45 mins
Some Fundamental Facts about the Infinite

And I had thought that Salvia-Space was weird...

If someone would have said to me before that the infinite set of integer numbers is the smallest one you can find, I would have though this person is ill or something. Now this professor comes along to shake up my tiny world
It's not only the talk content, but also the way he presents it, his style is perfectly cut to bring such matter forth IMHO.


It struck me about these impossible looking, seemingly contra intuitive situations coming to truth, something of that can be truly felt in some experiences we have, like e.g. the living of the fractal rather than a theoretic representation of it.

Also wondering how these theories are going to create spin offs in 'real' life due human cognition. It really hints to extra possibilities, the probing of the boundaries ( <-- who probably were never there to start with).

The problem with this argument though, is that it only works with an infinite list of infinite numbers.

One could argue that such a list itself, cannot exist.

If that is so, the argument is invalid, as it depends on there being such a list.

I would go with the first answer mentioned: one cannot talk about infinity, without eventually ending up with a paradox. The diagonal agrument is in my view eaxactly such a paradox.

Sure it can. That list is aleph naught integers long. It's as real as any other mathematical concept.

It's probably hyperbolic, but they say that Cantor's relentless contemplation of infinity played a part in his eventual collapse into madness. I often wonder if maybe it doesn't work the other way.

Blessings
~ND

This seems to be the deep culprit which is now more than 100 years old, and a prize that Georg Cantor was not able to pay with sane condition. How much the conflict stress was relevant? Science + philosophy is eating dust in quarrel, and the glassing on the cake (in those times not to be underestimated) a condemn from religious standpoints. Infinity = God you know, don't touch that
(source wikipedia)

For me it was a refreshing gate toward 'paradoxes'. Even if we might be able to un-proof them all as human erroneous constructs, then still life itself, the fractal, breaths paradox all over and deep. Like paradox is its fuel for that matter. It just feels like the impossible is, after all.

But even if you would indeed have such a list...there still would be the question if infinity minus one is less than infinity.

Let's make One compressible

You're treating infinity as if it is a number, which it is not. Consequently, questions of arithmetic on infinity are nonsensical. Transfinite numbers are really their own beast and don't behave like normal integers all the time.

Now, if your question is, is there a bijection between the sets [1, 2, 3, 4...] and [2, 3, 4, 5...], then yes, I believe they are both countably infinite sets with a cardinality of Aleph Naught. 1 -> 2, 2 -> 3, etc, for infinity. You could model it with the function f(x) = x + 1.

Blessings
~ND

My point is this: The 'argument' for there even being infinite numbers or sets is that, the list of numbers or set-elements is never finished. You could Always add another number, as you've basically shown with f(x)=x+1.
So firstly, the list is never complete. Nor in the amount of elements, nor in the size of the elements, individually.
Now, the diagonal-argument says that no matter how long the list is, there is Always an element that's not in it.

The first problem with this is that the assumption that such a list could truly exist, rests on the idea of the list existing more or less as a concept. But that assumption on it's turn, rests on the idea that we know something about the nature of things like numbers or sets. But, say that we would be living in some sort of matrix type of thing, and that everything that we think exists (including numbers and sets), purely exists as some sort of simulation. In that case, there could not ever realy be such a list because it would simply be physically impossible to ever simulate it. The only thing that could exist then, is merely the thought of there being such a list.

I would say that this assumption about logic, sets or numbers is as valid as say, intuïtionism or platonism. You could even argue that this is exactly what intuïtionism, eventually leads to: concepts, thoughts and logic only exist in the physical sense, as ions or electrons that flow through membranes or wires.

But the second problem is this: the diagonal argument only works when the list is infinite and the elements on the list are infinite as well. But then it would only show that if you'd had such a list, that you could Always come-up with some new element that is not on the list.
But how is that different from saying that there is Always some element that is not in the set of all odd numbers?
How is the addition of this diagonal number different from adding an extra decimal to the numbers on the list, or and extra number on the list itself?

The reason why we call all of these sets infinite is the same..we assume that the list is never finished, that you can Always add yet another odd number/set/positive integer, etc.
And therefore no one is saying that the set of all odd numbers is smaller than the set of all numbers.

Having said all of this though....maybe when my mind is cleared from the current cannabis fog i'll come to a different conclusion.

I think you're thinking about these things in the wrong way. Pure math doesn't care, at all, whether it's structures can be represented in the 'real' world. For a lot of pure mathematicians, the total lack of application is a mark of pride. I can totally create a discrete, manipulable set with infinite elements. We'll say that N = {x in R : x > 0}. That right there is the set of all positive real numbers. Infinite, but contained.

The point of these transfinite numbers is not to describe something in the real world, but rather, see how these cognitive objects behave according to the rules laid down by the axiomatic system.

It might be impossible to create an infinitely long list of numbers in reality, but I can use the *idea* of such a list to prove that some infinite sets are larger than others, which is the gist of Cantor's famous diagonalization argument. What Cantor showed was that it was impossible to create a list of all real numbers, by creating a number that COULD NOT be on that list, thus proving that the set of real numbers was 'uncountable,' unlike the set of integers, which is countable. Your analogy with odd numbers is an inappropriate one because no one disputes that there are numbers that are not in the set of odd numbers.

The set of all odd numbers is smaller than the set of all real numbers, by the diagonalization argument. The set of all odd numbers (call it {O}) is the same size as the set of all integers (call it {I}) though, since you can create a bijection between them using a function. In this case, it would be f(x) = 2x-1 I believe. You can take any integer x (which is an element of {I} and map it to an odd number (which is an element of {O}). Becuase they have the same cardinality, the sets are the same 'size,' even though they are both infinitely large.

Blessings
~ND

I think Cantor's argument only shows that the set of real numbers is uncountable. Not that it is 'smaller'. I think that's an merely interpretation of it.

If you talk abou how cognitive objects behave according to axiomatic rules, don't you think that these cognitive objects are real in that your imagination of them is real?

So that was my point..it's not about describing something in the real world, it's about the nature of these objects.
What is the nature of cognitive objects? Aren't they supposed to be bound by the laws of physics in the sense that they would not exist without brains or computers?

If that is so, then the question whether an idea can be used to prove something about other ideas, depends very much on whether we can coherently and cosnistently represent such an idea. And infinity tends to be a tricky idea in that sense.

The reason why i used the analogy with odd numbers was exactly because no one disputes that there are numbers that are not in that set. But that doesn't mean that the set is smaller than the set of all integers.
Why not? because the concept of infinity regarding that set is of the same nature as the concept of the infinity of the set of all integers.
But does cantor's argument show that the kind of infinity of the set of all real numbers is different? It's infinite because you can Always add yet another element, and you're never finished. And considering the set of odd numbers compared to the set of all integers, the fact that you can find something that is 'missing', disputed or not, apparently doesn't seem to be of any influence here.

The sum of the set of all odd numbers < The sum of the set of all real numbers.

@dragon
I think you may be confusing summation of numbers with numbers of numbers.

EDIT
Actually come to think of it I'm not sure if (1+3+5+..n)<(1+2+3+..n)

If you do the first 8 terms, the sum of the odds is 64 while sum of the integers is 45. They both sum to infinity, but the odds seem to do it faster. I don't have a formal proof of that, but my gut and a little bit of back-of-the-envelope math suggests it's true. It doesn't make any sense to say that the sum of infinite odd numbers is greater than the sum of all the integers though, because they both sum to infinity. There's nothing they tend to.

Dragonrider: I use 'bigger' in the sense that the cardinality of the set is greater, in the same way that a set with cardinality 8 is 'bigger' than a set with cardinality '3,' even if the set w/ 8 members only includes tiny fractions whil each of the 3 members of the set w/ 3 is in the hundreds of billions. The cardinality of the integers is Aleph naught, while the cardinality of the real numbers is the power set of the integers (2^Aleph naught), and thus greater.

Cantor's usage of infinity was actually highly specific and rigorously defined in the mathematical literature. These numbers are called transfinite numbers, and wiki has a pretty good coverage of what Cantor was doing when he introduced them.
https://en.wikipedia.org...first_set_theory_article

I think we're agreeing with the odd numbers thing? Both the integers and odds have cardinality of Aleph naught? I'm not sure where the disagreement comes from.


Yes and the difference is that one is a countable infinity and one is an uncountable infinity.

I don't know much about the nature of representations of mathematical constructs. Presumably they're just instatiations of certain patterns in the brain. They're as 'real' as a house made of Legos is 'real.' They're patterns made by the arrangements of things that we ascribe meaning to.

Blessings
~ND

There is no "sum of the set of all real numbers". Not in a practical sense, but more specifically it makes no sense conceptually. The "real numbers" are essentially and inextricably tied to the continuum. Countability and discrete summation as you use them are really only applicable to set of integers. The set of integers can be mapped onto the set of real numbers, but that does not truly represent real numbers. Practically speaking, you couldn't sensibly speak of a summation of the set of rational numbers already.

When in math class, they show you how the sets of integers, rationals and reals are embedded, that looks nice, but it doesn't actually do much justice to the essential conceptual differences of these concepts.

I kind of disagree with that. If you look at the concept of 'subsets,' you can pretty clearly see that the integers are a subset of the rationals are a subset of the reals. It's true, you loose some of the nuance, but I don't think it's entirely inaccurate.

Blessings
~ND

My bad I meant to say natural numbers instead of real numbers.

Just thinking haphazardly haha, you'll have to excuse me I'm still learning.

This is more what I meant.
The sum of the ratios of the set of natural numbers to the set of natural odd numbers?
N {0,1,2..n} / O{1,3,5..n}
Element to Element I mean.

N/O = 0/1 + 2/3 + 3/5 + 4/7 + 5/9 + 6/11 + 7/12...+ n/m.

I'd imagine that number to be one of those weird decimal numbers that appear to go on forever.

Something keeps spooking in my head: despite numbering was born out of the desire to manage size, how much of numbering is solely about "size"?

Is the axioma of numbering = size?

For example if numbering indicates discrepancies in quantum states, the concept of size as we know it start to slip trough the fingers. Then the numbers start to be more connected to energy states than the literal size it occupies.

If I could squeeze out the thumb sucking reflex 'how big/much' and have them numbers cleared from connotations, they might be free to live in the abstract on their own terms. Then they don't necessarily have to make sense anymore in the real world. Then 'sensibility' can no longer be feeding a prove/disprove discourse.

Feeling puzzled about this

You could say that numbers are in a sort of superposition to borrow from particle mechanics.

1 exists, it has a finite quantity. But 1 by itself may aswell be the entire observable universe without another quantity to compare with.
1 has a quantity of n when it's relation to another element is observed.
1=n if and only if the set it exists in has the element {1} and another element {x} to compare with.

We'll have to disagree then. You can see them as sets, but that does very poor justice to their conceptual essence. To me it's like equating an electric oven and a glass of water, and calling them "things" while disregarding all differences as nuances. The glass will obviously fit into the oven, but to what practical end would you want to do that?


IMHO without the "connotation" of intended meanings, the discourse would also stop making sense. Just my opinion.

In general, talking about mathematical infinities makes little practical sense, especially when regarding them as "something very big". Mathematicians actually use these infinity concepts to express the infinitely small. In order to systematically describe the real numbers, converging sequences are used. These sequences do not ever expand into the infinite, but converge up to a limit value. This is the framework of analytical calculus.

So there. What to the uninitiated seems to be very big is actually very small.