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Lord Antares

comparing infinities

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Let's say the collection of numbers A contains all number higher than 3 and lower than 4, including infinitesimals. This group is infinite.

Let's say that group B contains all numbers higher than 3 and lower than 5. This group is also infinite.

 

My question is if group B contains more numbers than group A?

Although B is a more ''extensive'' infinity, i.e. technically includes double the amount of numbers than infinity A, either one of them are infinite and therefore nothing can have a higher value than any one of them.

 

I would imagine this has been asked a trillion times over the course of history but I don't know of a definite conclusion.

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Both A and B contains infinitely many numbers. i.e B does not contain double many numbers as in A. It's because definition of infinity.
PS: There are as many numbers within 0 and 1 as there are in set of Real Numbers.

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Here is my (raw) point of view.


Either group is only infinite as a mathematical object.


As real world objects they are groups of elements that can be added to without end.


If (as real world groups) we stop the counting process at any stage* it is possible to add one more element to either group ,making that group temporarily larger.


As mathematical objects both groups need not concern themselves with real world practicalities and so the latter group can be seen in those terms as "twice the size " of the former-according to the mathematical convention.


*at a stage when both groups contain an equal number of elements.

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Let's say the collection of numbers A contains all number higher than 3 and lower than 4, including infinitesimals. This group is infinite.

Let's say that group B contains all numbers higher than 3 and lower than 5. This group is also infinite.

 

My question is if group B contains more numbers than group A?

Although B is a more ''extensive'' infinity, i.e. technically includes double the amount of numbers than infinity A, either one of them are infinite and therefore nothing can have a higher value than any one of them.

 

I would imagine this has been asked a trillion times over the course of history but I don't know of a definite conclusion.

There are different sizes of infinity. There are exactly as many positive numbers as there are even positive numbers, but there are more irrational numbers than there are even positive numbers despite both sets being infinite. For your question, I think those sets are the same size, but I'm not sure. I'd have to come up with a morphism between the two to check.

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I'd have to come up with a morphism between the two to check.

x -> 2(x-3)+3 . It's a bijective mapping, so in a sense the two sets have the same number of elements.

 

Sidenote: I don't know what the "infinitesimals" between 3 and 4 are, so I am assuming this thread refers to the real or the rational numbers.

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You can pair every number between each set with a single number from the other set, I wouldn't say they have the same amount of numbers in each set but they certainly have the same cardinality.

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Let's say the collection of numbers A contains all number higher than 3 and lower than 4, including infinitesimals. This group is infinite.

Let's say that group B contains all numbers higher than 3 and lower than 5. This group is also infinite.

 

My question is if group B contains more numbers than group A?

Although B is a more ''extensive'' infinity, i.e. technically includes double the amount of numbers than infinity A, either one of them are infinite and therefore nothing can have a higher value than any one of them.

 

I would imagine this has been asked a trillion times over the course of history but I don't know of a definite conclusion.

 

Please use the word set for your collection of numbers - this is the accepted correct term. Older names are aggregate or collection or sometimes class.

A 'group' is a very important particular type of set in mathematics and only some collections (sets) of numbers form groups.

 

OK so the set of whole numbers has nothing between each member of the set.

 

However the set is 'open ended' (has no beginning or end) as we can always add or subtract another 1 from any proposed first or last number.

 

We say that the set runs from negative infinity to positive infinity, although infinity itself is not a member of the set. That is infinity is not a whole number.

 

We use the positive whole numbers for counting (posh math word - enumeration) things.

 

In particular we can (try to) count the number of members of any given set. 1,2,3,4.. etc

We use this 'count' to measure the size of a set and compare the size of one set with another.

This works just fine for the number of members in a finite set.

So the set {1,3,5,7} which has 4 members is bigger than the set {1.3.5} which has only 3.

 

But we have already noted that there is no end to the process of adding 1 to the count. We never actually reach 'infinity'

 

So the number of members in the set of positive whole numbers is not a positive whole number.

 

It is in fact a (the first) transfinite number or 'infinity'.

 

Another way of looking at counting is the idea that we are putting the members of the counted set into one-to-one correspondence with the positive whole numbers.

 

1 2 3 4

W X Y Z

 

So in considering just the set of positive whole numbers we have found an infinity.

 

But we haven't included any of the fractional numbers in between, let alone those that can't be expressed as fractions.

 

So we are forced to the conclusion that more transfinite numbers are needed to place infinite sets into one-to-one correspondence.

 

How are you doing so far?

Edited by studiot

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From the fact that there exists an infinite number of numbers (real) between 0 and 1, can we conclude that the distance between 0 and 1 on the real line is also infinite, if one infinitesimal were to be considered as one unit of measurement?

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From the fact that there exists an infinite number of numbers (real) between 0 and 1, can we conclude that the distance between 0 and 1 on the real line is also infinite, if one infinitesimal were to be considered as one unit of measurement?

 

Infinitesimals are not numbers.

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Please use the word set for your collection of numbers - this is the accepted correct term. Older names are aggregate or collection or sometimes class.

A 'group' is a very important particular type of set in mathematics and only some collections (sets) of numbers form groups.

 

I am sorry. English is not my first language, no need to be condescending. Maybe you are not trying to be, but that's what I got out of how the post is formatted.

Also, what I meant by infinitesimals are all the possible smallest increments between each number which there are an infinity of. Maybe they're called increments or something else; again, it's just a language thing. I can use English well for general purposes, but these specific things never came up since I discussed them in my native language.

 

 

How are you doing so far?

 

Again, you're confusing me. Is that supposed to be condenscending towards me? What's the meaning behind this question?

 

I knew these things you said in your post logically, but you never answered my question, unless your answer was ''it's a paradox, there is no answer''.

Edited by Lord Antares

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I am sorry. English is not my first language, no need to be condescending. Maybe you are not trying to be, but that's what I got out of how the post is formatted.

Also, what I meant by infinitesimals are all the possible smallest increments between each number which there are an infinity of. Maybe they're called increments or something else; again, it's just a language thing. I can use English well for general purposes, but these specific things never came up since I discussed them in my native language.

 

 

 

Again, you're confusing me. Is that supposed to be condenscending towards me? What's the meaning behind this question?

 

I knew these things you said in your post logically, but you never answered my question, unless your answer was ''it's a paradox, there is no answer''.

 

Well I had no idea that English was not your first language, yours is very good.

 

So perhaps that is why you misread my purpose?

 

:)

 

Nor do I have any idea of your level of knowledge in mathematics.

 

All I have to go on is your use of words which (no offence) is imprecise.

 

Precision is very important in mathematics and I would have thought you might be interested in knowing the correct words.

I have put a lot of effort into avoiding the imprecise statements that are often made in answering questions about this subject.

 

But I did make a start on answering your question, I just did not finish it since there is a lot to take in.

 

So my question at the end was designed to find out if you had any problems with the necessary background.

 

So I'm glad to hear that you have followed it all.

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From the fact that there exists an infinite number of numbers (real) between 0 and 1, can we conclude that the distance between 0 and 1 on the real line is also infinite, if one infinitesimal were to be considered as one unit of measurement?

The number of members of a set is a different property than the "distance". What you're looking for is called "measure" or "metric", depending on what exactly you mean.

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Also, what I meant by infinitesimals are all the possible smallest increments between each number which there are an infinity of. Maybe they're called increments or something else; again, it's just a language thing. I can use English well for general purposes, but these specific things never came up since I discussed them in my native language.

There are no infinitesimals in the real numbers. This is not a language issue, it's a fundamental math issue.

 

An infinitesimal by definition is a nonzero quantity [math]x[/math] such that [math]x < \frac{1}{n}[/math] for each [math]n = 1, 2, 3, \dots[/math]

 

Clearly there is no such [math]x[/math] in the real number system. There are mathematical systems of numbers that do contain infinitesimals, but they are not the real numbers.

 

This is an important thing to understand about the reals.

 

The answer to your question is that there is a 1-1 correspondence between any two intervals of the real numbers, so we say they have the same number of points. But it's important to note that "same" is only a figure of speech. The mathematically significant fact is that there is a 1-1 correspondence.

Edited by wtf

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/cut

 

No harm done, it was just a misunderstanding. I understand that correct terminology is very important in math and I did consider the possiblity that you were just trying to teach me the correct usage, but I wasn't sure.

I am sure your knowledge of math is better than mine (assuming yours is solid), but in my opinion, this is stricly a logical matter of discussion and the benefit of knowing math here is being able to verify or form thoughts more precisely and also being able to deduce more correctly due to knowing accepted terminology and what it means, but it shouldn't be hard for me to follow what you are saying if I use only logic, if you know what I mean.

 

@wtf - yes, I used the term infinitesimal wrongly. That's not what I meant. I simply meant any real number which there is an infinity of between any given range of numbers because they can get exceedingly small the more you count them.

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@wtf - yes, I used the term infinitesimal wrongly. That's not what I meant. I simply meant any real number which there is an infinity of between any given range of numbers because they can get exceedingly small the more you count them.

This is incoherent. I say that not to be condescending or rude; but to emphasize that when you're trying to understand the real numbers, incoherent thinking is very unhelpful.

 

"... any real number which there is an infinity of between any given range of numbers because they can get exceedingly small the more you count them" simply does not express any thought or idea at all.

 

If you desire to understand the nature of the real number system, it's essential to think and speak clearly. These are not language issues, but rather conceptual misunderstandings you have about the real numbers.

 

I hope you can see that "... any real number which there is an infinity of between any given range of numbers because they can get exceedingly small the more you count them" does not express a coherent thought. What is the subject of the sentance? "Any real number?" That's only one number, there's nothing "between" it.

 

Can you explain more clearly what you are trying to say?

 

I am not trying to give you a hard time. I'm trying to help you to understand the nature of the real numbers.

Edited by wtf

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I undestand. I take no offense, it is only rational for you to try to understand exactly what I meant.

 

I simply meant to say: the amount of real (rational?) numbers is infinite between any two numbers.

For example, 3.9 is a real number. So is 3.99 and 3.999, 3.9999, 3.999932, 3.99999999 etc.

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I simply meant to say: the amount of real (rational?) numbers is infinite between any two numbers.

For example, 3.9 is a real number. So is 3.99 and 3.999, 3.9999, 3.999932, 3.99999999 etc.

Ok, no prob.

 

Now if we have the two intervals [math](3,4)[/math] and [math](3,5)[/math] from your original example, we can match up their points with a 1-1 correspondence via the function [math]y = 2x - 3[/math]. Each point of the first interval corresponds to exactly one point of the second, and vice versa.

 

So in this case we say that the two intervals have the same cardinality. Informally we can say they have the "same number of points," but that just means there is a 1-1 correspondence.

 

Of course the two intervals do in fact have different lengths. That shows that length and cardinality are different ways of assigning a notion of "size" to a set of points.

Edited by wtf

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This is where you need to be precise.

 

You are asking if your set B contains more numbers than your set A.

 

In order to approach this question you need to precisely define.

 

What you mean by more than?

Which in turn begs the question what do you mean by "How many numbers does either set have"?

 

Edit

Nice simple answer wtf.+1

Edited by studiot

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Although this answer was more along the lines of what I was looking for, it is obvious that it would give the same cardinality because both results are infinite.

I was still thinking since the range is double, it should affect the result but you are probably correct - every infinity is infinite in size and therefore nothing can be higher than it just by the nature of it.

 

But then aren't all sets of real numbers the same size (cardinality)? And isn't it impossible to list any one of them?
Also, it is interesting to think that set B has double the amount of members than A if you include only whole numbers. Also, if you include numbers up to one decimal point. And two, and three and however many you like except for infinite at which point B becomes equal to A. It's just a bit odd.

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Although this answer was more along the lines of what I was looking for, it is obvious that it would give the same cardinality because both results are infinite.

On the contrary, there are many different "sizes" of infinite sets and your question is a good one.

 

 

I was still thinking since the range is double, it should affect the result but you are probably correct - every infinity is infinite in size and therefore nothing can be higher than it just by the nature of it.

Any two intervals of real numbers are in bijection (synonym for 1-1 correspondence).

 

However the interval [math][3,5][/math] is indeed twice the length as the interval [math][4,5][/math]. The branch of math concerned with making that statement precise is called measure theory. https://en.wikipedia.org/wiki/Measure_(mathematics)

 

 

But then aren't all sets of real numbers the same size (cardinality)?

No, for example the set [math]\{1, 2, 3\}[/math] is clearly not the same size as the set of all real numbers.

 

Of course what you mean is, are all infinite sets of reals of the same cardinality? And the surprising answer is no! In the 1870's a brillian mathematician named Georg Cantor showed that some infinities are bigger than others; and that in particular, the infinity of all the counting numbers is strictly smaller than the infinite of all the real numbers. https://en.wikipedia.org/wiki/Cantor's_diagonal_argument

 

So now we have another question: Are all infinite sets of real numbers bijectable to either the set of all real numbers, or the set of counting numbers? This question is called the Continuum Hypothesis and it's one of the deepest questions in set theory. Nobody knows the answer, or even whether the question is meaningful; and if so, in what way. https://en.wikipedia.org/wiki/Continuum_hypothesis

 

I've given you a lot more info than you asked for. The tl;dr is that

 

* There are in fact many different sizes of infinity in modern math.

 

* Any two intervals of real numbers have the same cardinality.

 

* Even though they have the same number of points, two intervals can have different lengths. In effect, we can take the points in an interal of real numbers; rearrange or relabel them; and end up with an interval twice as large.

 

It's just a bit odd.

Poor Cantor was attacked for his ideas, and the stress drove him to be institutionalized for depression. Today his ideas are accepted as basic math. But yes, these are very odd ideas. We can be mathematically precise about infinity, and we can logically prove that there are infinitely many different levels of infinity.

 

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

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Great set of posts. Thanks to all so far in this thread - really good stuff.

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From the fact that there exists an infinite number of numbers (real) between 0 and 1, can we conclude that the distance between 0 and 1 on the real line is also infinite, if one infinitesimal were to be considered as one unit of measurement?

 

 

 

 

As the name suggests, the long line is a really long line, somehow “longer” than the regular number line. We can think of the regular number line as a bunch of unit-long intervals laid end to end. Specifically, there’s one interval for every integer. The long line is the same thing, except there’s one interval for every real number instead.

 

...

 

At least it would be nice if that were true.

https://blogs.scientificamerican.com/roots-of-unity/a-few-of-my-favorite-spaces-the-long-line/

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Since wtf has posted such an able discussion of numbers let us return to the other part of your question - that of infinitesimals.

 

Infinitesimals are not numbers and are not really used these days by pure mathematicians.

 

They are, however, of immense use in applied maths where numbers are given significance in some physical sense and called quantities.

 

Infinitesimals are quantities that are finite but small compared to the main bulk of the property or quantity being considered.

 

We can conceive of a sequence of these getting smaller and smaller and calculate what is known as a 'limit' for some compound property ( a quantity made up of more than one infinitesimal) which we regards as the 'value' of that property at a point.

 

A good example is density which is the ratio of mass to volume.

We call the density at a point the limit of this ratio as we shrink the infinitesimals of mass and volume.

Obviously they can never actually be allowed to reach zero or we would be trying to divide by zero.

 

At one time the differential calculus was predicated upon such a ratio but we adopt a different approach today.

Edited by studiot

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Since wtf has posted such an able discussion of numbers let us return to the other part of your question - that of infinitesimals.

 

Infinitesimals are not numbers and are not really used these days by pure mathematicians.

 

They are, however, of immense use in applied maths where numbers are given significance in some physical sense and called quantities.

 

Infinitesimals are quantities that are finite but small compared to the main bulk of the property or quantity being considered.

 

We can conceive of a sequence of these getting smaller and smaller and calculate what is known as a 'limit' for some compound property ( a quantity made up of more than one infinitesimal) which we regards as the 'value' of that property at a point.

 

A good example is density which is the ratio of mass to volume.

We call the density at a point the limit of this ratio as we shrink the infinitesimals of mass and volume.

Obviously they can never actually be allowed to reach zero or we would be trying to divide by zero.

 

At one time the differential calculus was predicated upon such a ratio but we adopt a different approach today.

 

I have never really seen infinitesimals used formally - are they the same as when a physicist talks about epsilon where epsilon is of arbitrary smallness ie a tiny addition / subtraction way smaller than the main variable

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I have never really seen infinitesimals used formally - are they the same as when a physicist talks about epsilon where epsilon is of arbitrary smallness ie a tiny addition / subtraction way smaller than the main variable

 

You may not have realised it but they are all over the place.

 

Take a cube [math]\delta x,\delta y,\delta z[/math] in continuum mechanics.

 

You can consider the electric/magnetic/fluid/heat flux through the cube and use the engineer's most popular equation

 

Input = output plus accumulation

 

to derive all sorts of useful stuff.

 

Or you can look at each face and note that there is (could be) a shear and normal stress associated with each face.

 

Flux or stress is something per unit area [math]\delta x\delta y[/math] etc and it makes no sense to have zero area in the denominator of a definition.

 

But for engineering purposes you still need a table of the stress at a point, the flow at a point etc.

 

 

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