A Possible Contradiction regarding the Set of All Natural Numbers and Its Size

[ryan]

It is said that there is an infinite number of elements in the set of natural numbers. Every natural number is succeeded by the next by adding 1. So then I could say that the number of times I add 1 to 1 will not only be the value of that natural number minus 1, but it will also be how many elements there are. For example, 5 elements in a set of successive natural numbers starting from 1 will mean that I added one 4 times.

Now think about the converse of the above idea. If I have 10 elements in a set of natural numbers that started from 1, then I know that there will be an n = 10. We would have added one 10 minus 1 times.

Finally, because there is an infinite number of elements in the set of all natural numbers, wouldn't it be completely reasonable to say that there must be an n that we had to add 1 to aleph 0 minus 1 times?


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For this not to be true, it is not enough to point out a contradiction for the negative in a different context because if there is an underlying flaw, we couldn't be sure it did not use the flaw to get there. It is only relevant to show what I did wrong mathematically.

 

[Juma]

wouldn't it be completely reasonable to say that there must be an n that we had to add 1 to aleph 0 minus 1 times?

No. Because infinite sets doesnt have a size. They are infinite.

 

[ryan]

wouldn't it be completely reasonable to say that there must be an n that we had to add 1 to aleph 0 minus 1 times?

No. Because infinite sets doesnt have a size. They are infinite.

Does this mean I couldn't say that the set of all natural numbers is larger than the set of natural numbers from 1 to 10? Or what about the fact that the infinite set of natural numbers is smaller than the set of real numbers; doesn't that imply size, at least in some sense of the word?

 

[beero1000]

It is said that there is an infinite number of elements in the set of natural numbers. Every natural number is succeeded by the next by adding 1. So then I could say that the number of times I add 1 to 1 will not only be the value of that natural number minus 1, but it will also be how many elements there are. For example, 5 elements in a set of successive natural numbers starting from 1 will mean that I added one 4 times.

Now think about the converse of the above idea. If I have 10 elements in a set of natural numbers that started from 1, then I know that there will be an n = 10. We would have added one 10 minus 1 times.

Finally, because there is an infinite number of elements in the set of all natural numbers, wouldn't it be completely reasonable to say that there must be an n that we had to add 1 to aleph 0 minus 1 times?


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For this not to be true, it is not enough to point out a contradiction for the negative in a different context because if there is an underlying flaw, we couldn't be sure it did not use the flaw to get there. It is only relevant to show what I did wrong mathematically.

All you're showing here is that your size measuring procedure is ill-defined on the set of all natural numbers. If you require that the only legal outputs are natural numbers, you can't complain about problems if you input a set that doesn't have size equal to a natural number.

 

[ryan]

It is said that there is an infinite number of elements in the set of natural numbers. Every natural number is succeeded by the next by adding 1. So then I could say that the number of times I add 1 to 1 will not only be the value of that natural number minus 1, but it will also be how many elements there are. For example, 5 elements in a set of successive natural numbers starting from 1 will mean that I added one 4 times.

Now think about the converse of the above idea. If I have 10 elements in a set of natural numbers that started from 1, then I know that there will be an n = 10. We would have added one 10 minus 1 times.

Finally, because there is an infinite number of elements in the set of all natural numbers, wouldn't it be completely reasonable to say that there must be an n that we had to add 1 to aleph 0 minus 1 times?


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For this not to be true, it is not enough to point out a contradiction for the negative in a different context because if there is an underlying flaw, we couldn't be sure it did not use the flaw to get there. It is only relevant to show what I did wrong mathematically.

All you're showing here is that your size measuring procedure is ill-defined on the set of all natural numbers.

I feel like one could say the same thing about Cantor's diagonal argument in that the size of the reals is ill-defined when it comes to using the naturals to measure it. There are not enough natural numbers to correlate to each element of aleph 0, just like there aren't enough natural numbers to correlate to each real number.

If you require that the only legal outputs are natural numbers, you can't complain about problems if you input a set that doesn't have size equal to a natural number.

But how can any output not be a natural number in the set of natural numbers?

 

[ryan]

Maybe, just maybe, there is a number of elements that is larger than any natural number but smaller than aleph 0. We could call this number, ryan's number.

 

[beero1000]

All you're showing here is that your size measuring procedure is ill-defined on the set of all natural numbers.

I feel like one could say the same thing about Cantor's diagonal argument in that the size of the reals is ill-defined when it comes to using the naturals to measure it. There are not enough natural numbers to correlate to each element of aleph 0, just like there aren't enough natural numbers to correlate to each real number.

If you require that the only legal outputs are natural numbers, you can't complain about problems if you input a set that doesn't have size equal to a natural number.

But how can any output not be a natural number in the set of natural numbers?

Because that is assuming that there is an upper bound on the natural numbers. There isn't.

Maybe, just maybe, there is a number of elements that is larger than any natural number but smaller than aleph 0. We could call this number, ryan's number.

Every infinite set has a countably infinite subset.

 

[ryan]

I feel like one could say the same thing about Cantor's diagonal argument in that the size of the reals is ill-defined when it comes to using the naturals to measure it. There are not enough natural numbers to correlate to each element of aleph 0, just like there aren't enough natural numbers to correlate to each real number.

If you require that the only legal outputs are natural numbers, you can't complain about problems if you input a set that doesn't have size equal to a natural number.

But how can any output not be a natural number in the set of natural numbers?

Because that is assuming that there is an upper bound on the natural numbers. There isn't.

I don't understand how that is an answer to my question.

Maybe the upper bound is aleph 0 but does include it in the set. Although this still doesn't help me understand this problem.

Maybe, just maybe, there is a number of elements that is larger than any natural number but smaller than aleph 0. We could call this number, ryan's number.

Every infinite set has a countably infinite subset.

Well, I am just trying to find something that makes sense to me because something does not seem right.

 

[beero1000]

I feel like one could say the same thing about Cantor's diagonal argument in that the size of the reals is ill-defined when it comes to using the naturals to measure it. There are not enough natural numbers to correlate to each element of aleph 0, just like there aren't enough natural numbers to correlate to each real number.

If you require that the only legal outputs are natural numbers, you can't complain about problems if you input a set that doesn't have size equal to a natural number.

But how can any output not be a natural number in the set of natural numbers?

Because that is assuming that there is an upper bound on the natural numbers. There isn't.

I don't understand how that is an answer to my question.

Maybe the upper bound is aleph 0 but does include it in the set. Although this still doesn't help me understand this problem.

Your output is either "the largest number in the set" or "the number of additions it takes to add all the numbers". Neither of those is defined if we try and apply your procedure to the set of all of the natural numbers. Your procedure does not expose a problem with set theory, it is just not well-defined.

Maybe, just maybe, there is a number of elements that is larger than any natural number but smaller than aleph 0. We could call this number, ryan's number.

Every infinite set has a countably infinite subset.

Well, I am just trying to find something that makes sense to me because something does not seem right.

There can be no set with cardinality greater than any finite number but less than that of the integers, so there's no ryan's number. Sorry.

 

[ryan]

I feel like one could say the same thing about Cantor's diagonal argument in that the size of the reals is ill-defined when it comes to using the naturals to measure it. There are not enough natural numbers to correlate to each element of aleph 0, just like there aren't enough natural numbers to correlate to each real number.

If you require that the only legal outputs are natural numbers, you can't complain about problems if you input a set that doesn't have size equal to a natural number.

But how can any output not be a natural number in the set of natural numbers?

Because that is assuming that there is an upper bound on the natural numbers. There isn't.

I don't understand how that is an answer to my question.

Maybe the upper bound is aleph 0 but does include it in the set. Although this still doesn't help me understand this problem.

Your output is either "the largest number in the set" or "the number of additions it takes to add all the numbers". Neither of those is defined if we try and apply your procedure to the set of all of the natural numbers. Your procedure does not expose a problem with set theory, it is just not well-defined.

But is it really any less defined than the concept of an infinite set with aleph 0 elements?

 

[beero1000]

But is it really any less defined than the concept of an infinite set with aleph 0 elements?

Very much so. Ill-definedness is not a 'this is counter-intuitive' check, it is 'are the rules given enough to determine an answer' check. The standard approach to cardinality is well-defined - for any two sets, either there exists a bijection between them or there is no bijection between them. This is unambiguous and completely defined for every pair of sets. The result may be counter-intuitive or confusing, and you might decide to reject the definition as unsuitable, but the important this is that there IS an answer, no matter which two sets are given as input.

Your procedure simply has no well-defined output when applied to the set of all natural numbers. It tries to find as output a natural number that is greater than all other natural numbers - something that does not exist.

 

[Juma]

No. Because infinite sets doesnt have a size. They are infinite.

Does this mean I couldn't say that the set of all natural numbers is larger than the set of natural numbers from 1 to 10? Or what about the fact that the infinite set of natural numbers is smaller than the set of real numbers; doesn't that imply size, at least in some sense of the word?

That is why cantors invented infinite numbers and extended the definition of size to include these. He wanted to measure the "infinite size" of infinite sets and to be able to order infinite sets by their size.

The smallest of the infinite numbers are the infinite size of the natural numbers, called aleph0.

 

[Juma]

Maybe, just maybe, there is a number of elements that is larger than any natural number but smaller than aleph 0. We could call this number, ryan's number.

Every nonempty set has elements. If you agree that you can pick elements you will see that every set contains a subset that has a bijectiv mapping to a set of natural numbers of same size. (That is your counting of the elements)

In this way you see that natural numbers are the most "sparse" of all infinite sets.

 

[ryan]

But is it really any less defined than the concept of an infinite set with aleph 0 elements?

Very much so. Ill-definedness is not a 'this is counter-intuitive' check, it is 'are the rules given enough to determine an answer' check. The standard approach to cardinality is well-defined - for any two sets, either there exists a bijection between them or there is no bijection between them. This is unambiguous and completely defined for every pair of sets. The result may be counter-intuitive or confusing, and you might decide to reject the definition as unsuitable, but the important this is that there IS an answer, no matter which two sets are given as input.

Your procedure simply has no well-defined output when applied to the set of all natural numbers. It tries to find as output a natural number that is greater than all other natural numbers - something that does not exist.

Can the input be infinite, then I would have a defined output? Can I divide all of the naturals by all of the naturals: {1,2,3, ...}/{1,2,3, ...} = {1,1,1, ...} = 1?

 

[beero1000]

Very much so. Ill-definedness is not a 'this is counter-intuitive' check, it is 'are the rules given enough to determine an answer' check. The standard approach to cardinality is well-defined - for any two sets, either there exists a bijection between them or there is no bijection between them. This is unambiguous and completely defined for every pair of sets. The result may be counter-intuitive or confusing, and you might decide to reject the definition as unsuitable, but the important this is that there IS an answer, no matter which two sets are given as input.

Your procedure simply has no well-defined output when applied to the set of all natural numbers. It tries to find as output a natural number that is greater than all other natural numbers - something that does not exist.

Can the input be infinite, then I would have a defined output? Can I divide all of the naturals by all of the naturals: {1,2,3, ...}/{1,2,3, ...} = {1,1,1, ...} = 1?

I can't tell you if you can do that or not, because I have no idea what that is supposed to mean.

Mathematics is a language with defined vocabulary and grammar, so the symbols and syntax you use matter. By ignoring all conventions and making up your own, you are doing yourself a major disservice, especially when you don't know enough to understand why the standard definitions are chosen the way they are. And to be honest, being on the other end isn't a great experience either.

I keep telling you this and you keep ignoring me and plowing ahead anyway. At this point, if you don't recognize that "{1,2,3, ...}/{1,2,3, ...} = {1,1,1, ...} = 1" is meaningless gibberish, I don't know what else to say. To have any chance of it making any sense, you'd need to completely redefine sets, division, and equality. More importantly, you'd need to tell other people those definitions for them to have any idea of what you're talking about. Have you done either of those things?

 

[Juma]

Very much so. Ill-definedness is not a 'this is counter-intuitive' check, it is 'are the rules given enough to determine an answer' check. The standard approach to cardinality is well-defined - for any two sets, either there exists a bijection between them or there is no bijection between them. This is unambiguous and completely defined for every pair of sets. The result may be counter-intuitive or confusing, and you might decide to reject the definition as unsuitable, but the important this is that there IS an answer, no matter which two sets are given as input.

Your procedure simply has no well-defined output when applied to the set of all natural numbers. It tries to find as output a natural number that is greater than all other natural numbers - something that does not exist.

Can the input be infinite, then I would have a defined output? Can I divide all of the naturals by all of the naturals: {1,2,3, ...}/{1,2,3, ...} = {1,1,1, ...} = 1?

Of course, you can define memberwise division for infinite dimensional vectors. But {1,1,1,1...} (an infinite dimensional vector) is not = 1 (since that is a scalar)

 

[beero1000]

Can the input be infinite, then I would have a defined output? Can I divide all of the naturals by all of the naturals: {1,2,3, ...}/{1,2,3, ...} = {1,1,1, ...} = 1?

Of course, you can define memberwise division for infinite dimensional vectors. But {1,1,1,1...} (an infinite dimensional vector) is not = 1 (since that is a scalar)

He was talking about division of sets, not vectors. You can try to add a vector space structure to get an ordering to the elements to do the division component-wise, or try a group structure to get a quotient set or index, or a bunch of other attempts. None of them really fix all the problems.

 

[ryan]

Can the input be infinite, then I would have a defined output? Can I divide all of the naturals by all of the naturals: {1,2,3, ...}/{1,2,3, ...} = {1,1,1, ...} = 1?

I can't tell you if you can do that or not, because I have no idea what that is supposed to mean.

Mathematics is a language with defined vocabulary and grammar, so the symbols and syntax you use matter. By ignoring all conventions and making up your own, you are doing yourself a major disservice, especially when you don't know enough to understand why the standard definitions are chosen the way they are. And to be honest, being on the other end isn't a great experience either.

I keep telling you this and you keep ignoring me and plowing ahead anyway. At this point, if you don't recognize that "{1,2,3, ...}/{1,2,3, ...} = {1,1,1, ...} = 1" is meaningless gibberish, I don't know what else to say. To have any chance of it making any sense, you'd need to completely redefine sets, division, and equality. More importantly, you'd need to tell other people those definitions for them to have any idea of what you're talking about. Have you done either of those things?

I will try this. As n goes to infinity (n E N) n/n, does this equal 1?

If it's allowed, then the input is infinite and the output is finite. Does this solve the problem you had with my "contradiction"?

 

[beero1000]

I can't tell you if you can do that or not, because I have no idea what that is supposed to mean.

Mathematics is a language with defined vocabulary and grammar, so the symbols and syntax you use matter. By ignoring all conventions and making up your own, you are doing yourself a major disservice, especially when you don't know enough to understand why the standard definitions are chosen the way they are. And to be honest, being on the other end isn't a great experience either.

I keep telling you this and you keep ignoring me and plowing ahead anyway. At this point, if you don't recognize that "{1,2,3, ...}/{1,2,3, ...} = {1,1,1, ...} = 1" is meaningless gibberish, I don't know what else to say. To have any chance of it making any sense, you'd need to completely redefine sets, division, and equality. More importantly, you'd need to tell other people those definitions for them to have any idea of what you're talking about. Have you done either of those things?

I will try this. As n goes to infinity (n E N) n/n, does this equal 1?

Yes.

If it's allowed, then the input is infinite and the output is finite. Does this solve the problem you had with my "contradiction"?

As far as I can tell, that would just define every set to have size 1. That's well-defined, but useless.

And it's not my problem with your contradiction. It's everyone's. Do you understand the necessity of having well-defined procedure for determining the size of sets? Do you understand that your procedure does not give a defined, unambiguous, and relevant result for each set?

 

[Juma]

I will try this. As n goes to infinity (n E N) n/n, does this equal 1?

Does what equal 1? You havent defined anything.

If it's allowed, then the input is infinite and the output is finite. Does this solve the problem you had with my "contradiction"?

You have actually no clue at all what you are doing.

How do you define the division of a set by another set? Why wiuld the result be a single number?

Clear these matters out first.