## Controversial concepts in mathematics

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### Controversial concepts in mathematics

How many iterations are there in a dead loop?

Code: Select all
while (true) { print('Lionel Richie - Hello'); }

One can say "There's an infinite number of them." And that's true but that merely tells us that the number of iterations is greater than one (and there isn't one such number), it does not give us an exact number. Hence, it's not really an answer.

Since the loop literally never ends, the number of iterations is greater than EVERY number, and since there is only one such number, that is the answer to our question. I call this number "largest impossible" where by "impossible" I mean "any number larger than the largest number" and by "largest number" I mean "a number larger than every other number".

Let that be an introduction to a thread that is entirely about controversial concepts in mathematics that people seem to love so much (and that they tend to discuss in inappropriate places.)

Let me introduce some.

Infinity refers to any number larger than every integer. There isn't one such number but many of them.

Smallest possible infinity refers to the smallest number larger than every integer. It's an instance of infinity and there's only one such number.

Largest number is a number larger than every other number. It's an instance of infinity and there's only one such number.

Standard infinity refers to some arbitrarily chosen infinite number that can be added to, subtracted from, multiplied and divided in order to get some other infinite number (hence, it must be greater than the smallest infinity and smaller than the largest number.) The purpose of this number is to serve as some kind of standard for doing arithmetic.

Impossible number is any number larger than the largest number. I call it impossible because it's an obvious contradiction. How can a number be larger than the largest number? It can't. Nonetheless, contradictory concepts can be useful (in their own limited way) so we're going to keep it.

Largest impossible is a number larger than every number. It's similar to the largest number, but unlike the largest number, this number is larger than EVERY number and not merely EVERY OTHER number which means it's also larger than itself.

Infinitesimal refers to a number greater than $$0$$ but smaller than every number of the form $$\frac{1}{n}$$ where $$n$$ is a natural number. An infinitesimal is thus a reciprocal of infinity. And since there is more than one infinity, there is more than one infinitesimal. (Thus, the reciprocal of the largest number is the smallest number greater than zero -- the smallest infinitesimal.)
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### Re: Controversial concepts in mathematics

Magnus Anderson wrote:Infinity refers to any number larger than every integer. There isn't one such number but many of them.

Doesn't "infinity" refer to the first point beyond the endless? - doesn't exist except as a direction - not a number?

Magnus Anderson wrote:Smallest possible infinity refers to the smallest number larger than every integer. It's an instance of infinity and there's only one such number.

Same as above - not an actual number. But aren't there infinite sets that are smaller than the ordinals - The ordinals minus every even number? I'm not sure there is a "smallest infinity".

Magnus Anderson wrote:Largest number is a number larger than every other number. It's an instance of infinity and there's only one such number.

I don't think there is such a number at all.

Magnus Anderson wrote:Standard infinity refers to some arbitrarily chosen infinite number that can be added to, subtracted from, multiplied and divided in order to get some other infinite number (hence, it must be greater than the smallest infinity and smaller than the largest number.) The purpose of this number is to serve as some kind of standard for doing arithmetic.

James' "infA".

Magnus Anderson wrote:Impossible number is any number larger than the largest number. I call it impossible because it's an obvious contradiction. How can a number be larger than the largest number? It can't. Nonetheless, contradictory concepts can be useful (in their own limited way) so we're going to keep it.

Ok.

Magnus Anderson wrote:Largest impossible is a number larger than every number. It's similar to the largest number, but unlike the largest number, this number is larger than EVERY number and not merely EVERY OTHER number which means it's also larger than itself.

I don't see the difference in this one and the last one.

Magnus Anderson wrote:Infinitesimal refers to a number greater than $$0$$ but smaller than every number of the form $$\frac{1}{n}$$ where $$n$$ is a natural number. An infinitesimal is thus a reciprocal of infinity. And since there is more than one infinity, there is more than one infinitesimal. (Thus, the reciprocal of the largest number is the smallest number greater than zero -- the smallest infinitesimal.)

I don't see why "n" has to be a natural number unless it is a "standard infinitesimal".
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### Re: Controversial concepts in mathematics

Observer wrote:Same as above - not an actual number.

That depends on how you define the word "number". If you define it in such a way so that it only refers to integers (which I'm sure you don't but I'm just giving you an example) then infinity isn't a number because it is defined as something different from integers.

On the other hand, it also depends on how you define the word "infinity". If you define it to mean "any number greater than every integer" then infinity is not a specific number but a category of numbers in the same exact way that "integer" is a category of numbers rather than some specific number.

In the end, what I'm saying is merely that to say that a queue consisting of people is infinite is to say that the number of people in that queue is greater than every integer. Do you disagree with this?

Doesn't "infinity" refer to the first point beyond the endless? - doesn't exist except as a direction - not a number?

I would say it refers to a quantity that is greater than every integer. I wouldn't use words such as "point" and "endless". And I wouldn't say it is a direction.

But aren't there infinite sets that are smaller than the ordinals - The ordinals minus every even number? I'm not sure there is a "smallest infinity".

Yes, there are infinite sets smaller than the set of natural numbers. But does that imply there is no such thing as smallest infinity? I wouldn't say so. Perhaps someone can present an argument in favor of your claim. Personally, I can't see it being the case.

I don't think there is such a number at all.

Why not?

James' "infA".

Exactly (:

I don't see the difference in this one and the last one.

The difference is that "largest number" is greater than every OTHER number which means it isn't greater than itself whereas "largest impossible" is larger than LITERALLY EVERY number includng itself.

I don't see why "n" has to be a natural number unless it is a "standard infinitesimal".

It can also be a positive integer or a positive real number. But it can't be a number greater than every integer because such would exlude certain numbers from the definition that shouldn't be excluded.
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### Re: Controversial concepts in mathematics

Again Magnus,

Infinity is not a number, it’s an operator.

Over and over again you describe infinity as greater than all integers.

That makes as much sense as saying plus is greater than all integers!
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### Re: Controversial concepts in mathematics

It might not be a number but it is certainly not an operator.
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### Re: Controversial concepts in mathematics

Magnus Anderson wrote:It might not be a number but it is certainly not an operator.

Ecmandu is right that it is an operator - note that all your descriptions are in terms of finites and "what to do" with them in order to achieve some specific status. This is what an operator does, and in practice this is happening when we define how to achieve "some kind of infinity" (in words if not in our most literal and stringently systemised conception).

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### Re: Controversial concepts in mathematics

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

Wikipedia wrote:Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number.

/ɪnˈfɪnɪti/

noun
1.
the state or quality of being infinite.
"the infinity of space"

Where do they say or how does what they say imply that infinity is an operator?

When people say that a queue consisting of people is infinite, what they mean is that the queue is endless which means that the number of people in the queue is larger than every integer. Nothing to do with operators.

Assuming that both you and Ecmandu think that infinity is a unary operator, can you tell me what $$\infty \enspace 5$$ means and what it equals to? And once you do, can you prove it's the standard definition i.e. how everyone else defines infinity?
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### Re: Controversial concepts in mathematics

https://www.ilovephilosophy.com/viewtop ... r#p2769195

Silhouette wrote:Just to wrap up what is meant when infinity is said to be an operator - it is only validly used in mathematics as part of an operation, more like an operator on an operator. This doesn't mean it's interchangeable with operators like addition and multiplication etc. - it means it can be used to apply to the number of times you add or multiply etc.
This isn't up for debate by the way, I'm just quickly saying what math does - nothing more nothing less, no should or shouldn't, just what math does.

https://www.ilovephilosophy.com/viewtop ... r#p2769191

Magnus Anderson wrote:
Ecmandu wrote:it’s an operator upon numbers.

Right. So "3 infinity 5" must be a perfectly sensible mathematical expression (: What's the result of it?

[..]

Ecmandu wrote:Well...

3 infinity 5 (without parentheses) would mean an infinite number of 3’s with a five that’s separate from the infinite number of 3’s.

Not so hard is it.

Where did Ecmandu got this idea that "3 infinity 5" means "an infinite number of 3's with a five that's separate from the infinite number of 3's"?

He made it up, right? So why should anyone care?
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### Re: Controversial concepts in mathematics

Magnus Anderson wrote:Where do they say or how does what they say imply that infinity is an operator?

When people say that a queue consisting of people is infinite, what they mean is that the queue is endless which means that the number of people in the queue is larger than every integer. Nothing to do with operators.

Assuming that both you and Ecmandu think that infinity is a unary operator, can you tell me what $$\infty \enspace 5$$ means and what it equals to? And once you do, can you prove it's the standard definition i.e. how everyone else defines infinity?

Well if wiki and google don't go into it I guess I'm wrong
Seriously though - they're generally great sources, but you'll forgive me if I expand on them in line with how infinity is actually used.

For example, while $$\infty \enspace 5$$ doesn't mean anything (and it's easy to use any operator in a meaningless way), $$\sum_{i=1}^{\infty}a_i$$ (as seen in wiki) is perfectly fine. You "infinitely sum" a starting from its 1st iteration to its 2nd and so on without stopping. It doesn't specify a stopping point, like you would do with a value. Instead it tells you a way in which to proceed - like a verb rather than a noun. Infinity is used as a process rather than a destination. Whenever it's casually used as an "answer in itself", it's being used as a non-ending process - or if you really want an answer it can at best mean "I don't know" or "indetermined" or "divergent" in some non-specified way like you would by contrast get with a convergence. Consistently, it can't be a specific value.

Good links by the way - looks like I already explained it, so I'm not sure why you're asking me to do it again.

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### Re: Controversial concepts in mathematics

Silhouette wrote:For example, while $$\infty \enspace 5$$ doesn't mean anything (and it's easy to use any operator in a meaningless way), $$\sum_{i=1}^{\infty}a_i$$ (as seen in wiki) is perfectly fine. You "infinitely sum" a starting from its 1st iteration to its 2nd and so on without stopping. It doesn't specify a stopping point, like you would do with a value. Instead it tells you a way in which to proceed - like a verb rather than a noun. Infinity is used as a process rather than a destination. Whenever it's casually used as an "answer in itself", it's being used as a non-ending process - or if you really want an answer it can at best mean "I don't know" or "indetermined" or "divergent" in some non-specified way like you would by contrast get with a convergence. Consistently, it can't be a specific value.

Alright, so you think that infinity is an operator (thereby agreeing with Ecmandu) but you don't think that expressions such as $$\infty \enspace 5$$ are meanigful (thereby most likely disagreeing with Ecmandu.)

I agree that $$\sum_{i=1}^{\infty}a_i$$ is a perfectly fine expression but I don't agree it's an expression where infinity is used as an operator. Of course, this depends on how you define the word "operator" but I'm assuming that what you mean by it is at the very least some kind of function. And I don't think that that expression is using the symbol of infinity to represent a function. If you insist otherwise, can you at the very least share with us which function it represents?

Moreover, I don't think the symbol of infinity merely specifies the direction of summation. (That also seems to be Observer's position.) It does but that's not ALL it does. (It wouldn't be much of a statement if that were the case. It would merely be saying "Go to some number greater than $$0$$" without saying exactly which one.) In fact, any other number in its place would do the same -- specify the direction of summation. $$\sum_{i=1}^{10}a_i$$ says "move upwards". $$\sum_{i=10}^{1}a_i$$ says "move downwards". But that's not all they do -- they also specify the stopping point. And that's also what $$\infty$$ does (but to a lesser extent.)

Finally, I don't think the symbol of infinity represents "I don't know" or "not specified". If that were the case, the probability of $$\sum_{i=1}^{\infty}a_i$$ representing $$a_1$$ would be the same as the probability of it representing $$a_1 + a_2$$.

It is my position that the symbol of infinity DOES specify the point at which the summation stops (and as a consequence, the direction of summation.) What it says is that the summation shouldn't stop if the number of terms summed up so far is an integer. Thus, it says that the total number of terms to be summed is greater than every integer. But since the number of numbers greater than every integer isn't equal to one (i.e. there is more than one infinity), it does not tell us the exact point at which the process of summation should stop (and that's why $$\infty$$ is less specific than integers.)
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### Re: Controversial concepts in mathematics

Magnus Anderson wrote:https://en.wikipedia.org/wiki/Infinity

Wikipedia wrote:Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number.

/ɪnˈfɪnɪti/

noun
1.
the state or quality of being infinite.
"the infinity of space"

Where do they say or how does what they say imply that infinity is an operator?

When people say that a queue consisting of people is infinite, what they mean is that the queue is endless which means that the number of people in the queue is larger than every integer. Nothing to do with operators.

Assuming that both you and Ecmandu think that infinity is a unary operator, can you tell me what $$\infty \enspace 5$$ means and what it equals to? And once you do, can you prove it's the standard definition i.e. how everyone else defines infinity?

I’ll tell you exactly what (infinity) 5 means:

Definition 1: 5 lasts forever

Definition 2: there are an infinite number of 5’s

Now that we have two definitions, we can invent a new symbol to delineate them. Very simple.

Actually, to make it easier, we could make different definitions by switching which side the infinity is on and agree to it

Or you could just say (infinity) 5 or (infinite) 5’s
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### Re: Controversial concepts in mathematics

The problem, Ecmandu, is that I don't think these are standard definitions but your own. If you think they are, can you prove it? The thing is that, if these are merely your own definitions, then they are irrelevant.
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### Re: Controversial concepts in mathematics

Magnus Anderson wrote:
Silhouette wrote:For example, while $$\infty \enspace 5$$ doesn't mean anything (and it's easy to use any operator in a meaningless way), $$\sum_{i=1}^{\infty}a_i$$ (as seen in wiki) is perfectly fine. You "infinitely sum" a starting from its 1st iteration to its 2nd and so on without stopping. It doesn't specify a stopping point, like you would do with a value. Instead it tells you a way in which to proceed - like a verb rather than a noun. Infinity is used as a process rather than a destination. Whenever it's casually used as an "answer in itself", it's being used as a non-ending process - or if you really want an answer it can at best mean "I don't know" or "indetermined" or "divergent" in some non-specified way like you would by contrast get with a convergence. Consistently, it can't be a specific value.

Alright, so you think that infinity is an operator (thereby agreeing with Ecmandu) but you don't think that expressions such as $$\infty \enspace 5$$ are meanigful (thereby most likely disagreeing with Ecmandu.)

I agree that $$\sum_{i=1}^{\infty}a_i$$ is a perfectly fine expression but I don't agree it's an expression where infinity is used as an operator. Of course, this depends on how you define the word "operator" but I'm assuming that what you mean by it is at the very least some kind of function. And I don't think that that expression is using the symbol of infinity to represent a function. If you insist otherwise, can you at the very least share with us which function it represents?

Moreover, I don't think the symbol of infinity merely specifies the direction of summation. (That also seems to be Observer's position.) It does but that's not ALL it does. (It wouldn't be much of a statement if that were the case. It would merely be saying "Go to some number greater than $$0$$" without saying exactly which one.) In fact, any other number in its place would do the same -- specify the direction of summation. $$\sum_{i=1}^{10}a_i$$ says "move upwards". $$\sum_{i=10}^{1}a_i$$ says "move downwards". But that's not all they do -- they also specify the stopping point. And that's also what $$\infty$$ does (but to a lesser extent.)

Finally, I don't think the symbol of infinity represents "I don't know" or "not specified". If that were the case, the probability of $$\sum_{i=1}^{\infty}a_i$$ representing $$a_1$$ would be the same as the probability of it representing $$a_1 + a_2$$.

It is my position that the symbol of infinity DOES specify the point at which the summation stops (and as a consequence, the direction of summation.) What it says is that the summation shouldn't stop if the number of terms summed up so far is an integer. Thus, it says that the total number of terms to be summed is greater than every integer. But since the number of numbers greater than every integer isn't equal to one (i.e. there is more than one infinity), it does not tell us the exact point at which the process of summation should stop (and that's why $$\infty$$ is less specific than integers.)

An operation is a function, yes. They tell you what to do with inputs. Perhaps more appropriately, "infinity" tells you what not to do - to "not stop". Operands are what you do the operations to (e.g. finite numbers), and you don't "do" anything to infinity - infinity is what informs the manner of doing. It's not a limiting factor, like a finite number is - it's the opposite of limiting, because that's what infinity literally means. Operands are limiting factors: finites.

This idea of a "direction" of summation isn't a thing. Addition is commutative: it doesn't matter what order you add in. I mean you could write a finite sum like $$\sum_{i=10}^{1}a_i$$ much to everyone's confusion, and just leave it to their common sense to infer that you want to add backwards for some reason, even though the order of addition doesn't affect the outcome. But because of commutativity, this is just redundant.

And when I say infinity can at best mean "I don't know" or "indetermined" or "divergent" (as I stated immediately before I said this) I was talking about when infinity is casually used as an "answer in itself" - like when you try to say some expression "equals infinity". I wasn't talking about when it's used legitimately, such as in an infinite sum or product. So comments like "the probability of $$\sum_{i=1}^{\infty}a_i$$ representing $$a_1$$" etc. don't apply to what I was talking about.

As for infinity specifying "the point where the summation stops", that's a clear contradiction in terms. And it doesn't matter "which infinity" you use to express something like an infinite sum, because the same limit is tended towards either way. Distinguishing the degree to which something isn't finite doesn't do anything - that whole notion is a product of flawed arguments like Cantor's diagonalisation argument, which I have disproven. It's just an intriguing "what if" to consider more than one infinity, to see if any interesting math can emerge from treating infinity like it can come in different types that have different magnitiudes of being "more non-finite" than some other non-finite.

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### Re: Controversial concepts in mathematics

Magnus Anderson wrote:
Observer wrote:Same as above - not an actual number.

That depends on how you define the word "number". If you define it in such a way so that it only refers to integers (which I'm sure you don't but I'm just giving you an example) then infinity isn't a number because it is defined as something different from integers.

On the other hand, it also depends on how you define the word "infinity". If you define it to mean "any number greater than every integer" then infinity is not a specific number but a category of numbers in the same exact way that "integer" is a category of numbers rather than some specific number.

In the end, what I'm saying is merely that to say that a queue consisting of people is infinite is to say that the number of people in that queue is greater than every integer. Do you disagree with this?

I think you missed the point.

To be infinite means to be endless - there is no end to be beyond. The word "infinity" refers to that point just beyond which doesn't exist, even in concept. So the word is used to refer to a direction or inaccessible pole, not a number that could be used in any maths way. A proposed process can be infinite in that it will not end but nothing can ever go beyond infinite to reach infinity. The very concept of infinity is invalid in logic and maths.

So "the smallest infinity" is like saying "the smallest extreme upward". That is not a number in any since. And it is fruitless to just arbitrarily define concepts differently than the current language.

What I disagree with is using the words "infinite" or "infinity" in conjunction with "number". They are not quantities but qualities. You might as well be saying - "n times yellow divided by 4 = ?" or "the smallest extreme yellow beyond yellow".

Magnus Anderson wrote:I would say it refers to a quantity that is greater than every integer.

Fine except that there is no quantity greater than any integer.

Magnus Anderson wrote:Yes, there are infinite sets smaller than the set of natural numbers. But does that imply there is no such thing as smallest infinity? I wouldn't say so. Perhaps someone can present an argument in favor of your claim. Personally, I can't see it being the case.

We get the smallest by dividing 1 by the largest. If there was a largest infinity then the smallest infinity would be 1 / (the largest infinity).

But there is no largest infinity. That is what Cantor was trying to say.

Because -
No matter how large the largest infinity is assumed to be, simply by raising it to its own power, it becomes unimaginably larger. And then the same can be done with that one - and again - and again. There is not any way to even express the largest possible infinity ("the highest cardinality"). It is the same as trying to find infinity - by definition such a quantity cannot exist.
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### Re: Controversial concepts in mathematics

Magnus Anderson wrote:The problem, Ecmandu, is that I don't think these are standard definitions but your own. If you think they are, can you prove it? The thing is that, if these are merely your own definitions, then they are irrelevant.

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?

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### Re: Controversial concepts in mathematics

Observer wrote:To be infinite means to be endless - there is no end to be beyond. The word "infinity" refers to that point just beyond which doesn't exist, even in concept. So the word is used to refer to a direction or inaccessible pole, not a number that could be used in any maths way. A proposed process can be infinite in that it will not end but nothing can ever go beyond infinite to reach infinity. The very concept of infinity is invalid in logic and maths.

If I understand you correctly, you are saying that the word "infinity" is an oxymoron (a contradiction, which is to say, something that is two or more opposing things at once) but that the word "infinite" is not. And I take it that what you mean by "infinity" is not what Google says it means (which is "the state or quality of being infinite") but rather "point at infinity".

And the reason you think that "point at infinity" is an oxymoron is because the word "infinite" means "endless" which means "no end" whereas "point at infinity" implies an end.

If that's an accurate representation of your position, I would like to ask the following question: what does the word "end" mean?

In order to answer this question, I would suggest defining the word "end" with respect to some specific concept e.g. the concept of sequence.

What does the word "end" mean with respect to sequences?

Let me provide a quick definition of the word "sequence". A sequence is first of all a set of places each one of which is occupied by an element (no unoccupied places are allowed.) But that's not all it is. It is also an ordered set which means that for every two places in the set one comes before the other and one comes after the other. Furthermore, if one place comes after some other place, it also comes after all places that come before that other place. Also, if one place comes before some other place, it also comes before all places that come after that other place. This means that each place can be indexed i.e. represented by a unique natural number.

The way I see it, the word "end" refers to the place with the highest index (one that comes after all other places) or to the place with the lowest index (one that comes before all other places.) (In some cases, which I believe we can safely ignore, the word "end" refers exclusively to the place with the highest index whereas the word "beginning" is used to represent the place with the lowest index.)

Thus, to say that a sequence is endless means that there is no place with the highest index and/or that there is no place with the lowest index. (The problem with this is that it leaves out sequences that have a beginning and an end but that are nonetheless infinite. Fortunately, this can be safely ignored.)

That said, the 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?

Perhaps you think that by saying that a sequence is infinite, we're also saying that the number of its places is equal to the number of places available in our imaginary universe or that the number of its places is equal to the largest number. I don't think that's what the word "infinity" means.

What I disagree with is using the words "infinite" or "infinity" in conjunction with "number". They are not quantities but qualities. You might as well be saying - "n times yellow divided by 4 = ?" or "the smallest extreme yellow beyond yellow".

But then again, when you say that a queue consisting of people is infinite, you're talking about how many people there are in that queue. Aren't you? (:

Fine except that there is no quantity greater than any integer.

Why do you think so? Wouldn't that imply that saying that a set is infinite is a contradiction?

But there is no largest infinity. That is what Cantor was trying to say.

Because -
No matter how large the largest infinity is assumed to be, simply by raising it to its own power, it becomes unimaginably larger. And then the same can be done with that one - and again - and again. There is not any way to even express the largest possible infinity ("the highest cardinality"). It is the same as trying to find infinity - by definition such a quantity cannot exist.

If we say that $$L$$ is the largest number, raising it to its own power results in a number larger than the largest number which is a contradiction.

In other words, if we say that $$L$$ is the largest number, we're also saying that we can't increase it without producing a conceptually impossible number.
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### Re: Controversial concepts in mathematics

Silhouette wrote:An operation is a function, yes. They tell you what to do with inputs.

Alright, so you think that the symbol of infinity represents a function of some sort. But what kind of function? Any function or some specific function? If it represents some specific function, what kind of specific function? How many arguments does it have, of what type and what kind of value does it output?

They tell you what to do with inputs. Perhaps more appropriately, "infinity" tells you what not to do - to "not stop".

Yes, every function tells us what to do with the inputs i.e. how to determine the output based on the given inputs. And if infinity is a function, that means it also tells us what to do with the inputs. The question is: what does it tell us? But before I can ask that question, you have to tell me what kind of inputs it accepts.

Operands are what you do the operations to (e.g. finite numbers), and you don't "do" anything to infinity - infinity is what informs the manner of doing.

Yes, that's because you said that infinity is an operation and not an operand. I am aware of that. I am merely interested in what kind of operation you think infinity is. For example, I want to know what kind of operands it operates on.

It's not a limiting factor, like a finite number is - it's the opposite of limiting, because that's what infinity literally means. Operands are limiting factors: finites.

I don't know what you mean by "limiting factor". But if you're saying that infinity cannot be an operand, then you're saying that infinity is an unusual kind of function that cannot be used as an input to some other function.

This idea of a "direction" of summation isn't a thing. Addition is commutative: it doesn't matter what order you add in. I mean you could write a finite sum like $$\sum_{i=10}^{1}a_i$$ much to everyone's confusion, and just leave it to their common sense to infer that you want to add backwards for some reason, even though the order of addition doesn't affect the outcome. But because of commutativity, this is just redundant.

That's all true but it does not address my actual point which is that infinity does not merely represent direction.

And when I say infinity can at best mean "I don't know" or "indetermined" or "divergent" (as I stated immediately before I said this) I was talking about when infinity is casually used as an "answer in itself" - like when you try to say some expression "equals infinity". I wasn't talking about when it's used legitimately, such as in an infinite sum or product. So comments like "the probability of $$\sum_{i=1}^{\infty}a_i$$ representing $$a_1$$" etc. don't apply to what I was talking about.

I think the word "infinity" means the same thing in both expressions: $$x = \infty$$ and $$\sum_{i=1}^{\infty}a_i$$. And in both cases it does not mean "I don't know". If $$x = \infty$$ means "I don't know what's the value of $$x$$" then that means that $$x$$ can be equal to $$5$$ and that's certainly not what it means.
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### Re: Controversial concepts in mathematics

Magnus Anderson wrote:
Observer wrote:To be infinite means to be endless - there is no end to be beyond. The word "infinity" refers to that point just beyond which doesn't exist, even in concept. So the word is used to refer to a direction or inaccessible pole, not a number that could be used in any maths way. A proposed process can be infinite in that it will not end but nothing can ever go beyond infinite to reach infinity. The very concept of infinity is invalid in logic and maths.

If I understand you correctly, you are saying that the word "infinity" is an oxymoron (a contradiction, which is to say, something that is two or more opposing things at once) but that the word "infinite" is not. And I take it that what you mean by "infinity" is not what Google says it means (which is "the state or quality of being infinite") but rather "point at infinity".

I guess it could be called an "oxymoron" if you think of separating the "infinit" part from the "y" part. It is that "y" that implies something that the "infinit" part denies.

And I don't know where Google get the idea that "infinity" is a state - "We are in a state of infinity"? - "It is at a state of infinity"? It doesn't make sense.

Magnus Anderson wrote:And the reason you think that "point at infinity" is an oxymoron is because the word "infinite" means "endless" which means "no end" whereas "point at infinity" implies an end.

If that's an accurate representation of your position, I would like to ask the following question: what does the word "end" mean?

In order to answer this question, I would suggest defining the word "end" with respect to some specific concept e.g. the concept of sequence.

What does the word "end" mean with respect to sequences?

It means the point at which the sequence stops - nothing further. What is confusing about that?

Magnus Anderson wrote:Let me provide a quick definition of the word "sequence". A sequence is first of all a set of places each one of which is occupied by an element (no unoccupied places are allowed.) But that's not all it is. It is also an ordered set which means that for every two places in the set one comes before the other and one comes after the other. Furthermore, if one place comes after some other place, it also comes after all places that come before that other place. Also, if one place comes before some other place, it also comes before all places that come after that other place. This means that each place can be indexed i.e. represented by a unique natural number.

I think a sequence is just a progressing pattern but if you prefer using 100 words instead - ok.

Magnus Anderson wrote:The way I see it, the word "end" refers to the place with the highest index (one that comes after all other places) or to the place with the lowest index (one that comes before all other places.) (In some cases, which I believe we can safely ignore, the word "end" refers exclusively to the place with the highest index whereas the word "beginning" is used to represent the place with the lowest index.)

Or the initiation of a sequence and the finale of a sequence (assuming the sequence has a finale).

Magnus Anderson wrote:Thus, to say that a sequence is endless means that there is no place with the highest index and/or that there is no place with the lowest index.

"End" has nothing to do with a beginning.

Magnus Anderson wrote: (The problem with this is that it leaves out sequences that have a beginning and an end but that are nonetheless infinite. Fortunately, this can be safely ignored.)

If it has a beginning and an end it is finite, not in-finite.

Magnus Anderson wrote:That said, the 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?

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?

Magnus Anderson wrote:Perhaps you think that by saying that a sequence is infinite, we're also saying that the number of its places is equal to the number of places available in our imaginary universe or that the number of its places is equal to the largest number. I don't think that's what the word "infinity" means.

I don't know what you meant by it so I can't say if that is what "infinity" means - but it doesn't seem right - "in our imaginary universe"? - "largest number"? I don't think it has anything to do with imaginary universes and there is no largest number - Google it. There is the largest calculated number for now but that isn't the largest possible - just add 1 to it.

Magnus Anderson wrote:
What I disagree with is using the words "infinite" or "infinity" in conjunction with "number". They are not quantities but qualities. You might as well be saying - "n times yellow divided by 4 = ?" or "the smallest extreme yellow beyond yellow".

But then again, when you say that a queue consisting of people is infinite, you're talking about how many people there are in that queue. Aren't you? (:

And you are saying that the queue is endless - there is no last person in the queue - an infinity of people. But NOT "an infinite number" - because there is no number involved.

Magnus Anderson wrote:
Fine except that there is no quantity greater than any integer.

Why do you think so? Wouldn't that imply that saying that a set is infinite is a contradiction?

I have explained that several times now. An infinite set is not a quantity of items. It is a quality of the set - a set that has no bounds - no maximum, highest item.

Magnus Anderson wrote:
But there is no largest infinity. That is what Cantor was trying to say.

Because -
No matter how large the largest infinity is assumed to be, simply by raising it to its own power, it becomes unimaginably larger. And then the same can be done with that one - and again - and again. There is not any way to even express the largest possible infinity ("the highest cardinality"). It is the same as trying to find infinity - by definition such a quantity cannot exist.

If we say that $$L$$ is the largest number, raising it to its own power results in a number larger than the largest number which is a contradiction.

In other words, if we say that $$L$$ is the largest number, we're also saying that we can't increase it without producing a conceptually impossible number.

That is right. And because we can ALWAYS increase the number - that largest number can't exist.
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### Re: Controversial concepts in mathematics

Silhouette wrote:As for infinity specifying "the point where the summation stops", that's a clear contradiction in terms.

I will have to ask you a question similar to the one I asked obsrvr524.

What does the word "stop" mean with respect to infinite sums such as $$a_1 + a_2 + a_3 + \dotso$$?

I am of the strong opinion that when we say that "an infinite sum never stops" that we're using the word "stop" in a figurative sense for the simple reason that the word "stop" only applies to things that exist in time whereas infinite sums don't.

When we say that an infinite sum never stops what we mean to say is that it has no end. So the question actually boils down to the one I posed to [b]obsrvr524[/b which is:

What does the word "end" mean with respect to sequences?

This is, of course, based on the assumption that infinite sums are sequences.
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### Re: Controversial concepts in mathematics

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.

And almost always the sum of an infinite set is called "the limit" - the point that any infinite process would never quite reach. It isn't actually a real quantity but an infinitesimally close quantity - an issue of just being practical.
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You have been observed.
Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
It's just same Satanism as always -
• separate the bottom from the top,
• the left from the right,
• the light from the dark, and
• blame each for the sins of the other
• - until they beg you to take charge.
• -- but "you" have been observed --
obsrvr524
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### Re: Controversial concepts in mathematics

Magnus Anderson wrote:[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 wrote: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?
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### Re: Controversial concepts in mathematics

obsrvr524 wrote: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.)
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### Re: Controversial concepts in mathematics

Ecmandu wrote: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.
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### Re: Controversial concepts in mathematics

Magnus Anderson wrote:
obsrvr524 wrote: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.
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You have been observed.
Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
It's just same Satanism as always -
• separate the bottom from the top,
• the left from the right,
• the light from the dark, and
• blame each for the sins of the other
• - until they beg you to take charge.
• -- but "you" have been observed --
obsrvr524
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### Re: Controversial concepts in mathematics

Sorry I missed this -
Magnus Anderson wrote: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 -
.
<----------------------->

The infinite line requires a different index marker to accommodate greater than an infinity of indexe numbers.
Member of The Coalition of Truth - member #1

You have been observed.
Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
It's just same Satanism as always -
• separate the bottom from the top,
• the left from the right,
• the light from the dark, and
• blame each for the sins of the other
• - until they beg you to take charge.
• -- but "you" have been observed --
obsrvr524
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