Trying to understand an apparent inconsistency (sets and subsets of natural numbers)

[NotJeffM]

Ok, I didn't know that.



Thanks, that makes sense. I am close to dropping this issue.



Here is just one last idea that your summation equation made me think of. This is also my attempted proof.


Assumptions:

- Every natural number in increasing order is greater than the previous number by 1.

- There is an infinite number of natural numbers.

Conclusion 1:

- 1 is being added to a previous natural number an infinite number of times.

Final Conclusion:

- There must be an infinite natural number


That is all I got. If you find something wrong with that (and I understand you), I will finally be at peace with all this.

We keep going round and round the same issue.

Let us say that the sum of an infinite sequence of 1’s is a number of some sort. (You understand, I hope, that a strict finitist would say that such a sum is a meaningless construct.)

You keep saying that such a sum x is a natural number. What are the definitionally required attributes of a natural number? Have you proved that x has each such attribute? If you have not defined what you mean by a natural number and logically proved that x satisfies the requirements, you are not justified in saying x is a natural number.

In fact, I can specify one of the attributes that I believe is necessary for a number a to be a natural number,
namely 2(a + 1) > a. But if x is a natural number, it is easily proved that 2(x + 1) = x. So I say that x is not a natural number.

In fact what I think you are doing is making an obvious error. There are many arithmetic operations on natural numbers that do not result in a natural number. 5 - 7. 7/3. sqrt(11). There is absolutely no a priori reason why adding up an infinite number of positive, natural numbers results in a natural number.

 

[Mates]

We keep going round and round the same issue.

Let us say that the sum of an infinite sequence of 1’s is a number of some sort. (You understand, I hope, that a strict finitist would say that such a sum is a meaningless construct.)

You keep saying that such a sum x is a natural number. What are the definitionally required attributes of a natural number? Have you proved that x has each such attribute? If you have not defined what you mean by a natural number and logically proved that x satisfies the requirements, you are not justified in saying x is a natural number.

Regarding what you are saying here, the only assumption that might be wrong is the first one. I thought it was a pretty safe assumption.

On Wikipedia, it says that there are 2 main formal methods to define natural numbers. One method uses "Peano axioms". The second axiom is of Peano axioms is, "Every natural number has a successor which is also a natural number". Then under the axioms, it says, "In ordinary arithmetic, the successor of x is x+1. But I am having trouble finding a source to confirm this in simple kinds of terms like that.

There is absolutely no a priori reason why adding up an infinite number of positive, natural numbers results in a natural number.

I know. That's why I am curious about all this. It seems as thought the definition of natural numbers allow it causing a contradiction.

 

[pka]

Regarding what you are saying here, the only assumption that might be wrong is the first one. I thought it was a pretty safe assumption.
On Wikipedia, it says that there are 2 main formal methods to define natural numbers. One method uses "Peano axioms". The second axiom is of Peano axioms is, "Every natural number has a successor which is also a natural number". Then under the axioms, it says, "In ordinary arithmetic, the successor of x is x+1. But I am having trouble finding a source to confirm this in simple kinds of terms like that.
I know. That's why I am curious about all this. It seems as thought the definition of natural numbers allow it causing a contradiction.

Instead of repeating the same statements over & over why not read
GEORG CANTOR His Mathematics and Philosophy of the Infinite by Joseph Dauben use the used market.

 

[NotJeffM]

Regarding what you are saying here, the only assumption that might be wrong is the first one. I thought it was a pretty safe assumption.

On Wikipedia, it says that there are 2 main formal methods to define natural numbers. One method uses "Peano axioms". The second axiom is of Peano axioms is, "Every natural number has a successor which is also a natural number". Then under the axioms, it says, "In ordinary arithmetic, the successor of x is x+1. But I am having trouble finding a source to confirm this in simple kinds of terms like that.

The article has a wealth of citations. More importantly, PKA, who has taught this stuff at the university level, has given you the name of a book to consult.

A property of natural numbers is that each has a successor.

[math]\text {Let } a \text { be a natural number.}\\ \therefore \text { the successor of } a \text { is a natural number.}\\ S(a) = a + 1.\\ \text {But } a + 0 = a.\\ \text {Assume, for purposes of contradiction, that } \exists \\ \text {a natural number } b \text { such that } S(b) = b.\\ \therefore \ b + 1 = b = b + 0 \implies 1 = 0, \text { which is false.}\\ \text {Thus, there is no natural number } b \text { such that } S(b) = b.\\ \text {But } \infty + 1 = \infty.\\ \text {Thus, } \infty \text { is not a natural number.} [/math]

I know. That's why I am curious about all this. It seems as thought the definition of natural numbers allow it causing a contradiction.

Would you please stop with “seems” and any such hand waving and give definitions and proofs.

 

[Mates]

The article has a wealth of citations. More importantly, PKA, who has taught this stuff at the university level, has given you the name of a book to consult.

Most responses to me by PKA just make me feel like crap. Unless it is a response directly addressing my post, I want nothing to do with that poster.

A property of natural numbers is that each has a successor.

[math]\text {Let } a \text { be a natural number.}\\ \therefore \text { the successor of } a \text { is a natural number.}\\ S(a) = a + 1.\\ \text {But } a + 0 = a.\\ \text {Assume, for purposes of contradiction, that } \exists \\ \text {a natural number } b \text { such that } S(b) = b.\\ \therefore \ b + 1 = b = b + 0 \implies 1 = 0, \text { which is false.}\\ \text {Thus, there is no natural number } b \text { such that } S(b) = b.\\ \text {But } \infty + 1 = \infty.\\ \text {Thus, } \infty \text { is not a natural number.} [/math]

I know that there are ways to prove that infinity is not a natural number. My problem is that I have come to a conclusion in my mind telling me otherwise.

Would you please stop with “seems” and any such hand waving and give definitions and proofs.

I gave you a definition to support my assumption. I put "seems" because I still don't believe that I found a contradiction within a definition of natural numbers. I gave you my "proof", and tried to support it; what else is left for me to show?

 

[NotJeffM]

Most responses to me by PKA just make me feel like crap. Unless it is a response directly addressing my post, I want nothing to do with that poster.



I know that there are ways to prove that infinity is not a natural number. My problem is that I have come to a conclusion in my mind telling me otherwise.



I gave you a definition to support my assumption. I put "seems" because I still don't believe that I found a contradiction within a definition of natural numbers. I gave you my "proof", and tried to support it; what else is left for me to show?

No, you have not given a proof. You say that [imath]\displaystyle \sum_{k=1}^{\infty} 1 = \infty[/imath] and then assert that that makes infinity a natural number. In the meantime, you say that you are aware that there are proofs that infinity does not meet the definition of a natural number. No one will take your assertion seriously unless you can show that every such proof is flawed.

Mathematicians like Cantor agree that that sum is a number (some mathematicians do not admit that it is a number at all). But no one contends that it meets the definition of a natural number. Nor have you made any effort to demonstrate that it does meet the definition of a natural number. You just say in essence that it is a natural number because you want it to be a natural number. “If wishes were horses, then beggars would ride.”

 

[Mates]

No, you have not given a proof. You say that [imath]\displaystyle \sum_{k=1}^{\infty} 1 = \infty[/imath] and then assert that that makes infinity a natural number. In the meantime, you say that you are aware that there are proofs that infinity does not meet the definition of a natural number. No one will take your assertion seriously unless you can show that every such proof is flawed.

I thought that if two proofs within the same system contradict each other, then the system is itself is flawed.

Nor have you made any effort to demonstrate that it does meet the definition of a natural number. You just say in essence that it is a natural number because you want it to be a natural number. “If wishes were horses, then beggars would ride.”

I tried to give you a reference in Post #22 from Wikipedia of a definition that would meet my first assumption in my attempted proof. It says "Every natural number has a successor which is also a natural number". Then under the axioms, it says, "In ordinary arithmetic, the successor of x is x+1".

 

[NotJeffM]

I thought that if two proofs within the same system contradict each other, then the system is itself is flawed.

You are correct

I tried to give you a reference in Post #22 from Wikipedia of a definition that would meet my first assumption in my attempted proof. It says "Every natural number has a successor which is also a natural number". Then under the axioms, it says, "In ordinary arithmetic, the successor of x is x+1".

Yes, but you are not talking about a number that is x = a + 1, are you? Any finite number plus 1 is still finite. It is not greater than any natural number.

Yes, if is a natural number, then a + 1 is a natural number. Specifically, 1 + 1 is a natural number. You then conclude that
b = 1 + 1 + 1 + … is a natural number. Prove it. Don’t assume it. If you do assume it, then infinity (if it even exists) is indeed a natural number. But there there are proofs that infinity is not a natural number. If you cannot show a flaw in each of those proofs, then your assumption and any conclusion dependent on that assumption are simply wrong.

I am sorry if I sound impatient. I sound that way because I am. You see an “inconsistency” because you feel a proposition “seems” true. But if that proposition is true, everyone else who has thought seriously about such problems for the last 140 years has been wrong; their proofs have a flaw. That of course is conceivable, but you need to PROVE that proposition and DEMONSTRATE those flaws. It’s not enough, not nearly enough, for you to say that adding an infinite number of positive values to a number is no different than adding a finite number of positive values to a number. It does not even begin to sound plausible. And it is inconsistent with logical proofs. Screaming “I‘m right” over and over again is not a compelling argument.

 

[Mates]

You are correct

Yes, but you are not talking about a number that is x = a + 1, are you? Any finite number plus 1 is still finite. It is not greater than any natural number. Yes, if is a natural number, then a + 1 is a natural number. Specifically, 1 + 1 is a natural number.

Then how does a sum of infinite 1's equal infinity? Going by what you say they are always being added to a finite number. This seems to mean that the sum of 1's from n=1 to infinity can only ever equal some n.

I am sorry if I sound impatient. I sound that way because I am.

You are very patient and respectful. That's all I can ask for. Again, I really do appreciate it.

You see an “inconsistency” because you feel a proposition “seems” true. But if that proposition is true, everyone else who has thought seriously about such problems for the last 140 years has been wrong; their proofs have a flaw. That of course is conceivable, but you need to PROVE that proposition and DEMONSTRATE those flaws. It’s not enough, not nearly enough, for you to say that adding an infinite number of positive values to a number is no different than adding a finite number of positive values to a number.

I hope that is not what an implication of what I am saying.

Screaming “I‘m right” over and over again is not a compelling argument.

Believe me, I do not think that I will be right about this. My "theories" are just where I am at with all of this.

 

[topsquark]

Then how does a sum of infinite 1's equal infinity? Going by what you say they are always being added to a finite number. This seems to mean that the sum of 1's from n=1 to infinity can only ever equal some n.

That's only if you add a finite number of 1s to each other. If you and an infinite number, you never reach a limit. So no natural number 1 + 1 + ... exists.

-Dan

 

[NotJeffM]

All I shall add to Dan’s succinct answer is this. There is a school of mathematicians (finitists), who simply say infinity is a meaningless concept. Most mathematicians accept the transfinite numbers as belonging to the family of numbers, but most sane people’s intuition about them tends to be wrong. It may be technically wrong to say it this way, but whenever you see “infinity,” try substituting ”without limit or end.” [imath]\aleph_0[/imath] is simply not like any other number we ever experience because we never experience anything that is without limit or end.

 

[Mates]

That's only if you add a finite number of 1s to each other. If you and an infinite number, you never reach a limit. So no natural number 1 + 1 + ... exists.

-Dan

I need to nail down some definitions. From what I understand, 1 + 1 + 1 + ... is not a natural number but rather something we call infinity? Is that accurate?

 

[topsquark]

I need to nail down some definitions. From what I understand, 1 + 1 + 1 + ... is not a natural number but rather something we call infinity? Is that accurate?

Not quite. Infinity is not a countable number, so it does not belong to the set of countable numbers. It is called a "cardinal number", which represent the "size" of sets. Finite cardinal numbers can be considered to be counting numbers, but the number of counting numbers is the "smallest" infinite cardinal number, [imath]\aleph _0[/imath], often called "countable infinity".

1 + 1 + ... is not actually anything. It's not even a number.

-Dan

 

[Mates]

Not quite. Infinity is not a countable number, so it does not belong to the set of countable numbers. It is called a "cardinal number", which represent the "size" of sets. Finite cardinal numbers can be considered to be counting numbers, but the number of counting numbers is the "smallest" infinite cardinal number, [imath]\aleph _0[/imath], often called "countable infinity".

1 + 1 + ... is not actually anything. It's not even a number.

-Dan

What about [imath]\displaystyle \sum_{k=1}^{\infty} 1 = \infty[/imath] ? Doesn't this mean that 1 + 1 + 1 ... is equivalent to infinty?

 

[NotJeffM]

What about [imath]\displaystyle \sum_{k=1}^{\infty} 1 = \infty[/imath] ? Doesn't this mean that 1 + 1 + 1 ... is equivalent to infinty?

It means that it is without limit, without end.

When we say [imath]\lim (\text {something}) = \infty[/imath], we mean that no natural number, no rational number, no real number can express it. In Cantor’s notation and thought, [imath]\aleph_0[/imath] is a transfinite number bigger than any natural number.

And ultimately the whole argument comes down to a simple definitional axiom about natural numbers. If a is a natural number, then a + 1 is a natural number greater than a. So if we assume that there is a number [imath]\aleph_0[/imath] (pronounced “aleph null”) that is the number of natural numbers, there are two possibilities, namely it is a natural number or it is not. If we assume that [imath]\aleph_0[/imath] is a natural number, then by definition the natural number [imath]\aleph_0 + 1 > \aleph_0[/imath], meaning the number of natural numbers is less than the number of natural numbers. That conclusion is nonsense on its face. So we must reject one of our two assumptions. We can logically conclude that either (a) there is no number that counts the natural numbers (Brouer’s position), or else (b) the number that counts the natural numbers is not itself a natural number (Cantor’s position). Either is logically consistent. What is not logically consistent is to say that [imath]\aleph_0[/imath] is a meaningful concept and that it is a natural number.

I wonder if your psychological impediment is simply that you do not know that you do NOT have to accept the idea that infinity is a meaningful concept. As far as I can see, most mathematicians accept Cantor’s position in abstract principle but mostly ignore it in what they actually do. (But my sight may be impaired because I am not a mathematician.)

I for one would be pleased if PKA pointed out the errors in this post.

 

[topsquark]

What about [imath]\displaystyle \sum_{k=1}^{\infty} 1 = \infty[/imath] ? Doesn't this mean that 1 + 1 + 1 ... is equivalent to infinty?

To go along with NotJeffM's comment, I usually write it something like this:
[imath]\displaystyle \sum_{k=1}^{\infty} 1 \to \infty[/imath]
to reinforce that the result is not a real number.

-Dan

 

[Mates]

It means that it is without limit, without end.

When we say [imath]\lim (\text {something}) = \infty[/imath], we mean that no natural number, no rational number, no real number can express it. In Cantor’s notation and thought, [imath]\aleph_0[/imath] is a transfinite number bigger than any natural number.

And ultimately the whole argument comes down to a simple definitional axiom about natural numbers. If a is a natural number, then a + 1 is a natural number greater than a. So if we assume that there is a number [imath]\aleph_0[/imath] (pronounced “aleph null”) that is the number of natural numbers, there are two possibilities, namely it is a natural number or it is not. If we assume that [imath]\aleph_0[/imath] is a natural number, then by definition the natural number [imath]\aleph_0 + 1 > \aleph_0[/imath], meaning the number of natural numbers is less than the number of natural numbers. That conclusion is nonsense on its face. So we must reject one of our two assumptions. We can logically conclude that either (a) there is no number that counts the natural numbers (Brouer’s position), or else (b) the number that counts the natural numbers is not itself a natural number (Cantor’s position). Either is logically consistent. What is not logically consistent is to say that [imath]\aleph_0[/imath] is a meaningful concept and that it is a natural number.

I wonder if your psychological impediment is simply that you do not know that you do NOT have to accept the idea that infinity is a meaningful concept. As far as I can see, most mathematicians accept Cantor’s position in abstract principle but mostly ignore it in what they actually do. (But my sight may be impaired because I am not a mathematician.)

I for one would be pleased if PKA pointed out the errors in this post.

I don't have any confusion about you say here. It all makes good sense to me.

However, I should tell you what I am thinking. I am realizing that I have a lot more questions about all of this. But I don't get the feeling that people come here to help this kind of an issue that has gone on this long and may go on much longer.

My interest in all of this is still very high, so I am conflicted about whether or not to continue.

I am not sure what to do.

 

[topsquark]

I don't have any confusion about you say here. It all makes good sense to me.

However, I should tell you what I am thinking. I am realizing that I have a lot more questions about all of this. But I don't get the feeling that people come here to help this kind of an issue that has gone on this long and may go on much longer.

My interest in all of this is still very high, so I am conflicted about whether or not to continue.

I am not sure what to do.

Suggestion:

Study the construction of the natural numbers and then study cardinality. When you have questions, open up a new thread for them.

-Dan

 

[Mates]

Suggestion:

Study the construction of the natural numbers and then study cardinality. When you have questions, open up a new thread for them.

-Dan

Ok, thanks

 

[NotJeffM]

I don't have any confusion about you say here. It all makes good sense to me.

However, I should tell you what I am thinking. I am realizing that I have a lot more questions about all of this. But I don't get the feeling that people come here to help this kind of an issue that has gone on this long and may go on much longer.

My interest in all of this is still very high, so I am conflicted about whether or not to continue.

I am not sure what to do.

I strongly suggest that you get the book recommended by PKA on Cantor, who was the first person to give logical meaning to infinities as numbers. You are interested in topics that go to the foundations of mathematics and to the philosophy of mathematics. It all requires very careful thinking.

As a starter to show you that others have had problems accepting Cantor’s conclusions (but not his logic), you might also look at this wiki article

Finitism - Wikipedia

en.wikipedia.org

and this one

Constructivism (philosophy of mathematics) - Wikipedia

en.wikipedia.org

If there is a book in English on the history of the detailed development of the concept of real numbers and on the foundations of mathematics, you need to read and understand that book because Cantor was embedded in the time and milieu where that work was done. (One way to find such books is to get a membership in a university program that allows you access to a university library. They usually do not cost much. I think mine costs $120 per year.)

In short, if you seriously want to grapple with these ideas, you need to read, and think deeply about, quite a few books and articles.

I would not continue this thread. Instead, start doing some reading (including bibliographies, especially annotated bibliographies). Come back here each time you get stuck. As you proceed, you will quickly go past what I can explain, but the person here who is probably most knowledgeable about these topics is PKA. He does not have my patience, but he knows the subject.

EDIT: Dan wrote a post while I was writing mine. I agree with him that starting with the axiomatic basis for the natural numbers and arithmetic will be a great introduction to the broader scope I suggested.