You can do arithmetic with infinite. Defining an operator on the set of hyperreal numbers is pretty simple. It’s actually really cool, because it’ll teach you about scales of infinities and infinitesimals.
And, while you could define a set to includes your cat, that doesn’t make your cat a number. I can make a set of six apples, and none of those are numbers. That is categorically not how numbers work.
I mean, you can square root a negative in complex numbers, but you can’t in real numbers. That doesn’t mean that the square root of negative one isn’t a number.
The “problem” is defining what you mean by “infinite” specifically. Infinite is an adjective that you can assign to a set of numbers, and the “infinity” would be the summation of that set, but what set are we talking about? Is it all natural numbers? Rational numbers? Real numbers? Even numbers? Powers of 10? These are all different infinities with different properties. Some can give very odd results, especially with analytic continuation. The set of natural numbers {1,2,3,…} can be evaluated to infinity (ℵ₀) or -1/12. You have to get more specific when dealing with infinite numbers; you need to define what infinity is when you work with it.
The “problem” is defining what you mean by “infinite” specifically. Infinite is an adjective that you can assign to a set of numbers, and the “infinity” would be the summation of that set…
Incorrect.
example: א_0 is an infinity, specifically a size of the natural number set, and is not a sum of any set.
Another example: infinity in real function analysis, is a concept of unbounded growth, either positive or negative.
The set of natural numbers {1,2,3,…} can be evaluated to infinity (ℵ₀) or -1/12.
Incorrect based on previous mistake.
You’re describing a series sum, no series sum is 0_א, that’s a major mistake, as it is specifically a set size and not a natural number.
And the 2nd concept you’re referring to leads to a contradiction as a sum of positives must be positive, this means that in order to get -1/12 you must make a mistake.
These are all already well defined (except for (naive) set theory, but it’s irrelevant to this), and you don’t need to “define what infinity means when you use it”, that’s nonsense.
You are literally proving my point. You have used at least three different definitions when using the word “infinity”. THAT is what I mean when I say you need to define what is being referred to by “infinity”. It is not a single concept in mathematics.
To address your specific points:
ℵ₀ is the cardinality of countable infinities like natural numbers, rational numbers, etc.
If you attempt to find the summation of an infinite series, you approach infinity.
I never claimed that ℵ₀ is the summation of a set. You base so much of your commenr on a claim I never made.
I said that the natural numbers can be EVALUATED to either infinity or -1/12 and I made sure to define what I meant by infinity to be ℵ₀. If you think that it is incorrect that the natural numbers can be evaluated to -1/12, you have no place trying to correct others on mathematics. Just watch this eleven year old video by Numberphile for proof.
Your fundamental misunderstanding and flip-floppong between definitions of infinity male my point glaringly clear here.
Don’t be a dumbass and cite a fucking YouTube video to someone giving you definitions, i honestly guessed you were going to come with VSauce and Numberphile even before you made this reply because i watched them so many years ago.
I’ve studied these at uni, I’ve even cited the courses I’ve studied these from. So don’t go “your fundamental misunderstanding blah blah” bro you’re citing a YouTube video.
You’re only raging because the only defense you have is a YouTube video, which i already saw the proof of about 10 years ago.
At least give an actual insult instead of impotent “i guess you failed the course you don’t remember blah blah”, for a course I finished the second part of last semester. So no reason to forget it as I’m expected to use it still.
You should have read about the topic instead of whatever this response is.
You are the one devolving to ad hominem. You haven’t addressed a point I have made in your last two comments. You seem to think that a YouTube video is some lowly source that doesn’t warrant merit. How sad then that you were proven wrong by a youtube video. YOU are the one who lacks any defense because you KNOW you were wrong, and by failing to address my points with facts you are proving that point.
…? I believe I said the set of hyperreal numbers, which would contain the real numbers cross joined with the set of infinites and infinitesimals. Technically, the infinity that bounds the natural numbers would be any of those infinites. I can’t really point to one specific infinity.
Infinity wouldn’t be in the set of natural, rational, real, or even complex numbers. It acts as a boundary for all of those sets of numbers, but you could have a set that also includes infinity, in addition to those sets, making them the extended number set.
However, I want to point out an issue with what you said about those being different infinities… That isn’t strictly true. Natural numbers, rational numbers, and powers of ten are the same level of infinite. Crazy as it is to imagine, the sets can be functionally mapped to each other. They have the same infinite of elements to them. It isn’t until you add in those irrational numbers that the level of infinite increases. There is a higher order of infinite more numbers in the irrational numbers than in the rational numbers, so that infinite IS bigger.
ℵ₀, the infinity that represents the cardinality of natural numbers, would not be “any infinity” in the set of hyperreal numbers. You have a fundamental misunderstanding of the concept of infinitea if you cannot point to a specific number that “bounds the natural numbers” because that number is ℵ₀ and can be pointed to. It is the only countable infinity. Bring in irrationals and now it is uncountable, because there are an infinite number of numbers between 1 and 2. You can never reach 2 if you counted every number between 1 and 2. The cardinality of irrational numbers is ℵ1, a distinctly different and larger infinity than ℵ₀.
Sets like naturals and rationals may have the same cardinality, but they are not functionally the same. Cardinality is just one attribute they share. The powers of 10 cannot be analytically continued to -1/12 like the natural numbers can. Therefore they are functionally different.
Yes, obviously there are different ordinalities to infinites, but for the express purpose of this comic, the particular infinity does not functionally change the comic. The infinity that bounds the real numbers is technically the one that matters, which you are suggesting is somehow different depending on the collection of numbers used to calculate it, which it doesn’t. The collection of powers of tens is the exact same as the collection of natural numbers, at infinite scale. It is not some power more, or different. The Euler zeta function doesn’t work like that.
Also, there are also an infinite amount of rational numbers between any two rational numbers, even without irrational numbers. The ordinality of the infinite doesn’t matter about that, but the ordinality of irrational numbers between them is bigger.
Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms.
Sure you could probably apply these things to “operators on the set”, but that’s completely missing the point and pretending that something you’ve applied to infinity to get a number, is the same thing as infinity itself. Turning infinity into to a number by treating it in some other context, is not the same thing as infinity being a number.
It’s like trying to claim that 1 is an even number because look, all you need to do is define a set of rules that multiplies it by 2.
And, while you could define a set to includes your cat, that doesn’t make your cat a number. I can make a set of six apples, and none of those are numbers. That is categorically not how numbers work.
That was exactly my point. It’s not how infinity works either.
Hey, you know what, I’ll make this easier for you. Name a mathematical operator you can’t do on infinity.
You can add it, you can subtract it, you can multiply and divide it, square root it, exponent it. You’ll get some weird answers for some of those, like subtracting infinity from itself, but it’s still doable. I mean, unless you’re suggesting zero or negative one isn’t a number.
If you can’t do something in any meaningful or consistent way, you can’t do it.
You “can” add 1 to infinity, but it doesn’t give a result that is measurably 1 greater, and therefore has not obeyed the basic axioms of arithmetic.
unless you’re suggesting zero or negative one isn’t a number.
There is no number you can add, subtract, multiply or divide infinity with, that results in zero. Unless, as I keep saying, you’re acting in some meta-contextual space in which infinity is counted as a countable “representation” of infinity, which isn’t the same thing as infinity being a number. I realize this is exactly what you are trying to do, and it’s not as clever as you seem to think it is, so you can stop repeating yourself.
Also, one divided by infinity is zero in an extended number line.
See, again with the added qualifications and sleight-of-hand redefinitions. It isn’t equal to zero, it’s just zero “in an extended number line”. You’re being argumentative and contrarian for its own sake, and I think you’re probably intelligent enough to know full well that that’s what you’re doing, but not quite intelligent enough to realize that you shouldn’t.
Also,
That seems pretty axiomatic, buddy.
you’re just wrong. I feel like you can’t accept that.
it’ll teach you about
Hey, you know what, I’ll make this easier for you.
Can you knock off the obnoxiously condescending phrases every other sentence? They don’t add to your argument, they just make you sound like a prick.
Infinity isn’t a real number, pal. So, I have to qualify it, because you’ll say that it’s impossible to divide by zero, chief. Unless you have an answer that isn’t infinity, friend. I can remove the qualifier if you’d like, and the answer is still technically correct, boss. I am willing to accept I’m wrong if you can answer which number one divided by zero is, I could very well be wrong.
Apologies for my upbeat tone, I’m Canadian, you see, and I believe that we should always use a friendly tone when teaching people new things, ya scamp. I realize this probably comes across to you as arrogant, but I think that might be more related to you thinking I’m wrong when I’m not.
Infinity plus any number besides negative infinity in the extended number line is always infinity. That seems pretty axiomatic, buddy. Infinity times a positive is infinity. Infinity times a negative is negative infinity. Are you suggesting that all numbers follow all rules of each other? That would be silly. There are always exceptions. Zero to the zeroth power, for example. Infinity is related to zero in a lot of ways.
As for your saying that there are no ways to apply infinity to equal zero, that’s also not strictly true. We can absolutely converge certain infinitely scaling functions to 0, even in real numbers or integers. You can even subtract infinities to get 0.
Infinity does not always represent a concept. It only represents a concept in real numbers because it’s not a real number. But neither is the square root of negative one. Unless you would like to show me 2_i_ of something. But 2_i_ is undoubtably a number.
It’s not clever, you’re just wrong. I feel like you can’t accept that.
Okay. All of those operations exist on the hyperreal numbers, or even just the extended real numbers, so I don’t see the issue. It’s not turned into a number, it IS a number, and it doesn’t represent anything other than itself in those contexts.
I could define a special class of numbers that includes my cat, and it wouldn’t make my cat a number either.
If you can’t do arithmetic with it, it’s a not a number.
You can do arithmetic with infinite. Defining an operator on the set of hyperreal numbers is pretty simple. It’s actually really cool, because it’ll teach you about scales of infinities and infinitesimals.
And, while you could define a set to includes your cat, that doesn’t make your cat a number. I can make a set of six apples, and none of those are numbers. That is categorically not how numbers work.
I mean, you can square root a negative in complex numbers, but you can’t in real numbers. That doesn’t mean that the square root of negative one isn’t a number.
The “problem” is defining what you mean by “infinite” specifically. Infinite is an adjective that you can assign to a set of numbers, and the “infinity” would be the summation of that set, but what set are we talking about? Is it all natural numbers? Rational numbers? Real numbers? Even numbers? Powers of 10? These are all different infinities with different properties. Some can give very odd results, especially with analytic continuation. The set of natural numbers {1,2,3,…} can be evaluated to infinity (ℵ₀) or -1/12. You have to get more specific when dealing with infinite numbers; you need to define what infinity is when you work with it.
Incorrect. example: א_0 is an infinity, specifically a size of the natural number set, and is not a sum of any set. Another example: infinity in real function analysis, is a concept of unbounded growth, either positive or negative.
Incorrect based on previous mistake. You’re describing a series sum, no series sum is 0_א, that’s a major mistake, as it is specifically a set size and not a natural number.
And the 2nd concept you’re referring to leads to a contradiction as a sum of positives must be positive, this means that in order to get -1/12 you must make a mistake.
These are all already well defined (except for (naive) set theory, but it’s irrelevant to this), and you don’t need to “define what infinity means when you use it”, that’s nonsense.
You are literally proving my point. You have used at least three different definitions when using the word “infinity”. THAT is what I mean when I say you need to define what is being referred to by “infinity”. It is not a single concept in mathematics.
To address your specific points:
ℵ₀ is the cardinality of countable infinities like natural numbers, rational numbers, etc.
If you attempt to find the summation of an infinite series, you approach infinity.
I never claimed that ℵ₀ is the summation of a set. You base so much of your commenr on a claim I never made.
I said that the natural numbers can be EVALUATED to either infinity or -1/12 and I made sure to define what I meant by infinity to be ℵ₀. If you think that it is incorrect that the natural numbers can be evaluated to -1/12, you have no place trying to correct others on mathematics. Just watch this eleven year old video by Numberphile for proof.
Your fundamental misunderstanding and flip-floppong between definitions of infinity male my point glaringly clear here.
Don’t be a dumbass and cite a fucking YouTube video to someone giving you definitions, i honestly guessed you were going to come with VSauce and Numberphile even before you made this reply because i watched them so many years ago.
I’ve studied these at uni, I’ve even cited the courses I’ve studied these from. So don’t go “your fundamental misunderstanding blah blah” bro you’re citing a YouTube video.
Me citing a youtube video proved your statement wrong and this is your response.
Guessing you failed the class you were studying this in? Definitely doesn’t sound like you remember much.
You’re only raging because the only defense you have is a YouTube video, which i already saw the proof of about 10 years ago.
At least give an actual insult instead of impotent “i guess you failed the course you don’t remember blah blah”, for a course I finished the second part of last semester. So no reason to forget it as I’m expected to use it still.
You should have read about the topic instead of whatever this response is.
You are the one devolving to ad hominem. You haven’t addressed a point I have made in your last two comments. You seem to think that a YouTube video is some lowly source that doesn’t warrant merit. How sad then that you were proven wrong by a youtube video. YOU are the one who lacks any defense because you KNOW you were wrong, and by failing to address my points with facts you are proving that point.
…? I believe I said the set of hyperreal numbers, which would contain the real numbers cross joined with the set of infinites and infinitesimals. Technically, the infinity that bounds the natural numbers would be any of those infinites. I can’t really point to one specific infinity.
Infinity wouldn’t be in the set of natural, rational, real, or even complex numbers. It acts as a boundary for all of those sets of numbers, but you could have a set that also includes infinity, in addition to those sets, making them the extended number set.
However, I want to point out an issue with what you said about those being different infinities… That isn’t strictly true. Natural numbers, rational numbers, and powers of ten are the same level of infinite. Crazy as it is to imagine, the sets can be functionally mapped to each other. They have the same infinite of elements to them. It isn’t until you add in those irrational numbers that the level of infinite increases. There is a higher order of infinite more numbers in the irrational numbers than in the rational numbers, so that infinite IS bigger.
ℵ₀, the infinity that represents the cardinality of natural numbers, would not be “any infinity” in the set of hyperreal numbers. You have a fundamental misunderstanding of the concept of infinitea if you cannot point to a specific number that “bounds the natural numbers” because that number is ℵ₀ and can be pointed to. It is the only countable infinity. Bring in irrationals and now it is uncountable, because there are an infinite number of numbers between 1 and 2. You can never reach 2 if you counted every number between 1 and 2. The cardinality of irrational numbers is ℵ1, a distinctly different and larger infinity than ℵ₀.
Sets like naturals and rationals may have the same cardinality, but they are not functionally the same. Cardinality is just one attribute they share. The powers of 10 cannot be analytically continued to -1/12 like the natural numbers can. Therefore they are functionally different.
Yes, obviously there are different ordinalities to infinites, but for the express purpose of this comic, the particular infinity does not functionally change the comic. The infinity that bounds the real numbers is technically the one that matters, which you are suggesting is somehow different depending on the collection of numbers used to calculate it, which it doesn’t. The collection of powers of tens is the exact same as the collection of natural numbers, at infinite scale. It is not some power more, or different. The Euler zeta function doesn’t work like that.
Also, there are also an infinite amount of rational numbers between any two rational numbers, even without irrational numbers. The ordinality of the infinite doesn’t matter about that, but the ordinality of irrational numbers between them is bigger.
https://en.wikipedia.org/wiki/Arithmetic
Sure you could probably apply these things to “operators on the set”, but that’s completely missing the point and pretending that something you’ve applied to infinity to get a number, is the same thing as infinity itself. Turning infinity into to a number by treating it in some other context, is not the same thing as infinity being a number.
It’s like trying to claim that 1 is an even number because look, all you need to do is define a set of rules that multiplies it by 2.
That was exactly my point. It’s not how infinity works either.
Hey, you know what, I’ll make this easier for you. Name a mathematical operator you can’t do on infinity.
You can add it, you can subtract it, you can multiply and divide it, square root it, exponent it. You’ll get some weird answers for some of those, like subtracting infinity from itself, but it’s still doable. I mean, unless you’re suggesting zero or negative one isn’t a number.
If you can’t do something in any meaningful or consistent way, you can’t do it.
You “can” add 1 to infinity, but it doesn’t give a result that is measurably 1 greater, and therefore has not obeyed the basic axioms of arithmetic.
There is no number you can add, subtract, multiply or divide infinity with, that results in zero. Unless, as I keep saying, you’re acting in some meta-contextual space in which infinity is counted as a countable “representation” of infinity, which isn’t the same thing as infinity being a number. I realize this is exactly what you are trying to do, and it’s not as clever as you seem to think it is, so you can stop repeating yourself.
Also, one divided by infinity is zero in an extended number line.
See, again with the added qualifications and sleight-of-hand redefinitions. It isn’t equal to zero, it’s just zero “in an extended number line”. You’re being argumentative and contrarian for its own sake, and I think you’re probably intelligent enough to know full well that that’s what you’re doing, but not quite intelligent enough to realize that you shouldn’t.
Also,
Can you knock off the obnoxiously condescending phrases every other sentence? They don’t add to your argument, they just make you sound like a prick.
For reference, saying “in an extended number line” is the equivalent of saying “including infinity”.
Like, if it’s the real number line, extended real number line means real numbers and infinity and negative infinity.
Infinity isn’t a real number, pal. So, I have to qualify it, because you’ll say that it’s impossible to divide by zero, chief. Unless you have an answer that isn’t infinity, friend. I can remove the qualifier if you’d like, and the answer is still technically correct, boss. I am willing to accept I’m wrong if you can answer which number one divided by zero is, I could very well be wrong.
Apologies for my upbeat tone, I’m Canadian, you see, and I believe that we should always use a friendly tone when teaching people new things, ya scamp. I realize this probably comes across to you as arrogant, but I think that might be more related to you thinking I’m wrong when I’m not.
Infinity plus any number besides negative infinity in the extended number line is always infinity. That seems pretty axiomatic, buddy. Infinity times a positive is infinity. Infinity times a negative is negative infinity. Are you suggesting that all numbers follow all rules of each other? That would be silly. There are always exceptions. Zero to the zeroth power, for example. Infinity is related to zero in a lot of ways.
As for your saying that there are no ways to apply infinity to equal zero, that’s also not strictly true. We can absolutely converge certain infinitely scaling functions to 0, even in real numbers or integers. You can even subtract infinities to get 0.
Infinity does not always represent a concept. It only represents a concept in real numbers because it’s not a real number. But neither is the square root of negative one. Unless you would like to show me 2_i_ of something. But 2_i_ is undoubtably a number.
It’s not clever, you’re just wrong. I feel like you can’t accept that.
Okay. All of those operations exist on the hyperreal numbers, or even just the extended real numbers, so I don’t see the issue. It’s not turned into a number, it IS a number, and it doesn’t represent anything other than itself in those contexts.