Is a number preceding infinity, finite?

[King]

Hi,

I'm not sure if this is the right section, but I'm talking about numbers :).

The questions is as written in the title: Is a number preceding infinity, finite?

 

[pwsnafu]

Hi,

I'm not sure if this is the right section, but I'm talking about numbers :).

The questions is as written in the title: Is a number preceding infinity, finite?

tl,dr version: no "number" (whatever that means) precedes infinity (which infinity?).

Long version: When you use the word "precede" it means there is a unique element which is less than it, but still larger than all others contenders. So on the integers, 3 precedes 4, because 3 < 4 but 3 > 1 and 2 . 10001 precedes 10002. And so on.

But on the rational numbers, nothing precedes, say, 2.
Why? Suppose there was, call it x. Then x &lt; \frac{2+x}{2} &lt; 2. So x can't precede it.

Similarly, take the smallest infinity \aleph_0. If n is a natural number that precedes it, when what about n+1?

What you can do, it look at larger infinities. So \aleph_0 precedes \aleph_1 which precedes \aleph_2 and so on.

 

[King]

But on the rational numbers, nothing precedes, say, 2.
Why? Suppose there was, call it x. Then x &lt; \frac{2+x}{2} &lt; 2. So x can't precede it.

Why is this? If x=0 the inequality makes sense.

I see it is as anything preceding infinity is finite. The reason is because inifinity is defined as a never ending value. Nothing can be greater than a never ending value, but I know plenty of values that are not never ending, and hence must be less than infinity. Of course infinity has some kind of strange philosophical aspect attached to it, and it sounds as though its separated from every other number.

 

[pwsnafu]

Why is this? If x=0 the inequality makes sense.

The inequality is true for any rational number less than two. That's the point.
If x=0, ie, we argue that that x=0 is the predecessor to 2, then the inequality gives 0<1<2, contradicting the definition of "preceding".
No matter what value of x we take, we can find a rational between the two.

I see it is as anything preceding infinity is finite.

"preceding" is not the same thing as "is less than". You are thinking of the latter.
To qualify for the former, there must be nothing in between the two values.

Edit: On second thoughts, scratch this. I'm thinking too much into it. Let's just use "less than" and be done with it.

The reason is because inifinity is defined as a never ending value.

That is not the definition of infinity. Importantly, you are using the word "infinity" in the singular. There are many infinities, some larger than others. \aleph_0 &lt; \aleph_1 &lt; \aleph_2 &lt; \ldots

We have two FAQs on this: https://www.physicsforums.com/showthread.php?t=510966"

Nothing can be greater than a never ending value,

Note that every real number has a never ending decimal representation. If we used your definition then every real number is infinite.

but I know plenty of values that are not never ending, and hence must be less than infinity.

Just because a property is true for some numbers does not mean it is true for every number.

 

[lavinia]

Hi,

I'm not sure if this is the right section, but I'm talking about numbers :).

The questions is as written in the title: Is a number preceding infinity, finite?

what other possibility would there be?

 

[pwsnafu]

what other possibility would there be?

Infinite. For example the infinity \aleph_0 is strictly less than the infinity \aleph_1, but none the less \aleph_0 is still an infinity.

I highly recommend you read the FAQ I linked.

 

[phinds]

what other possibility would there be?

So you believe pwsnafu is wrong, yes? why is that? What math do you use to say you are right and he is wrong?

 

[lavinia]

Infinite. For example the infinity \aleph_0 is strictly less than the infinity \aleph_1, but none the less \aleph_0 is still an infinity.

I highly recommend you read the FAQ I linked.

the posts here discuss the real numbers not the ordinals.

 

[lavinia]

So you believe pwsnafu is wrong, yes? why is that? What math do you use to say you are right and he is wrong?

I didn't say anybody was wrong. There needs to be some ordering of the extended real numbers before you can discuss what number precedes another number. this post seems to be discussing te extended reals rather than ordinals.

Maybe I don't understand your post. Explain.

 

[phinds]

I didn't say anybody was wrong. There needs to be some ordering of the extended real numbers before you can discuss what number precedes another number. this post seems to be discussing te extended reals rather than ordinals.

Maybe I don't understand your post. Explain.

The problem w/ answering the OPs question is, as explained by pwsnafu, the term "number preceding infinity". The point, as I see it, is that infinity minus 10 is STILL infinity, so I claim that "a number preceding infinity" is infinite, as does pwsnafu. You, on the other hand, seem to be saying that it is finite. I don't see how you can say that, and I don't see any confusion about my contending that you have said pwsnafu and I are wrong. We contend the answer is "infinite" and you contend that the answer is "finite". We can't both be right.

I'm not a mathematician, and I don't understand why you say "There needs to be some ordering of the extended real numbers before you can discuss what number precedes another number" in terms of THIS discussion, but since I DON'T understand it, maybe I'm missing something.

 

[lavinia]

The problem w/ answering the OPs question is, as explained by pwsnafu, the term "number preceding infinity". The point, as I see it, is that infinity minus 10 is STILL infinity, so I claim that "a number preceding infinity" is infinite, as does pwsnafu. You, on the other hand, seem to be saying that it is finite. I don't see how you can say that, and I don't see any confusion about my contending that you have said pwsnafu and I are wrong. We contend the answer is "infinite" and you contend that the answer is "finite". We can't both be right.

I'm not a mathematician, and I don't understand why you say "There needs to be some ordering of the extended real numbers before you can discuss what number precedes another number" in terms of THIS discussion, but since I DON'T understand it, maybe I'm missing something.

Infinity minus 10 has no meaning. When one says that a number precedes another number it must preceded it in some ordering.

Technically infinity is not a number unless you are talking about the ordinals. But in this post the numbers seem to be the reals plus infinity. But then why not minus infinity as well? And is minus infinity less than plus infinity? Are the finite numbers greater than minus infinity? Is minus infinity equal to plus infinity?

There is no arithmetic for the numbers plus infinity.

 

[pwsnafu]

There is no arithmetic for the numbers plus infinity.

Firstly, we have many ways of treating infinity as a number: ordinals, extended reals, real projectives, surreals, and the Riemann sphere to name a few. All of these define operations with it, so it is an arithmetic. And by the axiom of choice, there always is a well ordering. Whether we can write that ordering down is a another matter.

The problem with the OP was that he didn't specify which infinity he was talking about. If he wanted to talk about the extended reals so be it. But he must learn there are other possibilities.

Now, if we are to restrict the discussion to the extended reals, the answer is still no. The http://en.wikipedia.org/wiki/Extended_real_number_line" are the real numbers adjoined with positive and negative infinity. We define -\infty &lt; x &lt; \infty for all x \in \mathbb{R}. So negative infinity is an infinite that is less than positive infinity.

 

[King]

Regarding the definition of preceding:
1. Come before (something) in time.
2. Come before in order or position.
It doesn't have to be one position before. However, it should be safe to say that a number preceding another (where the numbers are ordered from lowest value to greatest value) has a lesser value. Hence, is the same as saying that it is less than.

Let me be a little bit more specific about my question. Given a set of real numbers, the largest possible number would be the (base - 1) recurring - using decimal, this would be 9 recurring. Because 9 is recurring indefinitely, surely one can state that it is infinite? I suppose using this definition the same could be said about 1 recurring, but one would argue that this value is still less than 9 recurring. I suppose this answers my question...1 recurring is infinite and is less than 9 recurring, hence a number less than infinity is not necessarily finite...

Does anyone agree with what I just said?

Hmm...perhaps I should be talking about natural numbers to avoid the negative numbers which do not apply in my scenario. What do the books of mathematics state about infinity in a set of natural numbers?

 

[ramsey2879]

Infinite. For example the infinity \aleph_0 is strictly less than the infinity \aleph_1, but none the less \aleph_0 is still an infinity.

I highly recommend you read the FAQ I linked.

The links you cited don't give any information on infinity. I agree that infinity \aleph_0 and \aleph_1 are both infinity, but it seems to me to be wrong to say that one infinity is less than another since infinity is not a number and it makes no sense to talk about infinity plus or minus a numerical value. Where do you get that one positive infinity is less than another positive infinity?
As I see it \aleph_0 and \aleph_1 are the same value.

 

[ramsey2879]

Regarding the definition of preceding:
1. Come before (something) in time.
2. Come before in order or position.
It doesn't have to be one position before. However, it should be safe to say that a number preceding another (where the numbers are ordered from lowest value to greatest value) has a lesser value. Hence, is the same as saying that it is less than.

Let me be a little bit more specific about my question. Given a set of real numbers, the largest possible number would be the (base - 1) recurring - using decimal, this would be 9 recurring. Because 9 is recurring indefinitely, surely one can state that it is infinite? I suppose using this definition the same could be said about 1 recurring, but one would argue that this value is still less than 9 recurring. I suppose this answers my question...1 recurring is infinite and is less than 9 recurring, hence a number less than infinity is not necessarily finite...

Does anyone agree with what I just said?

Hmm...perhaps I should be talking about natural numbers to avoid the negative numbers which do not apply in my scenario. What do the books of mathematics state about infinity in a set of natural numbers?

If you are talking of never ending decimal representatations with a fixed decimal point, you are talking of a finite limit which is not an infinite value, but a finite value.

 

[pwsnafu]

To King: in mathematics the term "successor" really does mean "one position after", for example, in Peano Axioms the successor function takes a number and returns the next number. "Predecessor" would mean something something similar, and in fact, it does in my field of work. So please, don't use the term.

Does anyone agree with what I just said?

That is not the definition of infinity people on this thread have been using. But if that is the definition you want to use, you are correct. Every real number has an infinity long decimal expansion. But that is not a useful definition

The links you cited don't give any information on infinity. I agree that infinity \aleph_0 and \aleph_1 are both infinity, but it seems to me to be wrong to say that one infinity is less than another since infinity is not a number and it makes no sense to talk about infinity plus or minus a numerical value. Where do you get that one positive infinity is less than another positive infinity?

The definition of \aleph_1 is the cardinality of the set of ordinal numbers. This is uncountable hence is strictly greater than \aleph_0. Applying the axiom of choice we can conclude there is nothing in between. http://en.wikipedia.org/wiki/Aleph_number"

 

[phinds]

As I see it \aleph_0 and \aleph_1 are the same value.

Well, the thing is, math operations don't really CARE what you think and they do not follow your "thought".

Aleph null and Aleph one are NOT the same value. I suggust you read up on them.

 

[ramsey2879]

To King: in mathematics the term "successor" really does mean "one position after", for example, in Peano Axioms the successor function takes a number and returns the next number. "Predecessor" would mean something something similar, and in fact, it does in my field of work. So please, don't use the term.



That is not the definition of infinity people on this thread have been using. But if that is the definition you want to use, you are correct. Every real number has an infinity long decimal expansion. But that is not a useful definition



The definition of \aleph_1 is the cardinality of the set of ordinal numbers. This is uncountable hence is strictly greater than \aleph_0. Applying the axiom of choice we can conclude there is nothing in between. http://en.wikipedia.org/wiki/Aleph_number"

Fine. Now we both know what we are talking about. If the cardinality of the countable natural numbers is \aleph_0 then \aleph_0 plus or minus 10 is also countable; you just count 10 more or 10 less. Therefore, \aleph_0 + 10 is the same value as \aleph_0. That is what I was trying to say.

 

[King]

I have concluded that I am working with integers. Can infinity be a member of the set of integers? As you may be able to tell, I am no mathematician, hence I have not been trained to understand infinity as some of you have. If infinity is not viewed as a number, then what is it? How else would you describe a number of unbounded size?

I'm truly trying to get my head around it, so please bare with me!

 

[lavinia]

Firstly, we have many ways of treating infinity as a number: ordinals, extended reals, real projectives, surreals, and the Riemann sphere to name a few. All of these define operations with it, so it is an arithmetic. And by the axiom of choice, there always is a well ordering. Whether we can write that ordering down is a another matter.

So what is the arithmetic of the extended reals?

What is the arithmetic of the Riemann sphere?

 

[lavinia]

I have concluded that I am working with integers. Can infinity be a member of the set of integers? As you may be able to tell, I am no mathematician, hence I have not been trained to understand infinity as some of you have. If infinity is not viewed as a number, then what is it? How else would you describe a number of unbounded size?

I'm truly trying to get my head around it, so please bare with me!

the ordinal numbers start with the integers. The next number following the integers is Aleph0. Unlike the integers it has infinitely many predecessors. One continues to generate new ordinals and finally reaches the first uncountable ordinal, Aleph1. It has the property that it is the first ordinal with uncountably many predecessors. Continuing further one generates a well ordered sequence of ordinals in which ordinals of ever increasing size are encountered.

two sets have the same size if there is a 1 - 1 and onto map between them. Thus the even integers have the same size as all of the integers. The size of an ordinal is taken to be the size of the set of its predecessors. It is a theorem that there is no largest ordinal although many have the same size. For instance, Aleph0 and Aleph0 + 1 have the same size though they are different ordinal numbers.

The real numbers have a size larger than Aleph0. The Continuum Hypothesis says that the reals have the size of the first uncountable ordinal. For years this was thought to be a theorem but was finally shown to be only one of the possibilities

 

[SteveL27]

I'm truly trying to get my head around it, so please bare with me!

I'd rather keep my clothes on, but thanks anyway.

 

[phinds]

I have concluded that I am working with integers. Can infinity be a member of the set of integers? As you may be able to tell, I am no mathematician, hence I have not been trained to understand infinity as some of you have. If infinity is not viewed as a number, then what is it? How else would you describe a number of unbounded size?

I'm truly trying to get my head around it, so please bare with me!

I'm not a mathematician, but I believe that lavinia's expanation is right, but I think it might be too abstruse an answer to your question, so here's my attempt: The problem w/ infinity vs a number is that the arithemeic just doesn't work the same, which is why it is dangerous to think of infinity as a number. If n is a number then it is never true that n +1 is the same as n but with infinity it is always true that infinity + 1 is the same as infinity.

Look up Hilbert's Hotel.

 

[SteveL27]

I'm not a mathematician, but I believe that lavinia's expanation is right, but I think it might be too abstruse an answer to your question, so here's my attempt: The problem w/ infinity vs a number is that the arithemeic just doesn't work the same, which is why it is dangerous to think of infinity as a number. If n is a number then it is never true that n +1 is the same as n but with infinity it is always true that infinity + 1 is the same as infinity.

For cardinals, yes. But for ordinals, x + 1 is not the same as x. Other posts in this thread have linked to the FAQ on ordinals. For this discussion, the ordinals are more important than the cardinals, because the OP asked about the predecessor of an infinite number.

If w is the ordinal corresponding to the usual order type of the natural numbers: 1, 2, 3, ..., then w + 1 is the ordinal successor to w; and it has a different order type; since w has no largest element, but w + 1 does.

Now the immediate predecessor of w + 1 is w. This is a good example of an infinite number whose immediate predecessor is also infinite; and this example does not depend on uncountable cardinals.

 

[phinds]

For cardinals, yes. But for ordinals, x + 1 is not the same as x. Other posts in this thread have linked to the FAQ on ordinals. For this discussion, the ordinals are more important than the cardinals, because the OP asked about the predecessor of an infinite number.

Thanks for that correction. As I said, I'm not a mathematician, so appreciate the information.

 

[King]

the ordinal numbers start with the integers. The next number following the integers is Aleph0. Unlike the integers it has infinitely many predecessors. One continues to generate new ordinals and finally reaches the first uncountable ordinal, Aleph1. It has the property that it is the first ordinal with uncountably many predecessors. Continuing further one generates a well ordered sequence of ordinals in which ordinals of ever increasing size are encountered.

Ok, now I think I have understood this, but of course I have understood only what I think I should have understood. Is the point being made here that Aleph is infinite, and that because of this, in order for infinity to be infinite it must have infinite predecessors? Hence, anything preceding infinity is infinite? I suppose infinity should be treated differently since there is no border between something being finite and something being infinite. It is not as though I can say finiteN+1=infinity because unlike infinity both finiteN and 1 are finite.

Have I finally made sense of the situation?

 

[lavinia]

Ok, now I think I have understood this, but of course I have understood only what I think I should have understood. Is the point being made here that Aleph is infinite, and that because of this, in order for infinity to be infinite it must have infinite predecessors? Hence, anything preceding infinity is infinite?
Have I finally made sense of the situation?

All infinite ordinals have infinitely many predecessors. All finite ordinals have finitely many predecessors.

The sequence looks like 0 1 2 3 4 ... n ... Aleph0, Aleph0 + 1, ..., Aleph0 + n, ... 2Aleph0 ... Aleph1 ...