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Infinity of a Different Size
- Thread starter The Hammer
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Heyo
Veteran Member
And humans have difficulties to even comprehend the smallest infinity
Since we are in Jokes & Stories, here's the story about Hilbert's Hotel.
And humans have difficulties to even comprehend the smallest infinity(read aleph-zero).
Since we are in Jokes & Stories, here's the story about Hilbert's Hotel.
I post these things in jokes because they are pictures and the forum is games/pics/jokes
This was intriguing.
It's my belief that we are unable to comprehend infinity. One can imagine they are there but it can always be more (or less) than ones imagination.
During a bad time in my life i spent many hours trying to imagine infinity in order to take my mind of what was really occuring. I couldn't get there but im sure i could see infinity on a clear day.
During a bad time in my life i spent many hours trying to imagine infinity in order to take my mind of what was really occuring. I couldn't get there but im sure i could see infinity on a clear day.
Speak fer yerself, toots!
I find infinity to be uderstandable & useful.
(If only I could learn to spell (understandable".)
Bet you cannot describe what you see there.
In maths infinity is used as a concept to solve otherwise unsolvable problems.
In science, invoking infinity means something has gone seriously wrong.
Edit: thats pure maths, not applied maths my the way
I can & have.
Think in terms of properties, function, & utility.
These things are easily understood.
Even @Polymath257 grasps the basics.
Or seriously useful, eg, imagining how our universe
could exist in an infinitely larger multi-verse.
I can do this with half my brain tied behind me back.
Yes. And? No joke here. It is an established fact of modern math.
And humans have difficulties to even comprehend the smallest infinity(read aleph-zero).
Since we are in Jokes & Stories, here's the story about Hilbert's Hotel.
I find it easier to comprehend aleph_0 than to comprehend some of the large *finite* numbers like Grahams's number:
Graham's number - Wikipedia
Not true. Infinite sets are common. Dealing with different sizes of infinity is necessary for even an undergraduate in math.
And I would bet you will find infinity in the form of limits in almost every advanced physics book.
Or so you think. But what to you see there?
I guess that depends somewhat on what you mean by the word 'comprehend'. I understand the properties; I can predict new characteristics; I teach about it on a regular basis; I can distinguish different sizes of infinity from others; etc.
What else is required for comprehension?
Sure but they are imaginary infinities that cannot be applied to the real world.
I bet you wouldn't. I certainly have never seen such a limited.
Thinking is what it's all about.
I see definitions, descriptions, function, & utility.
What I don't do is approach infinity this way....
"Just where is the end of this thing!"
"I can't see that far!"
"Just how small is the smallest part!"
Jolly good, do these things exist in reality, i.e. outside your mind?
What do you mean by 'imaginary infinities'?
Even in the definition of integration, there is an infinite limit used. And integration is necessary for any physics that uses calculus.
A book on solid state physics will typically talk about an infinite lattice of atoms, for example. This simplifies the calculations and gives accurate results, It is certainly NOT a sign that something is wrong.
In cosmology, it is typical to have space infinite in extent, especially for flat or negatively curved space.
In studies of wave propagation, it is typical to consider the far-field, which is done by taking infinite limits. Again, this is not a sign that something has gone wrong.
Once again, I would bet that almost every advanced physics book uses some sort of infinite limit in its calculations.
Mathematics is useful to model of reality.
It works when it works.
I'm no expert on reality though.
Heyo
Veteran Member
Vision is a good topic to help understand infinity. You know what parallax is? Usually your eyes lines of sight cross at the point you're viewing. Now imagine they don't, no matter how far away the object is. Mathematicians (like @Polymath257) are always looking into a distance you can't seem to see. That's how you spot them in a crowd. They also are unable to focus on tasks or objects right in front of them.
As has been pointed out there are different sizes of infinity. Those of aleph_0 and the size of the real numbers are very *small* infinities. That 0 as a subscript is a sign of more to study.
So, aleph_0 is the first possible size of an infinite set. The second is called aleph_1, then aleph_2, etc. One question is where the set of real numbers fits in this hierarchy.
For any infinite set, the collection of subsets of that set is a larger infinity than the original. This means there is an infinite hierarchy of sizes of infinity. There is actually more than just that.
You can discuss *large cardinals*, such as the inaccessible and Mahlo cardinals. You can ask about cofinality, or what happens to sums, products, and exponents involving infinities.
We can ask what additional properties sets of different sizes can have. We can talk about group theory, topology, etc and ask which sizes of infinity a group can be or what sizes a dense subset of a topological space and be. And it turns out that there are restrictions between the sizes of certain subsets and the size of the whole.
This material has been studied for well over 150 years at this point. Much of it can be taught to undergraduates and some is absolutely required for graduate students in math, This is not esoteric or mysterious stuff any longer.
Does the number 2? No. But they are very useful for modeling what we see and predicting new things about the universe around us.