Clarifying Infinity

[scientificphilosophe]

I have been challenged on another thread about the nature of infinity.
I have argued that infinity cannot be exceeded or added to.
If something is infinite it is without end in all possible directions and therefore covers everything everywhere and potentially more if the 'theoretical' can go beyond the actual.

I was challenged by reference to Hilberts Hotel

Maybe we should visit Hilbert's Hotel, to better examine this notion....

http://world.mathigon.org/Infinity

This was proposed as a mathematical proof, but to my thinking anything which argues that you can add to infinity is fundamentally flawed.

You can only have infinity or less.

If I am wrong, what is my error?

[someguy1]

You can start by answering my question about the counting numbers 1, 2, 3, 4, ... I think we'd all agree it's an infinite set. (Unless we're finitists, in which case it's an infinite collection but not a set. Or ultrafinitists, in which case there's no such thing as the collection of all the counting numbers).

Can't we add 0? In which case, didn't we just add to an infinite set? Next I'll ask about adding the negative numbers, the rationals, the reals (cardinality jump, but nobody knows how big a jump), the complex numbers, the quaternions, etc. We have an upward hierarchy of common mathematical systems of numbers. Agree? Disagree? Alternate point of view?

Hilbert's hotel is not a mathematical argument. It's a popularization of a mathematical argument.

[scientificphilosophe]

Hi someguy1

I think you pointed out the example of infinitely sub-dividing the 'space' between 0 and 1. This is a good example of illustrating that infinities apply in a direction or a dimension and that in other directions or dimensions the sequence is finite/truncated/limited.

In the case of the sequence 1,2,3 4 etc it is infinite in the positive direction and truncated in the opposite direction.
When you add zero you are not adding to the infinite but adding to the finite, because you are adding it in the limited direction. You are not touching the infinite.

The problem with Hilbert's Hotel is that it suggests you can add to the infinite.

[someguy1]

Anyone who believes they have a theory of the entire universe should be willing to educate themselves on the modern mathematical conception of infinity. You don't have to agree with these ideas, but you do need to be aware of them.

Here are some web resources, and of course you can Google others.

https://en.wikipedia.org/wiki/Cardinality

https://en.wikipedia.org/wiki/Infinity

https://en.wikipedia.org/wiki/Cantor%27 ... l_argument

https://nrich.maths.org/2756

If you have specific questions I'd be glad to answer them, but I prefer not to get into a dispute about technical material that's already accepted in standard math.

In the case of the sequence 1,2,3 4 etc it is infinite in the positive direction and truncated in the opposite direction.
When you add zero you are not adding to the infinite but adding to the finite, because you are adding it in the limited direction. You are not touching the infinite.

We can add a number at the high end too. We just go: 1, 2, 3, 4, ..., 0. All we've done here is put the 0 at the end. We can formalize this by defining a new "less than" relation on the counting numbers which is the same as the usual <, except that everything is less than 0. It's just a rearrangement, putting the same set of elements into a different order. In math this is called the order type or ordinal ω + 1, where that's the Greek letter lower-case omega.

Again, I'm only here to report what mathematicians do. There's no right or wrong to it. If you say, "I don't like it," I won't argue the point. But you can't ignore the fact that these ideas are widely accepted in modern math.

The problem with Hilbert's Hotel is that it suggests you can add to the infinite.

Hilbert's hotel is not a mathematical argument. It's a popularized account of a mathematical argument. It's like using a bowling ball on a rubber sheet to illustrate how Einstein's gravity works. It's not physics, it's a popularization.

But there's no difficulty adding an element to an infinite set. If you show me the integers (infinite in both directions in your terminology) I'll toss in a rational. If you show me the reals, I'll toss in the square root of -1. If you show me the complex numbers I'll toss in a quaternion. You can always add an element to an infinite set.

That's not a matter of opinion, it's a matter of accepted math. You are entitled to reject it, but you can't claim it doesn't exist.

Not every person needs to know about the mathematical theory of infinity. But people with an interest in putting forth their own ideas about the universe should at least have a glancing knowledge of mathematical infinity.

[scientificphilosophe]

As you say, this is what mathematicians do to avoid some awkward principles about infinity that break mathematical rules... such as infinity+1 is still infinity, not infinity +1.

You are also correct in saying that there is no right or wrong to this, as long as there are other logical interpretations that don't break the facts.
There are.
We start from the basics - infinity must be unlimited in at least one direction and in that direction it therefore can't be added to.
We then recognise that infinity may be limited 'along' or within some parameters by making them finite.
Those aspects/directions which are limited must be finite.

If infinity exists then it must have a direction towards an open/unlimited sequence. That is the direction of infinity.
You can have a 'forwards' form of infinity from a start point, or a 'negative' direction from an end point. By limiting infinity in other parameters it may be squeezed into a channel with a flow in one or mire directions.

It doesn't matter whether you add different 'dimensions' (new directions) in which to seek a new level of infinity, or close down more directions from an infinite set of directions, the basics don't change. Infinity can only exist without an end in the direction of travel.

In all of your examples you are adding to a sequence in directions that are not yet infinite.
I say again - Hilbert Hotel was presumably not pursued because is fundamentally incorrect in trying to add levels in the direction that is already infinite.

[vivian maxine]

someguy, are we adding numbers to that already infinite set? Or are we merely adding words for naming those far out numbers that we have not yet approached and had to make use of? Seems to me, that if the set is infinite, we cannot add to it. We can only start applying names farther out than we currently do.

[Lomax]

We start from the basics - infinity must be unlimited in at least one direction and in that direction it therefore can't be added to.
We then recognise that infinity may be limited 'along' or within some parameters by making them finite.
Those aspects/directions which are limited must be finite.

If infinity exists then it must have a direction towards an open/unlimited sequence. That is the direction of infinity.

Infinity does not need to have a "direction" at all. You are talking specifically about countable sets. (A sequence is different to a set, by the way.)

It's fine to have an opinion on whether there is more than one infinity but someguy1 is right: until you understand the difference between cardinality and ordinality you will never be able to persuade the mathematically informed.

[Lomax]

Anyway, the infinity of infinities is provable, and valid even within ZFC.

Cantor (or was it Frege? I forget...) noticed that the (set of even integers) and the (set of all integers) have a one-to-one correspondence:

2, 4, 6, 8, 10, 12...
1, 2, 3, 4, 5, 6...

If I may borrow from a more eloquent man:

It was formerly believed that this was a contradiction; even Leibnitz, although he was a partisan of the actual infinite, denied infinite number because of this supposed contradiction. But to demonstrate that there is a contradiction we must suppose that all numbers obey mathematical induction. To explain mathematical induction, let us call by the name “hereditary property” of a number a property which belongs to n +1 whenever it belongs to n. Such is, for example, the property of being greater than 100. If a number is greater than 100, the next number after it is greater than 100. Let us call by the name “inductive property” of a number a hereditary property which is possessed by the number zero. Such a property must belong to 1, since it is hereditary and belongs to 0; in the same way, it must belong to 2, since it belongs to 1; and so on. Consequently the numbers of daily life possess every inductive property. Now, amongst the inductive properties of numbers is found the following. If any collection has the number n, no part of this collection can have the same number n. Consequently, if all numbers possess all inductive properties, there is a contradiction with the result that there are collections which have the same number as a part of themselves. This contradiction, however, ceases to subsist as soon as we admit that there are numbers which do not possess all inductive properties.

Anyway, here's where it gets interesting: suppose an infinite sequence (with a given first term) of binary numbers. We can write this out as:

S1: 0,0,0,0,0,0,0,0,0,0...

Suppose another. And another. Make a few. Write them out together. For instance:

S1: 0,0,0,0,0,0,0...
S2: 1,1,1,1,1,1,1...
S3: 1,0,1,0,1,0,1...
S4: 0,1,0,1,0,1,0...
S5: 1,1,0,0,1,1,0...
S6: 0,0,1,1,0,0,1...
S7: 1,1,1,0,0,0,1...

And so on. Now take the first term from the first row, the second term from the second row, and so on:

S1: 0,0,0,0,0,0,0...
S2: 1,1,1,1,1,1,1...
S3: 1,0,1,0,1,0,1...
S4: 0,1,0,1,0,1,0...
S5: 1,1,0,0,1,1,0...
S6: 0,0,1,1,0,0,1...
S7: 1,1,1,0,0,0,1...

Make the highlighted terms into a new sequence (in this example: 0,1,1,1,1,0,1...), then make each one of them its opposite (so: 1,0,0,0,0,1,0...). Call this new set S0.

S0 is therefore different to every set we already had. (To explain why: suppose some set Sn. The nth term of Sn will be different to the nth term of S0, because that is how we deliberately constructed S0). Add S0 to your list of possible sequences and repeat the process, and you will always yield a new infinite set that you didn't already have on your list*.

This is called Cantor's "diagonal argument". What it shows us is that, not only is infinity as infinite as it seems, it is more so; there are an infinity of infinities.

Lomax

___

* don't actually do this, it would literally take forever. Just get a monkey and a typewriter.

[someguy1]

I'm not sure how to respond to this. I offered to discuss mathematical infinity. You have a personal theory, which is fine; but your theory makes up terms without defining them and doesn't make much sense to me. This forum has a Personal Theories section, which is where personal theories go. I see no discussion of mathematical infinity here.

I'll offer some specific comments here, but if you desire to learn about mathematical infinity you should start with the links I gave earlier. Else there's no place to go with this.

As you say, this is what mathematicians do to avoid some awkward principles about infinity that break mathematical rules... such as infinity+1 is still infinity, not infinity +1.

I don't recall saying anything like that. In any event, in the transfinite ordinals, adding to an infinite ordinal sometimes gives you a new one. You should add this to your reading list.

https://en.wikipedia.org/wiki/Ordinal_number

You are also correct in saying that there is no right or wrong to this, as long as there are other logical interpretations that don't break the facts.
There are.

I have no problem with this in theory. I don't think you've presented much of an alternative.

We start from the basics - infinity must be unlimited in at least one direction and in that direction it therefore can't be added to.

I have three separate problems here.

* Basics of what? Certainly not the basics of the mathematical theory of infinity. Rather, the basics of your own personal theory. In which case this thread goes in Personal Theories.

* You are confusing boundedness with cardinality. There are infinitely many real numbers between 0 and 1, but they form a bounded set. You can draw a finite circle around them. In fact a circle contains finite area, goes in every possible direction, yet contains infinitely many points.

* You can certainly add to a "directional" infinity as you call it. Consider the counting numbers 0, 1, 2, 3, ... Now just rearrange them in this new order: 1, 2, 3, ..., 0. The second order represents a different ordinal number than the first. You can see that because there's an order-preserving bijection between 0, 1, 2, ... and 1, 2, 3, ...; but in the second case we have 0 left over at the end. We DID just add an element at the unbounded end of an infinite set. Ordinals are really cool, it's too bad people don't hear about them much.

We then recognise that infinity may be limited 'along' or within some parameters by making them finite.
Those aspects/directions which are limited must be finite.

I don't know what that means. I think you're confusing boundedness with cardinality, or with order, or with something.

If infinity exists then it must have a direction towards an open/unlimited sequence.

Evidence? Examples? What about the infinite set of points between 0 and 1? What about a circle in the plane?

That is the direction of infinity.

You'll need to offer a formal definition else you're making word salad. "Direction of infinity?" That's not defined by you at all.

If you have vague ideas in your head, you can read Wiki and ask questions online to help clarify your ideas. There's little point in doubling down on your own confusions.

You can have a 'forwards' form of infinity from a start point, or a 'negative' direction from an end point. By limiting infinity in other parameters it may be squeezed into a channel with a flow in one or mire directions.

Squeezed into a channel with a flow? Does this word salad come with croutons?

It doesn't matter whether you add different 'dimensions' (new directions) in which to seek a new level of infinity, or close down more directions from an infinite set of directions, the basics don't change. Infinity can only exist without an end in the direction of travel.

You're just making stuff up. No law against it, but you're not giving me anything sensible to work with.

In all of your examples you are adding to a sequence in directions that are not yet infinite.

If 0, 1, 2, 3, 4, ... isn't infinite, I don't know what is. I already showed how to add something to the unbounded end.

I say again - Hilbert Hotel was presumably not pursued because is fundamentally incorrect in trying to add levels in the direction that is already infinite.

When the bowling ball distorts the rubber sheet, what pulls the bowling ball down? Meta-gravity? No, it's that the rubber sheet model is a fable, a simplified popularization. Same with the Hilbert hotel.

[someguy1]

Anyway, the infinity of infinities is provable, and valid even within ZFC.

Cantor (or was it Frege? I forget...) noticed that the (set of even integers) and the (set of all integers) have a one-to-one correspondence:

2, 4, 6, 8, 10, 12...
1, 2, 3, 4, 5, 6...

Galileo, although what's now called Galileo's paradox was already known in the early middle ages.

https://en.wikipedia.org/wiki/Galileo%27s_paradox

Today we take the property of a set being in bijection with a proper subset of itself as one of the definitions of an infinite set.

[someguy1]

someguy, are we adding numbers to that already infinite set? Or are we merely adding words for naming those far out numbers that we have not yet approached and had to make use of? Seems to me, that if the set is infinite, we cannot add to it. We can only start applying names farther out than we currently do.

Interesting question. Suppose I have the integers, ..., -3, -2, -1, 0, 1, 2, 3, ... Surely this is an infinite set. It's infinite in "both directions" as even the OP would agree. A more precise phrase is that it's unbounded in both directions.

Now suppose we want to add in the rational numbers like 1/2, 3/47, -97/12, etc. Would you say we are just adding words?" Or are we finding one infinite set properly contained within another?

After all, aren't there infinitely many even numbers? And don't they live inside the set of all whole numbers, which live inside the set of all rational numbers, which live inside the set of all real numbers?

Whether you consider this math or word games is a matter of personal viewpoint. You wouldn't be the first to suggest that math is really only word games. Philosophically there is something to that point of view. After all, we use finite strings of symbols to talk about infinity.

Do any of these examples help clarify your question? It's a good question, but mathematically it's clear that infinite sets can live inside other infinite sets.

[scientificphilosophe]

Lomax » April 13th, 2017, 5:12 pm

Infinity does not need to have a "direction" at all. You are talking specifically about countable sets. (A sequence is different to a set, by the way.)

It's fine to have an opinion on whether there is more than one infinity but someguy1 is right: until you understand the difference between cardinality and ordinality you will never be able to persuade the mathematically informed.

Hi Lomax

I can't claim to be a mathematician - as you've probably guessed - but I don't think my comments confuse cardinality and ordinality. Neither do I say that infinity has to have a direction, but where one is imposed then the direction should be recognised.

The full set of positive and negative integers will not be limited, but if you begin a sequence from zero and then just proceed with the positive sequence, you will have established a direction. If you move along the sequence away from 0 then it will be unlimited, but if you move towards zero it will be finite as it will not extend beyond 0.

If we look at Time running up to the present moment then it is finite in the 'forwards direction' because we cannot go further than the current moment, but if we look back into the past then it is potentially unlimited/eternal in that direction.

If you sub-divide the gap between 0 and 1 you are effectively establishing a new dimension for the numbers because you are no longer dealing with integers, and in the 'direction' of that new dimension you can find another infinity.

At headline level I agree with the infinity of infinities, but I also recognise that people or circumstances can limit an infinite sequence - thereby establishing a direction

[scientificphilosophe]

Someguy

Please read my last reply to Lomax. That will set a context.

Your examples do set limits because you start from zero and do not move in the negative direction.

You talk about re-arranging sequences but I don't see how that adds anything if you are just dealing with the same numbers, and not going negative. I think you also have to define what this re-arrangement of numbers represents if it is to reflect anything real. I have given examples with Time etc. - what are yours?

Sub-dividing adds a different dimension to the sequences you were originally working with.
I don't have a problem with an infinite set of sub-divisions - but it isn't the same set-up that we started with - Integers. You are very ford of changing the parameters of your examples. Deal with the basics.

If you start from zero you have applied a restriction/limit/direction.

I honestly don't see the relevance of your circle other than another opportunity for sub-division. Was that it?

A circle/ring (2 dimensions) is only infinite in its potential for sub-division - a direction - in other respects it is finite - as you said yourself. I think you missed my point when you try to say that a circle covers all directions.

[someguy1]

I'm replying here to your reply to Lomax, which you asked me to read.

I can't claim to be a mathematician - as you've probably guessed - but I don't think my comments confuse cardinality and ordinality.

Indeed they do, as we shall see.

Neither do I say that infinity has to have a direction, but where one is imposed then the direction should be recognised.

The fundamental entity is the set. Sets are disorderly collections of things. No order, no structure, no arithmetic, etc. All those other things are imposed on top of sets.

So if we have an infinite set, it doesn't have an order unless we give it one. And if we give a set an order, we can instead give it a different order. The same set may have may different orders put on it.

Think of it as a classroom full of unruly kids running around. That's a set. Now we tell them to line up in order of height. That's one way to order a set. Then we ask them to line up in alphabetical order of last name. That's the same set with a different order. Then we ask them to line up by reverse-alphabetical order of middle name. That's yet another example of the same set with a different order. The raw set consisting of 47 screaming kids is a cardinal concept. "How many." When we put an order on the kids, it's a different ordered set.

The full set of positive and negative integers will not be limited, but if you begin a sequence from zero and then just proceed with the positive sequence, you will have established a direction.

You've taken a set (cardinal) and imposed an order on it (ordinal). We could order the natural numbers in the usual way 1, 2, 3, 4, ... Or we could order them in reverse: ..., 4, 3, 2, 1. Or evens before odds. Or primes before composites. There are lots and lots of ways to order the natural numbers. They are all the same underlying set but they are all distinct ordered sets.

If you move along the sequence away from 0 then it will be unlimited, but if you move towards zero it will be finite as it will not extend beyond 0.

In that particular order.

If we look at Time running up to the present moment then it is finite in the 'forwards direction' because we cannot go further than the current moment, but if we look back into the past then it is potentially unlimited/eternal in that direction.

I don't know anything about time. That's physics. One of the biggest sources of confusion is to confuse math with physics. We are just talking about sets, their cardinality, and the many different ways we can impose order on them.

Models of time are really something else entirely. You can't reason about the nature of time by contemplating the usual order on the integers. Mathematically, one order is just as good as another and none of them are necessarily the way the universe works.

In fact when people start talking about the nature of time, and using as evidence the standard order on the natural numbers, I'm immediately reminded of the religious sophistry of William Lane Craig and his Kalam cosmological argument. It sets my alarm bells flashing.

If you sub-divide the gap between 0 and 1 you are effectively establishing a new dimension for the numbers because you are no longer dealing with integers, and in the 'direction' of that new dimension you can find another infinity.

That statement I don't understand. Can you clarify?

At headline level I agree with the infinity of infinities, but I also recognise that people or circumstances can limit an infinite sequence - thereby establishing a direction

"Establishing a direction" is precisely what we mean by imposing an order on a set.

I'll leave your other post for a little later, since I think this post of yours raised a lot of important issues.

The moment you talk about "direction" in a set, you are talking about order properties.

[scientificphilosophe]

Someguy - If you are so isolated from the world that you don't understand the basic concept of Time that we experience then I can't explain it to you.

Equally, both cardinality and ordinality are present in each mathematical debate - that doesn't mean they are confused.

To say that you can simply re-order numbers in a set with impunity is possibly valid if you are using those numbers as a code or a set of names which are equal - but to deny the intrinsic value of each integer seems ridiculous. Those numbers are not equal, and their inherent value determines their position in the sequence.

If you do deny the natural order then mathematics has no rationale.

By limiting your set to positive integers only you are placing an order on them because you are limiting that set in one direction only. That limitation sets a boundary and therefore a direction.

On your point about fractions - eg. the space between 0 and 1 - they are not integers, but they may have their value place within the natural order. By not being integers they are not part of the integer set. If you wish to position them within the integer set then must represent something else - a different dimension of numbers. I don't think that's a hard concept.

If you operate in a different dimension then you have a new direction to explore/travel in.

That is my point - what's yours?

[someguy1]

If you operate in a different dimension then you have a new direction to explore/travel in.

That is my point - what's yours?

To be perfectly honest I no longer understand what you're talking about. You initially had some questions about Hilbert's hotel and I see that I haven't been very helpful. Your most recent post seems rather far afield from anything I can understand. You might read up on ordinal numbers to see how we can put different orders on the same set.

If you can frame some specific questions I might be able to answer them.

Let me go back to your first post.

This was proposed as a mathematical proof, but to my thinking anything which argues that you can add to infinity is fundamentally flawed.

You can only have infinity or less.

If I am wrong, what is my error?

Would you agree that the set of even positive integers is infinite? This is the set {2, 4, 6, 8, 10, 12, 14, ...}.

Now what if I add in the odd numbers? I've added to an infinite set, right?

[scientificphilosophe]

If your set is of even numbers only then odd numbers can't feature in that set.
Even numbers do continue to infinity.
If you add odd numbers it is a different set - it is no longer a set of even numbers but a set of all integers.

When you define a set by excluding a particular category of number, that is part of the definition of your set.
You can say that it is infinite within those parameters and you can't add to the set within those parameters..
When you change the definition of your set it is a new set.

They are apples and pairs.

[Lomax]

To say that you can simply re-order numbers in a set with impunity is possibly valid if you are using those numbers as a code or a set of names which are equal - but to deny the intrinsic value of each integer seems ridiculous. Those numbers are not equal, and their inherent value determines their position in the sequence.

This belief is your confusion between cardinality and ordinality.

[Heavy_Water]

I agree that Infinity cannot be added to.

That's why, uh, it's called Infinity.

LOL

Tis a pity the homo sapien mind is totally incapable of truly understanding the term Infinity.

Just as we cannot fully grasp the term "Nothingness."

The latter fact is why the godists invent, well, their gods.

And their Afterlifes.

Their Heavens.

To close, I once heard a nice and imagery-pleasing description of infinity. I really like it, and if you sit and take a moment and try to imagine it, you SORT OF get an inkling of the concept of Infinity. Though, as I said, we can't ever really get it. Since we of course have no experience with it during our short lives.


Anyway, here it is. Sorry that I cannot recall the author. I wish I had invented it! But, alas, no...........


"Imagine that the moon is not made of rock, but instead of dense and solid black iron. Then let us see a butterfly flutters away from the Earth, and navigate through space, traversing the entire 240,000 miles to this Iron Moon.

The Butterfly then flies around the moon once, As he does, his inside wing juuuust barely brushes the Iron Moon.

The Butterfly then begins his journey back to Earth. When he touches down on Earth, he immediately begins another trip to the Iron Moon. And again his wing brushes is as he orbits it. And he again returns to Earth.

He continues this process. Over and Over.

When the Iron Moon has been reduced to the size of a marble, from the countless brushes of the Butterfly's wing.......

INFINITY HAS JUST BEGUN."



Cheers.

[someguy1]

They are apples and pairs.

That's a pretty good pun, even if unintentional. Although in this case we're not using the pairing axiom, but rather the axiom of unions.

https://en.wikipedia.org/wiki/Union_(set_theory)

And speaking of pears ... what's yellow and equivalent to the axiom of choice?

Zorn's lemon.

[someguy1]

If your set is of even numbers only then odd numbers can't feature in that set.
Even numbers do continue to infinity.
If you add odd numbers it is a different set - it is no longer a set of even numbers but a set of all integers.

But by that logic you can never add a number to a finite set either, because it would become a different set. So a finite set is no different than an infinite set in this regard. Right?

[scientificphilosophe]

Apologies for the delay in replying - I've been travelling for a few weeks.

It was a deliberate pun - but unfortunately my smileys don't seem to work on this site. Glad you liked it though!

Re: adding to a finite sequence - yes of course you can add to it. It only becomes a different set if it breaks your definition of the set.

[someguy1]

Re: adding to a finite sequence - yes of course you can add to it. It only becomes a different set if it breaks your definition of the set.

You've substituted the word sequence for the word set. Let me clarify these notions.

A set is a collection of elements. That's all it is. A set is completely characterized by its elements. So the sets , , and are three representations of the exact same set.

A sequence is a function from the counting numbers to some set. For example is a sequence of rational numbers.

As a set, they could be listed in any order.

Now, a set need not have any particular "rule" associated with it. Likewise a sequence might have a rule but it might not. There are complete random sequences and sets that can not be described by rules.

In other words there is no such thing as a "definition" of a set in every case. Some sets can be described by a rule; and some sets can be described only by listing all their elements. And there are some sets whose existence is stipulated by the axioms of set theory, but whose elements can't be identified or listed or described at all.

Now when we say we want to "add something to a set," we are actually engaged in abuse of terminology. If we have the set and we "add an element" to create the set , we've created a completely different set.

What we are actually doing is making a NEW set from the original one by unioning it. In other words, .

Now this is true of finite sets and it's true of infinite sets.

Your original point was that we can't add an element to an infinite set. I pointed out that from a strictly pedantic point of view, we can't add an element to a finite set either.

But if we have the set of even numbers, an infinite set, we can certainly "add" an odd number to make a new set. Just as we can add an element to a finite set to create a new set.

Hope some of this is clarifying.

[scientificphilosophe]

Someguy1

If you define a set by a sequence of elements in order to make your point, then the sequence is part of the set definition. Eg. a set defined as 'all positive even integers' cannot be added to by adding a negative number, odd number, or a fraction - it would break the set definition. Neither can you add another positive even number if the set is already fully populated because it is an infinite sequence and all bases are already covered. It doesn't matter what order you place those numbers in.

If you define the set as being 'any number' but the set is only partly populated, (say - there are currently only even integers in there), then of course you can add to this finite/limited population. This seems to be the scenario you have been deploying.

But you cannot add to an infinite sequence that is already fully populated.

An infinite sequence may start somewhere and continue to infinity, which means that the potential set is only partly populated. In that 'infinite direction' you cannot add to it. The only way to add more to the population is at the start point - the finite end.

For example... if your set definition is all even integers but the population within the set only starts from 6 and then continues in the direction of 8, 10 etc. to infinity, you can only add to it by adding 4 or 2 or some negative even numbers... the opposite direction of travel.

I don't see why this is so difficult or deniable, and I don't understand your points when placed in this context.

[TheVat]

But you cannot add to an infinite sequence that is already fully populated.

How can an infinite sequence achieve full population? Isn't the very definition of infinite that it can be added to forever?

[someguy1]

Let me respond to your last point first, since it puts the rest in contrast.

I don't see why this is so difficult or deniable, and I don't understand your points when placed in this context.

There are two versions of set theory being discussed. I'm discussing the standard, official view of elementary set theory as agreed to by virtually all mathematicians of the last hundred years or so.

You're talking about your own private conception of sets, which is very much different than the standard version.

There is nothing wrong with you inventing your own private theory of these matters. But if so, the burden is on you to clarify your conceptual framework and state any axioms you assume.

However I suspect something else is going on. I believe that you think you understand basic mathematical set theory, and that we're having a disagreement about how standard mathematical sets work. In this you are wrong. But this is exactly why you can't understand what I'm saying. You think we are disagreeing about how sets work; when in fact I am explaining to you how sets work.

In any event let me give an example to illustrate what it means to add an odd number to a set of even numbers.

Suppose we form the set .

In set theory we are allowed to form the union of two sets. A set union is like taking a can of carrots, and a can of peas, and pouring their contents into a bowl. Now we have a bowl of carrots and peas. Nothing could be simpler or more familiar to everyday experience.

Now if I have the number , say, which we all agree is an odd number, there's an axiom of set theory that allows me to form the singleton set ; and then the union .

There is absolutely no rule or principle in set theory that restricts the sets that E may be unioned with. The axiom of unions says that we may union ANY sets together.

This is basic set theory. If we disagree on it in some way, let's stop here and see this example through.

So tell me, do you have any disagreements or issues with what I just did? Or is it clear to you?

I also want to mention that sets inherently have no order; and "sequence" is an association of each of the natural numbers with a particular element of some given set. You've been using "set" and "sequence" interchangeably and that is sometimes causing confusion.

[someguy1]

But you cannot add to an infinite sequence that is already fully populated.

How can an infinite sequence achieve full population? Isn't the very definition of infinite that it can be added to forever?

We cannot add any elements to the set of even natural numbers. We can UNION this set with a set of odd numbers. But the original point I made is still worth being pedantic about. Strictly speaking we can never "add an element" to an existing set, since a set is characterized exactly by its elements.

All we can do is union an existing set with some other second set to form a third set. [Or union together an infinite collection of sets to form a new set that's distinct from each original set].

In this sense I agree with the OP's point of view here. We really can't add anything to the set of even numbers. We can only union it with another set to make a brand new set. But we CAN do that latter procedure, always, which is the point you were making. We can always union a set with some other set. That's the axiom of unions.

This goes for sequences too. If we have the sequence we can stick the number in there anywhere we like, but then we'll have a different sequence that happens to have the same limit.

[scientificphilosophe]

How can an infinite sequence achieve full population? Isn't the very definition of infinite that it can be added to forever?

No it isn't. Infinity means that all numbers per the definition (eg. all even numbers) are already present without end. There is no add-to.

If there was a capability to add, the sequence wouldn't be complete and therefore it would be finite. You can only add to the finite, not the infinite.

[scientificphilosophe]

Hi Someguy1

As I said at the start, I am not a mathematician, and therefore not skilled in the definitions you apply but on the basis of what you said, I have no problem with the idea of a union between two sets - or any sets.

I have taken the view that if limits are applied to a set, then that will impose a boundary and direction which defines a particular type of infinity. If we say that a set could be populated by any integer plus any fraction, but it is currently only populated by positive even numbers to infinity, that marks a boundary, and in my terminology, a direction.

We can agree that we can only add to the contents of the set with numbers that are not already present. In the above example where a defined sequence has been deployed, the add-to elements must fall outside the sequence ie. with negative even integers, zero, odd numbers, or fractions.

Yet I do see your point. The way I would have phrased it is like this.
If the set definition is 'any number', and we have partially populated the set with an infinite quantity of random numbers, we will be able to add to that infinite population by filling in the infinite number of gaps from the 'complete' set.

We have previously been defining either the set, or the population of the elements which are present in the set, by a sequence. To that extent it is easy to determine boundaries. But in the case of adding to an infinite set of random numbers, then I agree any number which was introduced would seem to be adding to an infinite population.

The underlying question is how you would define a random sub-set of numbers. Can we say that it would be any number not yet present? If so, does that give us a sense of direction?

Put another way, does in-filling represent a boundary of sorts, something finite, if the infinite random sequence is without end? You couldn't add to the end of the sequence... if you see what I mean.

[Lomax]

How can an infinite sequence achieve full population? Isn't the very definition of infinite that it can be added to forever?

No it isn't. Infinity means that all numbers per the definition (eg. all even numbers) are already present without end. There is no add-to.

If there was a capability to add, the sequence wouldn't be complete and therefore it would be finite. You can only add to the finite, not the infinite.

BiV, I'll just add that you're wiser not to get your definitions of these complicated things from somebody who (I'm sorry to say) only pretends to understand them. A set can perfectly well be infinite without including every possible number. For the same reason it doesn't have to be "complete" either, since mathematical completeness is the property of having no "gaps" (for example, the set of all numbers between 1 and 2 is both infinite and "complete". The set of all integers is infinite and "incomplete").