Sorry about that.

Anyway, with infinity there are

1) cardinality

2) density

3) measure

It seems to me you have to consider all three of these in comparing uncountable to countable and all the other comparisons. Are we sure there are even only these 3 ways of assessing infinity?

It seems to me you have to consider all three of these in comparing uncountable to countable and all the other comparisons. Are we sure there are even only these 3 ways of assessing infinity? — Gregory

Oh there are lots of ways. There's the subset relation, we can say that the set of odd naturals is smaller than the set of naturals because the odds are a proper subset of the naturals. This way of thinking is incompatible with bijections, but you can use it as a definition if you like. Bijection gives you a more interesting theory so that's why it's so common.

There's natural density, that's the idea that the density of the even numbers in the naturals must be 1/2, because the limit of the number of evens in the first n naturals goes to 1/2 as n goes to infinity.

There are probably other ways.

Firstly to clarify: The whole numbers are the numbers 0, 1, -1, 2, -2, a.s.o. (and so on), and the natural numbers are the not-negative whole numbes 0, 1, 2, 3, a.s.o.

No, I don't agree with your argument. The odds numbers don't line up with the whole numbers (you say), but you say they are equal infinities. — Gregory

When did I say that the odd numbers don’t line up with the whole numbers? Never, and that’s because the not-negative odds, the naturals, the odds and the wholes do line up perfectly:
Not-negative odds: 1, 3, 5, 7, 9, 11, 13, 15, 17, ...
Naturals: 0, 1, 2, 3, 4, 5, 6, 7, 8, ...
Odds: 1, -1, 3, -3, 5, -5, 7, -7, 9, ...
Wholes: 0, 1, -1, 2, -2, 3, -3, 4, -4, ...

Therefore,

You can prove "uncountable" infinities don't line up with the whole numbers either, but maybe they are equal as well. — Gregory

does not follow from anything that I have said. In fact, “uncountable” means “cannot be aligned with the natural numbers” by definition, and “have the same cardinality” is defined to mean “can be lined up with each other”.

Until you prove that "uncountable" cannot be lined up with the wholes you haven't proven Cantor right. The diagonal shows that there are numbers not in the wholes, but there are evens not in the odds. I don't see the argument for why you can't just start at zero and line any infinity up with any other — Gregory

The diagonal proof does not show that there are numbers not it the wholes. That would be trivial. The diagonal proof shows that it is not possible to find a bijection between the naturals or the wholes and the reals, that is, it shows that you cannot line up the naturals or the wholes with the reals. By definition, that means that they have different cardinalities. More generally, it shows that there is no set for which a bijection between that set and its power set (the set of all subsets of the original set) exists.

Here is how it works. Assume that there is a bijection f from the set IN of all natural numbers to the set {0, 1}^IN of all functions from IN into {0, 1}. {0, 1}^IN is the set of all sequences of binary digits indexed by the naturals. Now we define the sequence s as the function from IN into {0, 1} which sends each natural number n to 1 – f(n)(n) (remember that f(n) is a function from IN to {0, 1}). Then for every natural number n, if f(n) = s, then f(n)(n) = s(n), which in turn implies that f(n)(n) = 1 – f(n)(n), giving f(n)(n) = ½. This contradicts the fact that f(n) sends every natural number to either 0 or 1 and never to ½. Therefore, there can be no natural number which is sent to s by f, contradicting the fact that s lies in {0, 1}^IN and the assumption that f is a bijection between IN and {0, 1}^IN. This contradiction shows that the assumption that there is a bijection between IN and {0, 1}^IN is untrue, since for any such bijection f, a contradiction follows.

There you have a beautiful proof that IN and {0, 1}^IN do not have the same cardinality. For the more general proof, you simply have to replace IN by an arbitrary set S and realize that there is a one-to-one-correspondence between the power set P(S) and the set {0, 1}^S; each member s in {0, 1}^S corresponds to the set of all members of S which are sent to 1 by s, and each subset R of S corresponds to the function from S into {0, 1} which sends every member of R to 1 and all other members of S to 0.

How much math must one know to understand this Catorian proof? It seems to me infinity is everywhere and nowhere, speaking of abstract infinity that is. You might not know how to start a bijection of the reals to the wholes, but I say start with any member, and then another and so we have bijection to 1 and 2. Send them off infinity like you do comparing whole to odd, and walla we have Aristotle's result — Gregory

And... voila, as we have shown above, Aristotle’s assumption leads to a contradiction. As Fishfry said, you can understand the above proof with almost no prior mathematical knowledge.

Mathematical points are purely conceptual entities, like justice; or fictional entities like chess pieces. — fishfry

Though I agree with you on many other points, I strongly disagree with you on this one. Mathematical points and mathematical objects in general, as well as all other abstract entities, are not conceptual at all. They are neither physical nor mental and exist independently of space and time. In fact, if anything, they are more real than any concrete entity. For example, Justice itself comes before every just individual, act and country, for if there were no Justice, nothing could be just, yet if all just individuals were slain, all just acts stopped and all just countries overthrown, Justice itself would still exist totally unaffected. In fact, without Justice itself, the very fact that all concrete just things have been destroyed couldn’t exist. Also, without Justice itself, it wouldn’t make sence to jointly call certain fellows, acts and countries just.

When frishfry asked me if it Galileo's conclusion sounded rational to me "without overthinking it", I really didn't know what to say. You have proven that the uncountable are not equal to the countable. But that doesn't prove every countable is equal to every other countable. I think we have to base this on geometry. The pineal gland has as many points as the body, but it is a lesser infinity. There is something you guys are missing in this. The bijection argument that the natural and the wholes and the odd numbers are all equal infinities forgets about density first of all, and probably many other things. I think we have to posit a "basic uncountable infinity" such that this infinity plus one is suddenly larger (by one). Lining a couple of units of infinity up proves nothing. Yes they then go to infinity, but which infinity? Again, when you compare the odd to the natural numbers you have to move the odd numbers back, disturbing the infinity in the direction to the right. It's an illegal move imo

There are many senses in which mathematicians use the term 'infinity'. But when we get to formal definitions, we distinguish among those senses.

These are basic definitions :

w = the set of natural numbers

x is finite <-> there is a 1-1 correspondence between x and a natural number

x is infinite <-> x is not finite

x is denumerable <-> there is a 1-1 correspondence between x and w

x is countable <-> (x is finite or x is denumerable)

x is uncountable <-> x is not countable

More good understanding about this and related topics is available in any introductory textbook in set theory.

x is finite if and only if there is a 1-1 correspondence between x and a natural number — GrandMinnow

I am saying that is an illegal move

Wait, let me correct. It's legal for finite, not infinite

It can't be an "illegal move" since it is merely a definition.

The best explantion (an exellent one) I have found of how mathematical definitions work is:

Introduction To Logic - Patrick Suppes

"It's legal for finite, not infinite" makes no sense. You can inform yourself about the subject with introductory textbooks.

"It's legal for finite, not infinite" makes no sense. You can inform yourself about the subject with introductory textbooks. — GrandMinnow

Can you briefly explain why it's a legal move to move 3's place to line up with 2 when it's realty in-between 2 and 4?

I guess you're referring to something you wrote previously in this thread. Whatever you have in mind, it does not contradict that definitions are not "illegal moves".

Whatever you have in mind, it does not contradict that definitions are not "illegal moves". — GrandMinnow

I think that pairing up odd numbers with natural numbers by sliding the former back is an illegal move. Prove otherwise

Define f as the function from the set of all naturals to the set of all odd naturals which sends each natural n to the odd natural 2n+1. Do you agree that this function is well-defined and a bijection, that is, do you agree that f sends each natural to exactly one odd natural, that it sends no two naturals to the same odd natural, and that to each odd natural it sends some natural?

If yes, then by definition the set of all naturals has the same cardinality as the set of all odd naturals. That’s because having the same cardinality is defined as the binary relation on sets which for all sets A, B relates A to B if and only if there is a bijection between A and B. You can easily see that having the same cardinality is an equivalence relation.

By saying that

pairing up odd numbers with natural numbers by sliding the former back is an illegal move — Gregory

, you must mean that f doesn’t exist, which you believe is not the case if you answered “yes” to my question above.

I don't see the relationship between your equation and moving all the odd numbers back without changing its infinity. I mean I get the equation, but it has no operative power to slide back the odds without consequence. The odd numbers are a specific infinity, half in density than the naturals. I am shocked professional mathematician try to compare otherwise by bijection. None of it means anything. You can't put anything you like in place of odd numbers. Maybe the naturals can be defined geometrically, so that the odd of half in density like a banana is bigger than its peel. MAYBE the peel has as many points as the banana, but it's clear which is larger, and that has more truth

The odd numbers, just to recap, are eternally half the natural numbers, as they go to infinity. Moving 3 to place 2 ect is basically just adding the even numbers to the odd, if you think of this series with your imagination, or geometrically. The argument from Cantor is that all geometrical objects have the same infinity inside them. If this is true, it upsets saying for sure that the odd numbers are half the naturals. But then I don't see a clear reason why we couldn't say, considering that a circle inside a circle has the same points within it as the outside one, why countable infinities can't be equal to uncountable. If the part can be equal to the whole, as Cantor implies, then anything seems possible. Anyone?

I don't see the relationship between your equation and moving all the odd numbers back without changing its infinity. I mean I get the equation, but it has no operative power to slide back the odds without consequence. — Gregory

We aren’t literally sliding back the odd naturals. After all, they’re abstract entitities and thus can’t be changed. What we can very easily prove is this:
1. The binary relation ~, defined for all sets A, B by
A ~ B if and only if there exists a bijection between A and B,
is an equivalence relation on the class of all sets. Cardinalities are defined to correspond to the equivalence classes generated by the relation ~. So, we can call ~ by the name “having the same cardinality”.
2. The set of all odd naturals is related to the set of all naturals by ~. This I have proved above with my function f.
Therefore, ‘sliding back the odd naturals’ has no consequece with respect to ~, that is, is keeps the set in question in the same equivalence class generated by ~. Of course, that doesn’t mean that is keeps the set in the same equivalence class generated by some other equivalence relation. For example, it obviously doesn’t keep the set in the same equivalence class generated by the equivalence relation of equality, or the equivalence relation of having the same density.

The odd numbers are a specific infinity, half in density than the naturals. I am shocked professional mathematician try to compare otherwise by bijection. None of it means anything. — Gregory

It’s similar to when I say, “The Cologne Cathedral is higher than St. Peter's Basilica”, and then you say, “No, St. Peter's Basilica is longer and wider than the Cologne Cathedral. I’m shocked professional architects compare buildings by their height. Nothing of this means anything.” Your statement about length and width is just as true as mine about height, but as it turns out, height is one of the most useful and important characters of buildings. Same with sets; cardinality is applicable to every set and, as it turns out, gives us very useful information and a rich theory, whereas some others, like measure or density, are only applicable in specific situations. Comparing sets by the subset relation is, of course, universally applicable, too, but it doesn’t give us a totally ordered hierarchy of infinities, unlike cardinality (the latter can be proved, but needs some work).
As a matter of fact, it means very much whether or not two sets have the same cardinality, th.i. (that is) are related by ~. The rich theory of cardinal numbers is proof of that.

You can't put anything you like in place of odd numbers. — Gregory

Actually, you can. Let me give you the ordered pair (IN, nf), where IN is the set of all natural numbers and nf is the successor function n -> n+1 from IN into IN, and the ordered pair (IN’, nf’), where IN’ is the set of all odd natural numbers and nf’ is the odd successor function n -> n+2 from IN’ into IN’. Then you won’t be able to decide which pair is the ‘true’ structured set of the naturals. That’s because both have exactly the same structure. So, the odd naturals actually can ‘become’ the naturals. The same is true for the other direction.

MAYBE the peel has as many points as the banana, but it's clear which is larger, and that has more truth — Gregory

Again, saying that comparing by equality and the subset relation has more truth than comparing by cardinality is like saying that comparing by length and width has more truth than comparing by height. That is not true. Both are equally valid. That the set of the odd naturals is a proper subset of the set of the naturals is equally true as the proposition that the two sets have the same cardinality.

The argument from Cantor is that all geometrical objects have the same infinity inside them. — Gregory

Actually, Cantor’s proof has nothing to do with that. He proved that every set has a strictly smaller cardinality than its power set.

all geometrical objects have the same infinity inside them. If this is true, it upsets saying for sure that the odd numbers are half the naturals. — Gregory

No, since one set can have the same cardinality as another and still be a proper subset of it and have only half its density.

But then I don't see a clear reason why we couldn't say, considering that a circle inside a circle has the same points within it as the outside one, why countable infinities can't be equal to uncountable. If the part can be equal to the whole, as Cantor implies, then anything seems possible. — Gregory

Firstly, the part cannot be equal to the whole, and Cantor doesn’t imply that. A set is never equal to any of its proper subsets. Rather, some sets (the infinite ones, and no others) have the same cardinality as some of their proper subsets. That does not mean, however, that every two infinite sets can have the same cardinality. We have proven that above. Please don’t just take my word for it, but read Cantor’s proof, which I have given above, and understand every step of it. If you believe that something in there doesn’t seem sound, please tell me. I’d be happy to clarify.
By the way, countable infinities are by definition not equal to uncountable ones. What Cantor proved is that uncountable infinities exist.

I am truly interested in learning more. Saying that infinities have two aspects, cardinality and density, confounds me. I never have rearranged the odd numbers to biject them to the naturals in my imagination. If someone picks their favorite direction ( North south east or west) and imagines the naturals going off into the horizon, he can perceive that the odd numbers forever, eternally, objectively can never pass up the naturals. "If it's true for eternity, it's true for infinity" I'm thinken. I'll look into Cantor proof more today. I need to put down the tablet for now and get stuff done

Squaring the square, putting a square perpendicular to a square, in effort selecting an inverse ratio.

If you place a dot on the page and you place a second dot, in relation to each other, one is closer to a different edge of the object. Therefore I'm saying that when placing a dot, you use squaring agility.

Isn't measuring how many infinites there are depend on the potency of infinity? It's agility?



Isn't there also thus a multi-infinity?



Infinity usually takes form of an abyss. Infinite water infinite sound. Infinite human is a proposition at a certain ratio.

I'll look into Cantor proof more today. — Gregory

I suggest first establishing a firm understanding of the axioms and rules of inference of mathematical proof and definitions - whether in formal first order predicate logic or the informal techniques that are formalized by first order predicate logic. If one understands these conventions for mathematical proof, then one sees that the proofs of, for example, these theorems are incontestably correct:

the set of natural numbers is equinumerous with the set of odd numbers
the set of natural numbers is equinumerous with the set of rational numbers
the set of natural numbers is not equinumerous with the set of denumerable binary sequences
the set of natural numbers is not equinumerous with the set of real numbers
no set is equinumerous with its power set

Of course, one may choose to reject the axioms and rules of inference upon which the proofs depend. But it is incontestable that the theorems are entailed by those axioms and rules.

Saying that infinities have two aspects, cardinality and density, confounds me. — Gregory

As I mentioned before, the predicate 'is infinite' is defined as 'is not finite'. You then objected that that is "an illegal move". I explained that definitions are not "illegal moves" and I suggested a book that has an excellent discussion of the methods of mathematical definition. I don't know whether you understand now that definitions are not "illegal moves".

Meanwhile, 'is dense in the ordering' is a separate predicate.

I never have rearranged the odd numbers to biject them to the naturals — Gregory

There is no rearrangement in the proof.

As Tristan L has so generously and perspicaciously explained, the proof adduces a one-to-one function from the naturals onto the odds, and that is all that is needed.

Do you know the mathematical definitions of 'function', 'one-to-one', and 'onto'? Understanding those basic concepts, plus some grammar school arithmetic, leads to understanding the proof easily.

As Tristan L has so generously and perspicaciously explained, the proof adduces a one-to-one function from the naturals onto the odds, and that is all that is needed. — GrandMinnow

This should put an end to the issue. But it won't. :roll:

This thread makes me sad.

|>ouglas

Zeno's paradox has for half my life made me wonder how a figure can be both infinite and finite at the same time. Latter i realized that banach-tarski 's paradox stems directly from what Cantor said about points, even if this is not usually how it is derived. Does anyone deny that for Cantor the part of a figure has as many points as the whole? You can therefore, if this is true, take an uncountable infinity out of a figure and have a new figure. My mind blows up when people don't think this strange. If this can be true, I don't see how it might not be possible for the natural numbers to be twice the odd or perhaps for all infinities to be equal. It seems to me we know to little about infinity to say anything about it considering the numerous paradoxes that arise.

That's all I wanted to say. So Douglas can maybe cheer up