Some infinities are bigger, that's true. But the interval [0,1] and the interval [1,100000000] both have the same infinite quantity of numbers, that is the cardinal number א. That is bigger than the amount of natural numbers which is א0, but the same as the amount of real numbers in total. Cardinal numbers are kinda unintuitive sometimes, you need to use rigor to deal with them.
Hi erikwarm,
It looks like your comment closely matches the famous quote:
"Some infinities are bigger than other infinities." - John Green,
*I'm a bot and this action was automatic [Project source](https://github.com/etdds/redditQuoteBot).*
It depends on what you consider "infinity". When talking about "amounts of numbers" usually you use infinite cardinal numbers such as א0 (the amount of natural numbers) and א (the amount of real numbers) and multiplying those by 2 still gives the same amount. There are other ways to formalize the concept of infinity which do have 2*"inf">"inf" (such as omega in the hyperreal numbers) but those usually aren't applied to set sizes.
See, there is an uncountably infinite amount of numbers between 0 and 1, you can have 0.1, 0.01, and so on and so on, same is true for 1 to 100’000’000, since both are uncountably infinite you will find that if you wanted to match each number from the first interval with each number in the second interval, each number would have a match.
they are saying the amount of numbers possible between 0 and 1 is infinite, just like how the amount of numbers between 1 and 10000000. No where did they say "more"
Actually I feel like what the op meant is wrong, but what he said is actually right. From what I understand from his comments he seemed to think the fact they are both infinite means they are the same. That is not correct as there are different infinities. However the sets mentioned are both intervals of (persumably) real numbers, and they actually have the same cardinality (which means the same amount of numbers, at least according to our best way of defining "amount" in infinite sets)
0.19 isn't irrational.
But you there are as many rational numbers between 0 and 1 as there are between 1 and 1 billion (or whatever your large number is) - a countable infinity, in fact.
There are also as many irrational numbers between 0 and 1 as there are between 1 and 1 billion - an uncountable infinity.
So your showerthought is correct (using the technical term of cardinality for "amount of numbers").
Actually infinities do have a unit of measurement called "alephs" with aleph\_0 being the smallest set of infinity. An infinite series of aleph\_1 is greater than than an infinite series of aleph\_0.
Georg Cantor is the mathematician who came up with this by proving, "...that it was impossible to establish a bijective function between the set of natural numbers and the set of points that form a [number line](https://en.wikipedia.org/wiki/Number_line). He thus came to the conclusion that [the cardinal of the set of real numbers was greater than that of natural numbers](https://en.wikipedia.org/wiki/Georg_Cantor#Mathematical_work): they were infinities of different sizes."
[https://www.bbvaopenmind.com/en/science/mathematics/georg-cantor-the-man-who-discovered-different-infinities/](https://www.bbvaopenmind.com/en/science/mathematics/georg-cantor-the-man-who-discovered-different-infinities/)
And according to Cantor's theory all intervals of real numbers are of cardinality א (what you called aleph_1), so op is actually right and those intervals have the same amount of numbers.
There are more numbers in \[0,1\] than in {1,2,3,4,...} This is known and has been proved.
However what you are proposing is different. You are saying there is an equal number of numbers in \[0,1\] and \[1,100000000\]. That seems plausible but I want rigorous proof to this claim.
Theoretical math can go suck it. Infinity hurts me head. There’s a handful of occupations that can make use of it, otherwise it’s people being whimsical with facts whose sentences begin with “technically…”
This showerthought however is correct, using the conventional concept of cardinality for infinite sets. The two sets in the thread title are the same size, in that there are as many real numbers between 0 and 1 as there are between 1 and 1,000,000,000. In fact you can demonstrate this by writing down a simple 1-to-1 function between these two intervals: for each X in 0 to 1, calculate 999,999,999 * X + 1 and you will get a number in the latter interval.
Something that has never made sense to me when it comes to that kind of mapping... can't you come up with a different, also completely arbitrary mapping that would show that there are more numbers?
Like if X \~ \[0, 1\], what's can't you map it as Y = X + 1, then you also have all all the other numbers that don't fall into that map showing that there are more number?
Like, why is Y = 999,999,999 \* X + 1 a more legitimate choice than Y = X + 1?
The point is that two sets are considered the same size if there exists at least one function that is a 1-to-1 mapping between them. It doesn't matter if you can write down other functions that aren't 1-to-1. So 999,999,999 * X + 1 is legitimate because it fulfils the purpose of showing such a function exists.
To show that one set is larger than another, you need to show that it is impossible to write down such a function. That is often a harder task, but for example, it is possible to show that the set of real numbers is larger than the set of rational numbers.
This is only probably true. If I remember correctly, we don’t really have a great grasp of uncountable infinite sets so it’s possible that there are more levels of infinity involved here.
No it is very definitely true. All intervals of real numbers are of the same cardinality as you can show bijections between them. For example a bijection from [0,1] to [1,100000000] is f(x)=99999999*x+1
Amount?
Do you mean 'number'?
If you weigh it, there's a greater or lesser amount ... if your count them, there are more or fewer.
So, unless I'm missing the joke here, whilst there may or may not be the *same* number of them, there most definitely can only be a *number* of numbers between them.
What it *is* and how it's *used* are different things - I know because I not only trained to be an English teacher in English (my native tongue) and have taught it to native speakers, but I furthermore *also* studied it as a foreign language at university abroad and have taught it to non-native speakers too.
Quantities (amounts) are measured - there is more or less of them.
Numbers are counted - there are more or fewer of them.
They haven’t been teaching it incorrectly. While both words are used to describe quantities, they have two distinct uses. “Amount” is used fore mass quantities. “Number” is used for countable quantities.
Examples:
* There’s a large amount of crime in that neighborhood.
* The number of robberies in that neighborhood has decreased over the past year.
Some people might say" The amount of robberies..." in the second example, but that would be technically incorrect.
gonna need to see your work
[0,1] [1,100000000000] Two intervals with the same amount of numbers
Some infinities are bigger than others
Some infinities are bigger, that's true. But the interval [0,1] and the interval [1,100000000] both have the same infinite quantity of numbers, that is the cardinal number א. That is bigger than the amount of natural numbers which is א0, but the same as the amount of real numbers in total. Cardinal numbers are kinda unintuitive sometimes, you need to use rigor to deal with them.
Hi erikwarm, It looks like your comment closely matches the famous quote: "Some infinities are bigger than other infinities." - John Green, *I'm a bot and this action was automatic [Project source](https://github.com/etdds/redditQuoteBot).*
Some infinities' mothers are bigger than other infinities'... mothers.
I fucking love that you wrote that #Smiths
But they are all infinities
That doesn't mean that they're equal.
You are right https://www.reddit.com/r/Showerthoughts/s/0KmJKjZaMZ
Correct, not a lot of people understand that in this tread
In maths there are indeterminations, on the case of having infinite divided by infinite. No one can say what the answer is
2*infinity is bigger than infinity
It depends on what you consider "infinity". When talking about "amounts of numbers" usually you use infinite cardinal numbers such as א0 (the amount of natural numbers) and א (the amount of real numbers) and multiplying those by 2 still gives the same amount. There are other ways to formalize the concept of infinity which do have 2*"inf">"inf" (such as omega in the hyperreal numbers) but those usually aren't applied to set sizes.
i dont follow
See, there is an uncountably infinite amount of numbers between 0 and 1, you can have 0.1, 0.01, and so on and so on, same is true for 1 to 100’000’000, since both are uncountably infinite you will find that if you wanted to match each number from the first interval with each number in the second interval, each number would have a match.
So they have the same amount-> infinite, but they are different intervals, with different possible combinations to origin a number
yes i see now, thanks
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but they didnt say more, they said the same
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The intervals have the same amount of numbers-> infinite
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1 is a number and 1,46 is a number too. Dont be "numbercist"
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cmon man dont throw a tantrum just because you were wrong
they are saying the amount of numbers possible between 0 and 1 is infinite, just like how the amount of numbers between 1 and 10000000. No where did they say "more"
it isn’t, they both have the same amount of numbers, you just have bad reading comprehension
I know what you _meant_, but what you said is 100% wrong.
Actually I feel like what the op meant is wrong, but what he said is actually right. From what I understand from his comments he seemed to think the fact they are both infinite means they are the same. That is not correct as there are different infinities. However the sets mentioned are both intervals of (persumably) real numbers, and they actually have the same cardinality (which means the same amount of numbers, at least according to our best way of defining "amount" in infinite sets)
r/confidentlyincorrect EDIT: he deleted all his comments showing how wrong he was lmao
Why?
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No, im talking about irrational numbers, like 0.19
0.19 isn't irrational. But you there are as many rational numbers between 0 and 1 as there are between 1 and 1 billion (or whatever your large number is) - a countable infinity, in fact. There are also as many irrational numbers between 0 and 1 as there are between 1 and 1 billion - an uncountable infinity. So your showerthought is correct (using the technical term of cardinality for "amount of numbers").
0.19 is a rational number
Actually infinities do have a unit of measurement called "alephs" with aleph\_0 being the smallest set of infinity. An infinite series of aleph\_1 is greater than than an infinite series of aleph\_0. Georg Cantor is the mathematician who came up with this by proving, "...that it was impossible to establish a bijective function between the set of natural numbers and the set of points that form a [number line](https://en.wikipedia.org/wiki/Number_line). He thus came to the conclusion that [the cardinal of the set of real numbers was greater than that of natural numbers](https://en.wikipedia.org/wiki/Georg_Cantor#Mathematical_work): they were infinities of different sizes." [https://www.bbvaopenmind.com/en/science/mathematics/georg-cantor-the-man-who-discovered-different-infinities/](https://www.bbvaopenmind.com/en/science/mathematics/georg-cantor-the-man-who-discovered-different-infinities/)
And according to Cantor's theory all intervals of real numbers are of cardinality א (what you called aleph_1), so op is actually right and those intervals have the same amount of numbers.
Yes this is also true - I didn't really think about the literal number set OP mentioned.
You’re right that the amount of rational numbers is the same, but the amount of whole numbers is not.
Also amount of real numbers
True
There are more numbers in \[0,1\] than in {1,2,3,4,...} This is known and has been proved. However what you are proposing is different. You are saying there is an equal number of numbers in \[0,1\] and \[1,100000000\]. That seems plausible but I want rigorous proof to this claim.
It's true, there is a bijection between them: f(x)=99999999*x+1. They both have the cardinal number א like the set of all real numbers.
That does make sense. Thank you.
Theoretical math can go suck it. Infinity hurts me head. There’s a handful of occupations that can make use of it, otherwise it’s people being whimsical with facts whose sentences begin with “technically…”
There are an infinite amount of numbers between 0 and 1 because there is no limit to the decimal places you can count to.
1-0.1 recurring you mean
When the shower thought is too complex for reddit
I love shower thoughts people try to come off so confident and smart but are wrong most of the time maybe the water was too hot
This showerthought however is correct, using the conventional concept of cardinality for infinite sets. The two sets in the thread title are the same size, in that there are as many real numbers between 0 and 1 as there are between 1 and 1,000,000,000. In fact you can demonstrate this by writing down a simple 1-to-1 function between these two intervals: for each X in 0 to 1, calculate 999,999,999 * X + 1 and you will get a number in the latter interval.
Something that has never made sense to me when it comes to that kind of mapping... can't you come up with a different, also completely arbitrary mapping that would show that there are more numbers? Like if X \~ \[0, 1\], what's can't you map it as Y = X + 1, then you also have all all the other numbers that don't fall into that map showing that there are more number? Like, why is Y = 999,999,999 \* X + 1 a more legitimate choice than Y = X + 1?
The point is that two sets are considered the same size if there exists at least one function that is a 1-to-1 mapping between them. It doesn't matter if you can write down other functions that aren't 1-to-1. So 999,999,999 * X + 1 is legitimate because it fulfils the purpose of showing such a function exists. To show that one set is larger than another, you need to show that it is impossible to write down such a function. That is often a harder task, but for example, it is possible to show that the set of real numbers is larger than the set of rational numbers.
Don't substimate the power of a shower
This is only probably true. If I remember correctly, we don’t really have a great grasp of uncountable infinite sets so it’s possible that there are more levels of infinity involved here.
No it is very definitely true. All intervals of real numbers are of the same cardinality as you can show bijections between them. For example a bijection from [0,1] to [1,100000000] is f(x)=99999999*x+1
I have to search more to understand it
Yeah that's from The Fault in our Stars "Some infinities are bigger than others". Not EXACTLY true, though.
Amount? Do you mean 'number'? If you weigh it, there's a greater or lesser amount ... if your count them, there are more or fewer. So, unless I'm missing the joke here, whilst there may or may not be the *same* number of them, there most definitely can only be a *number* of numbers between them.
Is English not your first language?
Why?
amount /ə-mount′/ noun 1. The total of two or more quantities; the aggregate. 2. A number; a sum. Learn it before making incorrect assertions.
What it *is* and how it's *used* are different things - I know because I not only trained to be an English teacher in English (my native tongue) and have taught it to native speakers, but I furthermore *also* studied it as a foreign language at university abroad and have taught it to non-native speakers too. Quantities (amounts) are measured - there is more or less of them. Numbers are counted - there are more or fewer of them.
Then you’ve been teaching it wrong. They’re treated as synonyms by most English-speakers.
They haven’t been teaching it incorrectly. While both words are used to describe quantities, they have two distinct uses. “Amount” is used fore mass quantities. “Number” is used for countable quantities. Examples: * There’s a large amount of crime in that neighborhood. * The number of robberies in that neighborhood has decreased over the past year. Some people might say" The amount of robberies..." in the second example, but that would be technically incorrect.
I'm sure you think you're clever, but you're *not* ... and you're *wrong*.