Controversial concepts in mathematics

[Magnus_Anderson]

@Magnus Anderson

[T]he sequence of natural numbers ((1, 2, 3, \dotso)) is endless because there is no place within it with the highest index. And if we index the elements of this sequence using the set of natural numbers (like so: 1->1, 2->2, 3->3, and so on), there will be no number that will be indexed as infinity-th (occupying position with an index of infinity.) If you want to do something like that, you’d have to either increase the number of elements in the sequence (such that the set of natural numbers becomes insufficient to count all of the elements) or count beyond that sequence. Are you saying that such a thing is impossible? And if the answer is “yes”, why do you think it is?

@Observer

Why do I think the natural numbers have no highest value? - Seems like a silly question. No matter what number is assumed the highest, more can be added. When did that come into question?

There is no largest natural number. I never said such a thing. Where did you get that from?

I take it that you agree with the following statement of mine:

[T]he sequence of natural numbers ((1, 2, 3, \dotso)) is endless because there is no place within it with the highest index.

And I also take it that you agree with the following:

And if we index the elements of this sequence using the set of natural numbers (like so: 1->1, 2->2, 3->3, and so on), there will be no number that will be indexed as infinity-th (occupying position with an index of infinity.)

But what about the following statement?

If you want to do something like that, you’d have to either increase the number of elements in the sequence (such that the set of natural numbers becomes insufficient to count all of the elements) or count beyond that sequence.

It doesn’t seem like you understood it. I’ll try to explain it.

We have a sequence of natural numbers (S = (1, 2, 3, \dotso)).

Let the elements of this set be indexed like so: first → 1, second → 2, third → 3 and so on.

We can both agree that there will be no element (X) that will be indexed like so: infinity-th → X.

I think that we agree so far.

But let’s add another element to our set. Let this element be (a). What we have now is (S = (1, 2, 3, \dotso, a)).

Let’s say that the word “infinity” represents the number of natural numbers.

What follows is that (S) now has an element whose index is infinity. In other words, there’s now “place at infinity” within this sequence. And this place is occupied by (a).

Does that make sense? If so, do you disagree? If so, what do you disagree with?

[Magnus_Anderson]

@obsrvr524

An infinite SUM is not a sequence. We use sequences to see where the sum would end up IF we could do an infinity of additions. The term “infinite sum” means “the sum of the infinite set” - a single quantity.

An infinite sum is a particular type of representation of quantities. (0) and (0 + 0 + 0 + \dots) represent one and the same quantity but the first is not an infinite sum whereas the second is. An infinite sum consists of an infinite number of terms. You can think of these terms as belonging to a set but you can also think of them as belonging to a sequence (but since addition is commutative sequences aren’t the simpler way to do so.)

[Magnus_Anderson]

@Ecmandu

You’re going to go down the “your own definitions” track with people?

Don’t worry that I invented something… just look at the logic of it.

Like seriously, people can’t figure out shit nobody thought of before?

This thread is about how OTHER people define the word “infinity” and not about how YOU define it. Thus, whatever insights you arrive at by employing deductive reasoning, they will be irrelevant to the extent they are based on YOUR definition of the word “infinity” and not the standard one.

I don’t think there is ANYTHING about the standard definition of the word “infinity” that implies that “infinity 5” means “5 lasts forever”. That’s entirely your own creation.

[obsrvr524]

@Magnus Anderson

@obsrvr524

An infinite SUM is not a sequence. We use sequences to see where the sum would end up IF we could do an infinity of additions. The term “infinite sum” means “the sum of the infinite set” - a single quantity.

An infinite sum is a particular type of representation of quantities. (0) and (0 + 0 + 0 + \dots) represent one and the same quantity but the first is not an infinite sum whereas the second is. An infinite sum consists of an infinite number of terms. You can think of these terms as belonging to a set but you can also think of them as belonging to a sequence (but since addition is commutative sequences aren’t the simpler way to do so.)

(0 + 0 + 0 + \dots) - that - is an infinite SERIES. It has a final sum which is the “sum of an infinite series”.

Some people might say, “infinite sum” to mean the sum of an infinite series. It’s shorter. But it is misleading and technically wrong.

Remember you asked how I define those words - not you.

[obsrvr524]

Sorry I missed this -

@Magnus Anderson

But let’s add another element to our set. Let this element be (a). What we have now is (S = (1, 2, 3, \dotso, a)).

Let’s say that the word “infinity” represents the number of natural numbers. [wrong word but ok]

What follows is that (S) now has an element whose index is infinity. In other words, there’s now “place at infinity” within this sequence. And this place is occupied by (a).

Does that make sense? If so, do you disagree? If so, what do you disagree with?

No. When you added the “a”, you started a new sequence. “a” does not represent the next higher number - only the next item that cannot be indexed with any number unless you start with a new sequence of index numbers - “a1 , a2, a3, …” then “b1, b2, b3…”.

You have a split index.

And that is the same as my example of the infinite line -
<----------------------->

But then an extra point added above it -
[list].[/list:u]<----------------------->

The infinite line requires a different index marker to accommodate greater than an infinity of indexe numbers.

[Magnus_Anderson]

@obsrvr524

(0 + 0 + 0 + \dots) - that - is an infinite SERIES. It has a final sum which is the “sum of an infinite series”.

Some people might say, “infinite sum” to mean the sum of an infinite series. It’s shorter. But it is misleading and technically wrong.

That might be true but it doesn’t seem to be a relevant correction (it’s more of an effort to help me better express myself.) The point is that what we’re dealing with here is an expression involving an infinite addition of terms that exists outside of time rather than inside of time. What I am asking is “What does it mean for such an expression to never stop?” To say that such an expression stops or that it never stops is a figurative use of the word “stop” since the word “stop” can be literally applied only to things that exist in time. Since such an expression can be represented as a sequence of terms (not merely as a set of terms), the question can be reframed as “What does it mean for a sequence to have no end?” That’s the question I asked you, right? And it is better to ask Silhouette the same question if it’s possible to do so. So that’s what I did.

@Magnus

Let’s say that the word “infinity” represents the number of natural numbers.

@Observer

[wrong word but ok]

Actually, the truth value of that statement can be barely contested (: That’s because I provided a PROVISIONAL definition – basically my own – to be used for the duration of that particular argument I presented. I did NOT claim that’s the official definition. (The official definition, I believe, is more general than that.)

No. When you added the “a”, you started a new sequence.

We didn’t add (a) to a different sequence (one that was moreover empty.) Rather, we added it to the existing sequence. Thus, we didn’t really start a new one.

“a” does not represent the next higher number

(a) is not a number at all, it’s a letter. It’s a letter that we added to a sequence of natural numbers. The index of the place it occupies in the sequence, however, is a number and that number is basically the number of natural numbers. (The only thing I am not entirely sure about is whether that number is the number of natural numbers OR the number of natural numbers plus one. The reason being that the index of the first place in the sequence is (1) rather than (0). In fact, I am far more inclined to believe its actual index is infinity plus one.)

By the way, assuming that this can more easily change your mind, James had no problem using infinity as an index. (There was a discussion between him and Carleas on whether or not the set of even natural numbers is in one-to-one correspondence with the set of natural numbers – or something like that.)

[Magnus_Anderson]

One has to choose the method of indexing.

There are two methods that immediately come to mind:

1. the method where the index of any element is a number that is equal to the number of elements that come before that element

2. the method where the index of any element is a number that is equal to the number of elements that come before that element plus one

If we choose the first method, the index of the first element would be (0). The index of (a) in ((1, 2, 3, \dotso, a)) would then be the number of natural numbers (we can denote this with (\infty)).

If we choose the second method, the index of the first element would be (1). The index of (a) in ((1, 2, 3, \dotso, a)) would then be the number of natural numbers plus one (we can denote this with (\infty + 1)).

[obsrvr524]

@Magnus Anderson

The point is that what we’re dealing with here is an expression involving an infinite addition of terms that exists outside of time rather than inside of time. What I am asking is “What does it mean for such an expression to never stop?” To say that such an expression stops or that it never stops is a figurative use of the word “stop” since the word “stop” can be literally applied only to things that exist in time. Since such an expression can be represented as a sequence of terms (not merely as a set of terms), the question can be reframed as “What does it mean for a sequence to have no end?” That’s the question I asked you, right? And it is better to ask Silhouette the same question if it’s possible to do so. So that’s what I did.

Technically the word “stop” does imply time so a process stops or comes to an end - ends. But if you are talking strictly about the concept, void of time issues, the end merely means a highest number or position in the sequence above which there are none. Obviously that doesn’t apply to an infinite set or series because it doesn’t have an end point above which there is none.

I don’t understand why that is not clear.

@Magnus Anderson

@Magnus

Let’s say that the word “infinity” represents the number of natural numbers.

@Observer

[wrong word but ok]

Actually, the truth value of that statement can be barely contested (: That’s because I provided a PROVISIONAL definition – basically my own – to be used for the duration of that particular argument I presented. I did NOT claim that’s the official definition. (The official definition, I believe, is more general than that.)

And I “barely” did contested it.
I said, “but ok”.
And if you had said, “For this discussion let’s say that infinity refers to the number of naturals” I would have only said “ok”. But now we are clear.

And since we both know what “infA” means - why not just used that.

@Magnus Anderson

No. When you added the “a”, you started a new sequence.

We didn’t add (a) to a different sequence (one that was moreover empty.) Rather, we added it to the existing sequence. Thus, we didn’t really start a new one.

If you are using the 1to1 correspondence, you have to use a different sequence because all of your naturals are used up.

@Magnus Anderson

“a” does not represent the next higher number

(a) is not a number at all, it’s a letter. It’s a letter that we added to a sequence of natural numbers. The index of the place it occupies in the sequence, however, is a number and that number is basically the number of natural numbers. (The only thing I am not entirely sure about is whether that number is the number of natural numbers OR the number of natural numbers plus one. The reason being that the index of the first place in the sequence is (1) rather than (0). In fact, I am far more inclined to believe its actual index is infinity plus one.)

If you are free to change your index system starting at zero then you can change it to start at -2 and also add “b” and “c”. It is the same difference - use a1, a2, a3,… and b1, b2, b3… or use 1, 2, 3,… and -1, -2, -3. It doesn’t matter. All I said was that you have to modify your index method - and you did.

@Magnus Anderson

By the way, assuming that this can more easily change your mind, James had no problem using infinity as an index. (There was a discussion between him and Carleas on whether or not the set of even natural numbers is in one-to-one correspondence with the set of natural numbers – or something like that.)

Yes, I read that. He also had a long discussion on a different board concerning hyperreals and that 1=0.999… issue.

I don’t have an issue with the infA+1 notation. My issue is that when defining the terms you asked about like “largest number”, “infA+1” does not represent a natural number. It is a greater quantity than merely infA but there is no natural number that can represent it. That is why it has its own word.

James specifically stated that you have the freedom to set a chosen standard and must do so if you are going to add or do maths with infinity. I can see that makes sense. And as long as you make it clear that you have, it isn’t a problem.

On that other board (“Rational” something) they were telling James that he “doesn’t know shit” and that he should just accept what he is told (definitely not James’ style - much like Mr Trump in that regard). But then James pointed out that what was told back in 1947 or so by a Hewitt somebody is exactly what he was saying merely in different notation. James was explaining, with many indisputable arguments why he, that Hewitt bloke, and others was right. James was big on the “why” questions - to an extraordinary extreme. If you want to know the “why” behind just about anything - study up on James - maybe get that book Mithus published.

James understood, as do I now, that ALL of these confusions are merely about the words not being carefully defined. Once anyone carefully defines/explains the words all of these issues go away. The same is true concerning laws and politics but no one seems to want to fix that either (you can imagine why).

[Magnus_Anderson]

@obsrvr524

Technically the word “stop” does imply time so a process stops or comes to an end - ends.

Precisely (:

But if you are talking strictly about the concept, void of time issues, the end merely means a highest number or position in the sequence above which there are none. Obviously that doesn’t apply to an infinite set or series because it doesn’t have an end point above which there is none.

Right. So when talking about sequences, which do not exist in time, the word “end” refers to an element that comes after all other elements. (It can also be used to refer to an element that comes before all other elements but we can ignore that.) Thus, a sequence without an end (which is what an infinite sequence is) is a sequence that has no element that comes after all other elements. But does that imply that there are no elements OUTSIDE of that sequence? Not really. And does it imply that elements that are outside of that sequence cannot come after all of the elements of that sequence? Again, not really. Thus, if you have two sequences, say (A = (1, 3, 5, \dotso)) and (B = (2, 4, 6, \dotso)), and you join them conceptually into a super-sequence (C = (1, 3, 5, \dotso, 2, 4, 6, \dotso)), then the index of the place occupied by (2) would be equal to the number of elements in (A) plus one which is half the number of natural numbers plus one which, in James’s terms, equals (\frac{infA}{2} + 1). If you remove (1) from (C), the index of (2) would then be (\frac{infA}{2}). And if you add to it another (\frac{infA}{2}) elements of your choice, the index of (2) would be (infA) i.e. it would be located at “the point of infA”. In all three cases, (2) is located at “the point of infinity” because in all three cases its index is greater than every integer.

I don’t understand why that is not clear.

That’s clear and I actually agree with it. What’s not clear is how you derive “There is no point at infinity” from it. (Also, what you mean by “point at infinity” is not completely clear.)

And I “barely” did contested it.
I said, “but ok”.

Actually, you said “wrong word” which means you didn’t really contest it. Instead, you merely said that my use of words is wrong. I misread (I thought you said “wrong wrong” which would imply that what I’m saying is wrong.) Nonetheless, I am really only interested in hearing what’s wrong with what I am saying. I am not so interested in what’s wrong with what I am doing and especially not with how I use my words.

And if you had said, “For this discussion let’s say that infinity refers to the number of naturals” I would have only said “ok”. But now we are clear.

I can understand that (:

And since we both know what “infA” means - why not just used that.

I generally don’t use it because it’s specifically James’s term. As far as I am concerned, I am perfectly comfortable using the word “infinity” to mean two different things. Sometimes, I use it the way most people do, which is as a quantity greater than every integer; sometimes, I use it in a narrow sense to refer to some specific quantity greater than every integer (which is similar, but not necessarily the same, as James’s (infA), since James’s (infA) refers specifically to the number of natural numbers.)

But yes, I could have used that term.

If you are using the 1to1 correspondence, you have to use a different sequence because all of your naturals are used up.

True. But I am not using the set of natural numbers to count the resulting set.You can’t do that – the set of natural numbers is smaller. What I’m doing is I am using a set that contains every natural number plus (infA + 1).

Yes, I read that. He also had a long discussion on a different board concerning hyperreals and that 1=0.999… issue.

I don’t have an issue with the infA+1 notation. My issue is that when defining the terms you asked about like “largest number”, “infA+1” does not represent a natural number. It is a greater quantity than merely infA but there is no natural number that can represent it. That is why it has its own word.

James specifically stated that you have the freedom to set a chosen standard and must do so if you are going to add or do maths with infinity. I can see that makes sense. And as long as you make it clear that you have, it isn’t a problem.

On that other board (“Rational” something) they were telling James that he “doesn’t know shit” and that he should just accept what he is told. But then James pointed out that what was told back in 1947 or so by a Hewitt somebody is exactly what he was saying merely in different notation. James was explaining, with many indisputable arguments why he, that Hewitt bloke and others was right.

James understood, as do I now, that ALL of these confusions are merely about the words not being carefully defined. Once anyone carefully defines/explains the words all of these issues go away. The same is true concerning laws and politics but no one seems to want to fix that either (you can imagine why).

That’s pretty much it.

[obsrvr524]

Everything you said seemed right to me until you got to this conclusion -

@Magnus Anderson

it would be located at “the point of infA”. In all three cases, (2) is located at “the point of infinity” because in all three cases its index is greater than every integer.

Neither infA nor infA+1 is a “point” (as in a location on a number line). It is an amount - equal to or greater than infA (the standard infinite set of natural numbers). But it is not a number and it is not a point or location in a sequence. It implies a different sequence than the natural numbers alone. So it cannot be “the highest number” being referenced. That “plus 1” is just another item in an infinite list - no hierarchy can be applied. It is not the highest or lowest. It is just another item beyond our ability to count with numbers.

@Magnus Anderson

I generally don’t use it because it’s specifically James’s term. As far as I am concerned, I am perfectly comfortable using the word “infinity” to mean two different things. Sometimes, I use it the way most people do, which is as a quantity greater than every integer; sometimes, I use it in a narrow sense to refer to some specific quantity greater than every integer (which is similar, but not necessarily the same, as James’s (infA), since James’s (infA) refers specifically to the number of natural numbers.)

But that ambiguity is what is perpetuating the confusion. I don’t think James held any ownership to the term “infA” as long as anyone understands it. And it allows for an “infB” or “infC” to maintain distinctions while talking about varied infinities.

If you are not helping - you are probably hurting - or maybe just wasting time and effort.

[Magnus_Anderson]

“Point” is another word for “position” or “place”. Thus, “point at (infA)” merely means “place whose index is (infA)”. In the case of ((1, 2, 3, \dotso, a)), (a) is located at the point of (infA + 1) which means no more than that it occupies place whose index is (infA + 1). Thus, if you agree that it occupies place whose index is (infA + 1), you also have to agree that it is located at the point of (infA + 1). (And if it occupies place at (infA + 1), you will also have to agree that it occupies place at infinity since the word “infinity” means “a number greater than every integer” and that’s precisely what (infA + 1) is.)

Also note that we started with ((1, 2, 3, \dotso)) which is an infinite sequence that has no last element (= an element that comes after all other elements = an element with the highest index.) But once we add (a) to it, and add it after all of its elements, we get ((1, 2, 3, \dotso, a)) which is an infinite sequence with an end – it has the last element. (a) is now an element with an index that is greater than the index of every other element in the sequence, its index being (infA + 1).

That’s why it’s inaccurate to say that an infinite sequence is a sequence that has no last element. Indeed, it’s not even accurate to say that an infinite sequence is a sequence that has no last element and/or no first element. (What’s true is that every sequence that has no last element and/or no first element is an infinite one. But a sequence that has both is not necessarily finite.)

The proper definition of the term “infinite sequence” is a sequence whose elements can be put in one-to-one correspondence with one of the infinite sets. (And a set is said to be infinite if its number of elements is greater than every integer. Note that sets have no notion of “end”.)

[Magnus_Anderson]

@obsrvr524

But that ambiguity is what is perpetuating the confusion. I don’t think James held any ownership to the term “infA” as long as anyone understands it. And it allows for an “infB” or “infC” to maintain distinctions while talking about varied infinities.

If you are not helping - you are probably hurting - or maybe just wasting time and effort.

You may be right that my use of words isn’t optimal. The thing is that that’s a separate discussion that I am moreover not interested in at the present time.

And while I do welcome feedback of the form “I don’t understand what you mean by X”, I do not welcome criticism of how I use words nor an advice on how to use them. (Leave these things up to me, please (

[obsrvr524]

@Magnus Anderson

“Point” is another word for “position” or “place”. Thus, “point at (infA)” merely means “place whose index is (infA)”.

But there is no such place. infA is a quality of a set. It is NOT a point at the end of a sequence. There is no “end” to the sequence - no boundary for the set. You cannot ever be at the edge, boundary, or end - no such things exists. So there is no “one position or point beyond it”. There is only an addition to the entire infinite set that is not within the set or at an edge of the set or “after” the set but merely appended to it as a new set - infA and also 1 along side or in combination.

@Magnus Anderson

That’s why it’s inaccurate to say that an infinite sequence is a sequence that has no last element. Indeed, it’s not even accurate to say that an infinite sequence is a sequence that has no last element and/or no first element. (What’s true is that every sequence that has no last element and/or no first element is an infinite one. But a sequence that has both is not necessarily finite.)

Ok. I see what you are saying now. I’m not sure that I erred in that way, but we can assume that back there somewhere I did.

@Magnus Anderson

The proper definition of the term “infinite sequence” is a sequence whose elements can be put in one-to-one correspondence with one of the infinite sets. (And a set is said to be infinite if its number of elements is greater than every integer. Note that sets have no notion of “end”.)

I can’t accept that for similar reason as you just pointed out concerning an infinite sequence not necessarily not having end points. And “infinite” doesn’t mean “greater than every integer”. The infinite set of every even integer certainly isn’t greater than the set of all integers - or “greater than every integer”.

A sequence is a progressing pattern. It has a direction of increasing or decreasing. Each element in the pattern might have a 1to1 correspondence with another set. But that is not what defines it as a sequence (infinite or otherwise). And if you add something that is not part of that sequence, it is not appended to “the end”. It is merely a part of the total collection. It is not before or after anything because it is not part of any sequence.

@Magnus Anderson

And while I do welcome feedback of the form “I don’t understand what you mean by X”, I do not welcome criticism of how I use words nor an advice on how to use them. (Leave these things up to me, please (

That can lead to kickback.

[polishyouthgotipbann (polishyouthgotipbanned)]

are you two seriously still getting excited about high-school mathematical induction???this is what a sectarian does…takes a bit of information out of context out of bible or mathematics or whatever and strings his bizarre theories off it.

[obsrvr524]

@polishyouthgotipbanned

are you two seriously still getting excited about high-school mathematical induction???this is what a sectarian does…takes a bit of information out of context out of bible or mathematics or whatever and strings his bizarre theories off it.

What’s it to you?

[polishyouthgotipbann (polishyouthgotipbanned)]

its funny watching two insane goons falling over in a well lit room telling eachother they are the only ones that can see. reminds me of my neo-nazi friends.

[obsrvr524]

@polishyouthgotipbanned

its funny watching two insane goons falling over in a well lit room telling eachother they are the only ones that can see. reminds me of my neo-nazi friends.

What is more funny is that neither of us was telling eachother that we are the only ones who can see. You seem to be the one doing that. All wise and perceptive are you? - Now THAT would be funny.

And from what I have seen of many of your posts, “high-school” would be a step up for you.

[polishyouthgotipbann (polishyouthgotipbanned)]

insanity is not a joke…and you are clearly a kook. take care of yourself.

[Magnus_Anderson]

@obsrvr524

But there is no such place.

But I thought that you already agreed that there is such a place, if not explicitly, then at least implicitly (: In the case of ((1, 2, 3, \dotso, a)), that place is the one occupied by (a). Didn’t you agree that the index of that place is (infA + 1)? If so, that’s a “point at (infA)”.

Let me restate the entire logic:

1. “Point” is another word for “position” or “place”

2. Thus, “point at infinity” is another expression for “place at infinity”

3. “Place at X” is another expression for “place whose index is X”

4. Thus, “place at infinity” is another expression for “place whose index is infinity”

5. (infA) is an instance of infinity

6. Thus, “place whose index is (infA)” is also “place whose index is infinity”

7. The index of (a) in ((1, 2, 3, \dotso, a)) is (infA + 1)

8. Thus, (a) occupies “place whose index is (infA + 1)” which means “point at (infA + 1)” which means “point at infinity”

Please let me know what you disagree with.

infA is a quality of a set.

I don’t think that “quality” is an appropriate way to describe what (infA) is. You can’t even say that (infA) specifies that a set has no end because sets have no notion of “end”. (“Endless set” is an instance of figurative speech.) What (infA) does is it specifies how many elements there are in a set. Specifically, what it does is it states that the number of elements in a set is equal to the number of natural numbers.

It is NOT a point at the end of a sequence.

It is an index of the place (which is a point) occupied by (a) in the case of ((2, 3, 4, \dotso, a)).

There is no “end” to the sequence - no boundary for the set. You cannot ever be at the edge, boundary, or end - no such things exists.

I am not sure I understand what you’re saying here. Are you saying that there is no “end” to the sequence that is ((1, 2, 3, \dotso, a))? But there is. Remember how we defined the word “end”? It refers to a place that comes after all other places i.e. to a place with the highest index. And that place in the case of ((1, 2, 3, \dotso, a)) is the one occupied by (a).

So there is no “one position or point beyond it”.

If this is true then ((1, 2, 3, \dotso, a)) is an invalid sequence. Do you agree with that?

There is only an addition to the entire infinite set that is not within the set or at an edge of the set or “after” the set but merely appended to it as a new set - infA and also 1 along side or in combination.

I didn’t add (a) to a new sequence. That’s NOT the operation I performed. I added it to THE SAME sequence and I added it AFTER all of its elements. You are insisting that I performed an operation that I did not actually perform. (It’s akin to saying I multiplied (1) by (0) when in fact I divided it by (0).) You may want to argue that I’m performing an impossible operation instead (which is false but which is at least based on what I really did rather than something that I didn’t do.)

And note that when we say that a sequence has no end, we’re merely saying that it has no place with the highest index. This means we can’t add an element at the end of that sequence (which means we can’t add it to the place with the highest index – because there is no such place) which does not mean we can’t add it AFTER all of its elements (thereby creating a place with the highest index.)

And “infinite” doesn’t mean “greater than every integer”.

You’re probably aware of the fact that I believe that to be the best definition of the word “infinite” out there – better than “without an end”. The reason being very simple: sets have no notion of “end” and a sequence can be infinite even if it has a beginning and an end.

The infinite set of every even integer certainly isn’t greater than the set of all integers - or “greater than every integer”.

The set of even integers is NOT greater than the set of integers. Nonetheless, the number of even integers is CERTAINLY greater than every integer. Are you saying that the number of even integers is an integer? If so, which one?

A sequence is a progressing pattern. It has a direction of increasing or decreasing. Each element in the pattern might have a 1to1 correspondence with another set. But that is not what defines it as a sequence (infinite or otherwise).

“Progressing pattern” is a figurative description of what a sequence is. It’s not a strict mathematical definition. And since sequences do not exist in time, they are neither increasing nor decreasing.

And if you add something that is not part of that sequence, it is not appended to “the end”. It is merely a part of the total collection. It is not before or after anything because it is not part of any sequence.

You choose where you’re going to insert it. You can insert it between two existing places e.g. you can insert it between the first and second element (thereby becoming second element.) But you can also insert it between one non-existent and one existent place e.g. you can add it between the zero-eth place (which doesn’t exist) and first place (thereby becoming first element.) Finally, you can insert it after all other elements (thereby becoming the last element.)

When adding (a) to ((1, 2, 3, \dotso)), I added it after all of its elements. The result is ((1, 2, 3, \dotso, a)).

And note that I added it to THE SAME sequence. I didn’t start a new one. That’s NOT the operation I performed.

You may want to argue such an operation is an impossible one rather than saying I added it somewhere I didn’t.

[obsrvr524]

@Magnus Anderson

@obsrvr524

But there is no such place.

But I thought that you already agreed that there is such a place, if not explicitly, then at least implicitly (: In the case of ((1, 2, 3, \dotso, a)), that place is the one occupied by (a). Didn’t you agree that the index of that place is (infA + 1)? If so, that’s a “point at (infA)”.

No that isn’t what I agreed to. You have changed what I said. That extra “a” that has been included into the set is NOT “after” or “before” the sequence. It can be indexed as the “infA+1” item. But there is no “infA point”, position, or place. The infA set has no upper limit. The total relative volume is described as infA - but that is not an index to “the last element” - because there is no last element in that set. infA is larger or smaller than other infinite sets.

@Magnus Anderson

Let me restate the entire logic:

1. “Point” is another word for “position” or “place” - ok

2. Thus, “point at infinity” is another expression for “place at infinity” - not ok - there is no place “at infinity”

3. “Place at X” is another expression for “place whose index is X” - ok

4. Thus, “place at infinity” is another expression for “place whose index is infinity” - infA is not an index at infinity. It is an infinite quantity - not a pointer to locations. X cannot be "infinity"In maths nothing goes TO infinity. It only goes TOWARD infinity.

@Magnus Anderson

infA is a quality of a set.

I don’t think that “quality” is an appropriate way to describe what (infA) is.

You’re right. I was thinking of merely an infinite set. InfA is a specific set, so ok now it specifies a relative quantity (actually I think “volume” is a better word. James used “degree”).

@Magnus Anderson

You can’t even say that (infA) specifies that a set has no end because sets have no notion of “end”. (“Endless set” is an instance of figurative speech.)

Sets have boundaries. Those boundaries are commonly referred to as the set’s “ends” - initial and final. But in the case of an infinite set with no end point, the boundary is denoted as “no more than infinite”. The boundary is qualitative rather than quantitative (until the set is exactly specified). Once exactly defined or specified the quantity of the set is that of being “endless” - no end - no index pointing to an end - merely a relative amount compared to some other infinite set that also has no end. It is really just a relative size or volume rather than a quantitative amount.

Does that clear anything up? - [size=85]probably not[/size]