it's end, the limitation in comprehending how far infinite goes or when something will stop still makes whatever is finite, infinite in itself. And so perhaps finite and infinite are not mutually exclusive, but one in the same.

Thinking about what it means for anything to be infinite has proven to be difficult, obviously. If you illustrate your thinking with human analogies, God, Time, Space, Cells, grass, sand, etc, you will most likely come to see it as not possible to prove either, or, as become more confused. The quote Professor Hamman posted, "The Infinite! No other question has ever moved so profoundly the spirit of man" very much asserts this. I think that the topic of infinity inevitably raises spiritual and philosophical discussion. Where this may not be good enough for mathematicians because math solves problems and is governed by laws, infinity is above and beyond laws, and to "solve" or figure it out is almost foolish and will become an infinite project in itself!

I asked my father casually the other day, "What do you think of infinity?" He answered simply and calmly, "Eternal now." I was shocked and moved, and was satisfied with his answer.

Basically, what hearing that did for me was remind me to stop thinking about infinity, or, to stop mental chatter all together and just be present to my experience in the moment. This is very much the awakening and enlightened realization that many spiritual traditions such as Buddhism and Daoism speak of. It doesn't surprise me that my father would say that, coming from a traditional Buddhist upbringing. It also reminded me of Jessica's tattoo which had a clock with wings and in the middle of the clock it says, "NOW." Very cool, Jessica! I think the whole concept, perception, or reality of infinity is asking us to be present and accept what is happening NOW, and it does that by not allowing us to ever really know. I think that scientists who go mad ultimately surrender and accept that infinity is called infinity for a reason! But, maybe I'm wrong.

Infinity- What is it?

After a brief introduction of our classmates the big question was asked, "What is infinity?" An hour into our discussion we could still not figure this out. To explore this thought we took apart some thoughts that were used to try and describe infinity. Can infinity be counted? We analyzed this thought by relating infinity to something that's countable yet not knowable. Nobody knows how much grass there is on earth, but we do know that there is a finite number of it. Nobody knows how many humans there on earth, but for a half second there is a definite number around the globe. From this the class agreed that finite number of things could be "infinite" because although the finite number exists, it is not knowable. One classmate described infinity as the category that things are tossed into when they can never be knowable. Through this thought we explored the big bang theory. There is a finite amount of matter but how much of it is unknown. Therefore, infinity could maybe be measured. How do we measure it? This is still unknown because there's the counter argument that maybe it is not measurable. Like a circle, we can accept that it keeps going but we cant understand its infinite rotation. Because the human mind is meant to accept a beginning and an end through our own reality, it's hard to accept something as being infinite. And so we try to find it's end, or infer that a never ending process of change keeps something from going out of existence. Is change an attribute of infinity? Cause and effect is something all humans can relate to and accept. But to think that something had no cause and has always been caused further confusion in our class. How can you measure something if you don't know where it begins? If this is not explainable does it just get tossed into the category of infinity? Or maybe, the beginning is knowable and before we keep moving forward we need to go back and answer the start. As I write this I notice how many circles our class discussion went in. We started with not knowing, moved to analyzing why it is or isn't knowable, and ended with not knowing the answer to that as well. Have no fear classmates, we are not alone. No one has figured this out yet.

In conclusion, after our first discussion our class knows nothing about infinity and its attributes and knows little about the finite.