Reassessment of Italian Combat Prowess

[ArgusEye]

I didn't understand much of this.

Could someone please provide a crayon version interpretation? Many thanks.

I'll give it a try, but don't be fooled: this is only a sketch of reality, and by no means a proof of anything. There are multiple definitions of the sets, I use the simplest one to explain, which is antiquated.

Let's suppose we play a game. One of us says a number, the other one says a number, the one saying the higher number wins. This is a very tedious game. After all, the only thing the second speaker has to do, is to add one to the number the first speaker said. There is no highest number that cannot be beaten. The sequence of numbers stretches out infinitely, in both the positive and negative directions.

In many mathematical applications, one needs to take the entirety of the set of numbers one is working with, and perform some mathemagic with them. This kind of maneuver is called improper in most cases. Whether or not this succeeds, or is allowable whilst maintaining the truth of the calculation, depends very much on the circumstances one is computing in. Knowing something about the nature of infinity is therefore very useful.

It turns out infinity performs some really odd tricks. I will refer you to the Hilbert Hotel for some entry-level mind-blowing oddity. Sometimes one can perform operations using infinity, sometimes not. This was a problem for many years, and caused much upset and conflict until Georg Cantor decided to properly define some different kinds of infinity, and their properties. This goes quite deep, and I will only pick out the most pertinent bits to this discussion, a little later in this post.

Let us return to the stupid game of my first paragraph. Infinity is not a number, so we cannot call it as our play; what infinity, in its simplest form, boils down to in this context, is if our first player would say: I will not stick to one number, I will keep counting up, forever! This could be said to be a cheat, but then: so could the notion of infinity.

Now let us look at another problem. There is a safe, with a combination keypad, and a burglar with eternity on his hands. We don't know how long the key string of numbers is, but if we start to count at zero, and keep adding up, we are bound to find it before eternity runs out. Every little step of the way gets us that one step closer to the solution. The safe builder has infinite numbers to choose from, but we will hit that number, no matter how insanely huge it is.

This example is of something called 'countable infinity'. There is no limit to the amount of numbers to try, but we get there in the end. Countability is due to the set of numbers in use.

Let me explain what is meant by set in this context. If I'm counting, I'm using only the set N+, which contains all positive natural numbers. Those are the whole numbers, like 1, 2, 3, 59874, etc. It does not contain 2/3, Pi, -4, or i. I can extend this set to be N, the set of all natural numbers, including the negatives. This contains N+, but adds to this negative whole integers like -1, -5, -1008237, etc. I do this by performing a subtraction [a mathematical operation] on my set, and this opens up a new expanse of possible numbers. Now, I can also decide to use a bigger set. We can add the set Z [or Q], of the rational numbers. These are numbers where one natural number is divided by another. Examples are 2/3, -1/2, 18/19, 227/13, and so forth. This set of numbers is opened up to us by using a mathematical operation, division, on our natural numbers. It contains all the previous sets.

With that last set, you can get arbitrarily close to any number you want to reach. The fine-ness of your stripes along the number line can be as fine as you want it. But it will always be stripes, it will always be little steps. There will always be space between the stripes. And in that space, there is always room for a point along the line. No matter how fine you make it.

Coming back to our burglar, this is equivalent to him having to type in two numbers, one above the stripe and one below, giving him a lot more work, to get them both exactly right. But he can do it, in much the same way he can do it with whole numbers. It will take him idiotically longer, though.

If we play another stupid game, I could challenge you to give me two numbers so close to each other, that there is no space between them. I will always be able to give you a number in between. If you say 0 and 1, I will tell you 1/2. If you say 2/10000000000000 and 3/10000000000000, I'll tell you 25/100000000000000. And so on. There is always room.

This ends when we go to the set R, of the real numbers. Real numbers must be on the number line, between other numbers, but they don't have to be rational. They are allowed to have a value that you cannot express as a fraction. These numbers fill the space completely. This becomes pretty hard to explain further, and proof is usually left for advanced students, and to be honest I don't remember it anyway. Just take my word for it.

We now have a set R which contains all the previous sets, plus numbers like e, sqrt(2) and pi. We can expand further, but for the sake of sanity we won't. The set R is obtained from the set Z by a mathematical operation, to be exact: infinite series. This refers to a mathematical expression containing an endless series of fractions added to each other. Any two [or more] fractions can be added to make another single fraction, but an infinite series escapes this bound by dint of it being infinite itself. You have to go into infinity just to get one number, so to speak.

All sets before R were countably infinite. Even the rationals, which can be made as dense as you want them, are countable. I posted a YouTube movie about that earlier. But the real number set, the reals are different. You can make the width of the 'stripes' on the number line infinitely small. The width of each number becomes zero.

This makes them innumerable. After you counted any number of reals, you still have not advanced anything along the timeline. There are infinitely many points between any[/y] two points, no matter how close! And this is another kind of infinite than with the rational numbers, where we have the same story, of infinite possibilities between any two numbers.

Here we come to the meat of the matter. The countable infinities are made using non-infinite methods, of subtraction, addition and division. Reals are made injecting infinity into their creation using the infinite series. For some mathematical applications [mostly pure math though] this is an important difference. Therefore, Cantor cast the distinction between countable and uncountable sets into numberable infinity, Aleph Null, and innumerable infinities, Aleph 1 and up.

If the choices for the combination to the safe are numerable, the burglar will at one time get there, and it is an infinity Aleph null. If there is no way to go by all the possibilities, no matter how many, one by one, there is Aleph One or higher.

Useful? Only if you use infinity in finicky applications. I have never found a good use for them in my field.

From this you must see why I choked on pedantry about obscure mathematical expressions used wrongly. By our very nature we define graphs not by infinite series, but by inputting data within our tragically short lifetimes. Give me the same keyboard and mouse as 76mm, and infinity to go through the possibilities, and I will at one point create any graph he created. Not that I have the patience.

[BigDog944]

Many thanks, ArgusEye, for opening a window clear enough for me to see through into the world of set theory.

[JasonC]

It is not an error, he is still projecting and pounding wet straw.

A countable infinity of countable infinities gives you more than a countable infinity.

All (countably infinitely long) sequences of rationals (themselves countably infinite) comprise a set that is a countable infinity of countable infinities - and have the cardinality of the reals, and are not countable.

It is only the single most famous result in all of set theory, that all sequences of rationals cannot be enumerated - that is, put into one to one correspondence with the natural numbers.

I already pointed out this mistake to ArgusEye when I mentioned that he ignored the word "sequence" in my phrase that he quoted. All he has done since is double down on that mistake, and pretend I didn't say "sequences of rationals" but merely "rationals", when I simply didn't. He is making it up.

The rest of what he is retailing is not obscure, is not in dispute, and is utterly basic in this area.

[$Pec5]

So in the end, there is something larger than infinity?

[Michael Emrys]

So in the end, there is something larger than infinity?

Sure, a bigger infinity.

Michael

[dieseltaylor]

Yep, my mother-in-laws mouth.

[76mm]

Holy thread hijack, seriously, who cares? Open a separate math thread or better yet, a separate forum. This thread is reserved for important topics like whether the 1940 Germans were better/tougher/stronger/better equipped/bigger/more numerous/meaner/better led than the 1944 Germans.

[Erwin]

Thanks ArgysEye. Your explanation is so obvious when you put it that way. This thread has been a real education.

[costard]

Thanks Argus Eye - I was wondering how you could define a continuous function as made up of reals, then use this to prove a cardinality that it manifestly isn't. In my opinion, if that is a crayon version, it's the sort of work that hangs in the Louvre (and can't be seen under good light unless you are important enough for the museum to sacrifice some of the life span of the pigments). Thanks again.

[BigDog944]

Holy thread hijack, seriously, who cares? Open a separate math thread or better yet, a separate forum. This thread is reserved for important topics like whether the 1940 Germans were better/tougher/stronger/better equipped/bigger/more numerous/meaner/better led than the 1944 Germans.

I think you mean Italian rather than German.

[76mm]

I think you mean Italian rather than German.

Actually I was being a bit tongue-in-cheek...

[BigDog944]

Actually I was being a bit tongue-in-cheek...

Lol! So was I. Dry humour doesn't translate to this medium, which is pretty hilarious if you think about it!

[Yeknodathon]

The Germans were obviously stronger because they had better uniforms.

Graph that.

[76mm]

The Germans were obviously stronger because they had better uniforms.

Graph that.

\

Thank you Yeknodathon, back on track!

LOL. sorry BigDog, you're right, we're just too subtle for our own good I guess.

[LukeFF]

Thanks ArgysEye. Your explanation is so obvious when you put it that way. This thread has been a real education.

Yes, and an example of how to turn an interesting topic into one that got sidetracked into some boring discussion of mathematical theory.

[JonS]

Nah, it's an excellent example of how to turn a boring side tracked thread into an interesting one about a tangentially related topic.

[Yeknodathon]

So, was the boring thread at post 1 more boring by post 240 even though the quantity and quality of boredom has changed over time? Or has one side been able to produce more boredom than the other side while the other side has merely extended the pool of qualifying boredom?

[Spinoza]

So, was the boring thread at post 1 more boring by post 240 even though the quantity and quality of boredom has changed over time? Or has one side been able to produce more boredom than the other side while the other side has merely extended the pool of qualifying boredom?

Almost as Peng Challenge , just more fresh and smells better.

[Michael Emrys]

So, was the boring thread at post 1 more boring by post 240 even though the quantity and quality of boredom has changed over time? Or has one side been able to produce more boredom than the other side while the other side has merely extended the pool of qualifying boredom?

Huh? Did you say something?

Michael

[ArgusEye]

A countable infinity of countable infinities gives you more than a countable infinity.

Read the link I posted on the Hilbert Hotel. You're wrong again.

All (countably infinitely long) sequences of rationals (themselves countably infinite) comprise a set that is a countable infinity of countable infinities - and have the cardinality of the reals, and are not countable.

If you replace the word "sequence" with the word "series", then this is indeed the reals. But not a countable infinity of countable infinities, because that would constitute another countable infinity. You can't switch cardinalities by simply raising your domain to the nth power. Cantor diagonalization still works for nth powers.

It is only the single most famous result in all of set theory, that all sequences of rationals cannot be enumerated - that is, put into one to one correspondence with the natural numbers.

Cantor diagonalization is not the grand result of set theory, but it does blow your statement away. As for enumeration of rationals, I posted a nice YouTube video proving that they are numerable.

I already pointed out this mistake to ArgusEye when I mentioned that he ignored the word "sequence" in my phrase that he quoted. All he has done since is double down on that mistake, and pretend I didn't say "sequences of rationals" but merely "rationals", when I simply didn't. He is making it up.

If you don't want your terminology errors corrected, that's fine, but then your statement stays gibberish. Sequences of numbers do not make new numbers, you see.

The rest of what he is retailing is not obscure, is not in dispute, and is utterly basic in this area.

In fact everything I wrote is utterly basic. I don't know more than the very basics.

You remind me of people who crank their engines until their battery is dead, instead of reviewing their options. Repeating errors doesn't make you right.

And I do apologize for derailing the thread.

[sburke]

Now I get it!!! This is all proof that a thread can be infinitely boring while still amusing....sort of.

Can boring be measured to infinity? Is that mathematically possible?

*edit just giving you guys a little grief to keep your feet on earth. If it helps pass time till Gustav is released, it is okay by me.

[Cogust]

If may be inifinitely boring, but that doesn't necessarily make it real boring.

As long as the puns are countable, even when infinitely bad, they can't be real bad.

I think I'll stop there.

Sorry.

[Michael Emrys]

As long as the puns are countable, even when infinitely bad, they can't be real bad.

You are, of course, entirely wrong.

Michael

[Baneman]

You are, of course, entirely wrong.

Michael

You're in luck, Cogust - that's sig material right there

[Yeknodathon]

Well, I've just spotted the uber secret weapon of tedium that will change everything.