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Infinity of a Different Size

The Hammer

[REDACTED]
Premium Member
FB_IMG_1675576543150.jpg
 

ChristineM

"Be strong", I whispered to my coffee.
Premium Member
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.
 

ChristineM

"Be strong", I whispered to my coffee.
Premium Member
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
 

Revoltingest

Pragmatic Libertarian
Premium Member
Bet you cannot describe what you see there.
I can & have.
Think in terms of properties, function, & utility.
These things are easily understood.
Even @Polymath257 grasps the basics.
In maths infinity is used as a concept to solve otherwise unsolvable problems.
In science, invoking infinity means something has gone seriously wrong.
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.
 

Polymath257

Think & Care
Staff member
Premium Member

Yes. And? No joke here. It is an established fact of modern math.

And humans have difficulties to even comprehend the smallest infinity
8f2403fd8489b18cc392c10c099323a0d816c7cb
(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

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

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.
 

ChristineM

"Be strong", I whispered to my coffee.
Premium Member
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.

Or so you think. But what to you see there?
 

Polymath257

Think & Care
Staff member
Premium Member
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.

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?
 

ChristineM

"Be strong", I whispered to my coffee.
Premium Member
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.

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.
 

ChristineM

"Be strong", I whispered to my coffee.
Premium Member
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?

What properties has infinity got?
How can an infinity have new characteristics?
 

Polymath257

Think & Care
Staff member
Premium Member
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.

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.
 

Heyo

Veteran Member
Or so you think. But what to you see there?
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.
 

Polymath257

Think & Care
Staff member
Premium Member
What properties has infinity got?
How can an infinity have new characteristics?

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.
 
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