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.
“how sad” sure I’m at uni and you need a YouTube video to defend yourself because you don’t know the subject matter, andyou are trying to get my attention.
You have more important stuff to do than continuing this thread, might i suggest reading about the subject matter on Wikipedia?
Lmao you replied to me in the first place, exactly how am I trying to get your attention? I already had it from the beginning…
You gripe about the merits of a youtube video (which I only linked to because I’m not gonna spell the whole damn proof out for you here), and you tell me to go read wikipedia? I’m guessing you are just being sarcastic there, because if not… sheesh. Yikes. Oof, even.
…? 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.
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.
“how sad” sure I’m at uni and you need a YouTube video to defend yourself because you don’t know the subject matter, andyou are trying to get my attention.
You have more important stuff to do than continuing this thread, might i suggest reading about the subject matter on Wikipedia?
Lmao you replied to me in the first place, exactly how am I trying to get your attention? I already had it from the beginning…
You gripe about the merits of a youtube video (which I only linked to because I’m not gonna spell the whole damn proof out for you here), and you tell me to go read wikipedia? I’m guessing you are just being sarcastic there, because if not… sheesh. Yikes. Oof, even.
…? 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.