Infinity: The Limit Concept and Cantor Transfinites

[Frank Peters]

Supposedly, infininity has been purged from mathematics. Both the infinitely small and the infinitely large have been replaced by the idea of a "limit."

For example, a series x0+x1+x3+... is not considered to be a literal infinite sum with infinite terms but only the limiting value of an arbitrarily large number of terms.

In other words, an completed infinity makes no sense in the concept of limit.

However, the notion of infinity introduced by Georg Cantot does admit completed infinities. In Cantor's ideas, infinity does indeed exist an an actual quantity, and there are even a hierarchy of infinities.

How are these two seemingly opposed viewpoints reconciled? If we cannot define an actual sum of infinite terms because infinity has been replaced by the limit concept, how can we admit the actual infinities defined by Cantor?

I hope I am being clear. If not then let me know.

 

[stevendaryl]

Supposedly, infininity has been purged from mathematics. Both the infinitely small and the infinitely large have been replaced by the idea of a "limit."

For example, a series x0+x1+x3+... is not considered to be a literal infinite sum with infinite terms but only the limiting value of an arbitrarily large number of terms.

In other words, an completed infinity makes no sense in the concept of limit.

However, the notion of infinity introduced by Georg Cantot does admit completed infinities. In Cantor's ideas, infinity does indeed exist an an actual quantity, and there are even a hierarchy of infinities.

How are these two seemingly opposed viewpoints reconciled? If we cannot define an actual sum of infinite terms because infinity has been replaced by the limit concept, how can we admit the actual infinities defined by Cantor?

I hope I am being clear. If not then let me know.

Cantor's theory has actual infinite objects, but there are no infinite (or infinitesimal) reals in the standard way of doing real numbers. So it's still true that proofs involving series involve limits, not actual infinity.

 

[Mark44]

Supposedly, infininity has been purged from mathematics. Both the infinitely small and the infinitely large have been replaced by the idea of a "limit."

Maybe "infininity" has been purged, but "infinity" still has a prominent role. As for limits, virtually every calculus textbook has statements such as the following:
$\lim_{x \to 0} \frac 1 {x^2} = \infty$ and $\lim_{x \to 0^-} \frac 1 x = -\infty$
Here the infinity symbols mean that the first fraction grows large without bound as x approaches 0, and that the second fraction becomes unboundedly negative as x approaches 0 from the negative side. Both of these limits can be stated more precisely and rigorously using variants of $\delta-\epsilon$ statements.

For example, a series x0+x1+x3+... is not considered to be a literal infinite sum with infinite terms but only the limiting value of an arbitrarily large number of terms.

Not sure why your series is missing x2, but maybe that's just an oversight.

An expression such as $x_0 + x_1 + x_2 + \dots$, or $\sum_{n = 0}^\infty x_n$ is an example of an infinite series, and is understood to have an infinite number of terms. It's possible that for some values of the $x_i$ terms, the series will add up to (or converge to) some finite number, but for other values, the series will become unbounded or otherwise not converge to a specific value.

What you describe as "the limiting value of an arbitrarily large number of terms" might not exist at all. This infinite series, $\sum_{n = 0}^\infty \frac 1 {2^n}$ represents an infinite number of terms that add up to 1. This infinite series, $\sum_{n = 1}^\infty \frac 1 n$ is the so-called harmonic series. It can be proven that the sequence of partial sums is unbounded, meaning that this series diverges.

 

[FactChecker]

Supposedly, infinity has been purged from mathematics.

Who says that?

Both the infinitely small and the infinitely large have been replaced by the idea of a "limit."

I disagree. The concepts of infinity and limit have existed coherently for a long time.

 

[RPinPA]

Since mathematics is by nature picky in the usage of language, I'm going to be picky about some of your statements.

For example, a series x0+x1+x3+... is not considered to be a literal infinite sum with infinite terms but only the limiting value of an arbitrarily large number of terms.

There is a term for every natural number n. There are infinitely many natural numbers. That is, the set of natural numbers is recognized to be infinite, and an "infinite set" is a well-defined thing. We are perfectly happy with the idea that some sets are infinite and some are not.

What you've picked up is that "infinity" is not a real number. So we don't for instance define 1/0 but we can define the limit of $1/a_n$ as $n \rightarrow \infty$ and we may have that $a_n \rightarrow 0$, and when we define that, everything in the definition is a real, finite number. We define the limit as a trend as n gets large but is always finite. We don't talk about the value "at" infinity.

And so the value of that sum is defined in terms of a limit. We define a partial sum as a sum of a finite number of terms, and we analyze what happens as the number of terms in the partial sums gets larger and larger, but remains finite.

In other words, an completed infinity makes no sense in the concept of limit.

I don't know if I'd say "makes no sense". But I would say that the definitions are careful to only talk about what is happening with real numbers, which are by nature finite.

It doesn't have to be that way. You can define a rigorous mathematics where $\infty$ and $-\infty$ are added to the set of numbers. In that case it's called the "extended reals".

However, the notion of infinity introduced by Georg Cantot does admit completed infinities. In Cantor's ideas, infinity does indeed exist an an actual quantity, and there are even a hierarchy of infinities.

No, in Cantor's ideas, sets are infinite. He didn't claim any real numbers are.

How are these two seemingly opposed viewpoints reconciled?
If we cannot define an actual sum of infinite terms because infinity has been replaced by the limit concept, how can we admit the actual infinities defined by Cantor?

We can define an actual set of infinite terms, using Cantor's ideas. There is one term as I said for every natural number. We can define a bijection between the set of natural numbers and the set of terms. Therefore the set of terms has the same cardinality as the set of natural numbers.

And we can define the sum of those terms in terms of finite things.

Real numbers are finite. Sets can be infinite, including the set of real numbers. There is no contradiction between those things.

In an infinite sum, each term is finite. The number of terms is infinite. The sum of the terms is defined as a limit, in terms of finite sums of terms (partial sums). The set of partial sums is infinite, but each individual partial sum is finite.

 

[SSequence]

In other words, an completed infinity makes no sense in the concept of limit.

However, the notion of infinity introduced by Georg Cantor does admit completed infinities. In Cantor's ideas, infinity does indeed exist an an actual quantity, and there are even a hierarchy of infinities.

One thing is that Cantor's notion of infinite doesn't seem to be required to do much of "usual" mathematics. Apparently, there has been much work on it. Obviously trying to pursue/understand the exact detail (which I don't know very well) requires time commitment ... (not to mention that there are good number of variations on this). But still, I can include a number of pointers if someone is interested.

For example, it seems to be generally favoured (apparently with good reasons) that FLT is provable in PA. You can search for it and you will find good number of links.

====================

One thing though that is rarely talked about and, I feel, "might" be a point of concern. Consider the following proposition:
p: "There is no bijection between ℕ and ℝ"

Now one of the problems that I feel is that when we conclude (in ZFC) that:
q: "There is a bijection between ℝ and ω₁"

The jump from p to q just seems a bit too much to me. I don't know whether it leads to some kind of actual "syntactic" issues or not though. Now it seems that some mathematicians do believe that ω₁ shouldn't be considered a set (ref. if needed). So there seem to be two different conceptual lines of thinking here.

====================

And even the plausibility that ZFC is incorrect about some number theory statements (more specifically, incorrect about halting of some programs) is also expressed sometimes (ref. if needed). Note that this can also imply ~con(ZFC) (see next sentence). But even if one is wrong about halting and it is detectable via purely finite calculations (in the sense that we say a program loops when it actually halts), it might not be possible to detect it in any reasonable time scale (notably this also doesn't cover consistency doesn't imply soundness issue anyway).

For example, consider the abstract "possibility" that some statement (with comparable description length to FLT) is incorrect (but we consider it to be correct). If we express it as a reasonably small program (say around 100 lines), then in the "worst-case scenario" we would have to wait till a time-scale BB(100) to find out that the statement was actually wrong.

====================

Regarding my own opinion about this, I think something like con(PA) is true (because of its proof). If we find a similar proof for con(ZFC) ... after enough verification ... that would decisively settle its consistency at least. But ofc people who are expert on this say that this seems to be very far away (ref. if needed). But I don't really know. This is just what I have read.

There is something important (with regards to this topic) that I have quite recently learned. Consider the following two problems:
A-- There is a "path of natural notations" below ω_CK
B-- con(ZFC)
Of course both are very old problems (from a purely mechanical viewpoint, one is a conceptual "problem" ... if it is one ... and the other is a syntactic problem). A is closely related to "natural notations" problem and it seems that Godel has explicitly mentioned (will describe the specific quote if someone is interested) both A and its related problem (I read this about an year ago and mentioned this in one of threads I made here).

Now what I have learned quite recently (based on some discussion on mathSE ... will provide link if someone is interested), about a month ago, is that if we consider the following possibilities/combinations:
p1-- A is not a "real" problem and B is true
p2-- A is not a "real" problem and B is false
p3-- A is a real problem (with a solution) and B is true
p4-- A is a real problem (with a solution) and B is false

Then the possibility p3 is unlikely (but still possible apriori). The (informal) reason is that, for p3 to exist, any method for A would quite likely somehow have to draw a "boundary line" at identifying ω_CK ... and failing after that (still apriori possible but hard to see for me). Otherwise, without a distinguishing boundary line, such a method just seems to show that ω₁ should not be considered a set.
And the sound of it does appear too fictional/fantastic admittedly. Possibly that also (partly) explains the pessimism around problem A?

====================

P.S.
As I have mentioned before, I had tried A (without even realising that I was trying the same problem ... because I wasn't even aware of existence of absurd recursive notations at that time) about 5 years ago. But after realising that the methods at that time didn't work, I didn't think about it again until 2 years ago. Now after a break of many months, I finally had the time to look at my method (which I had thought of an year ago) again (in last 2 months).

In hindsight, in my own approach (which didn't really work) for A, if they did work hypothetically, possibility p4 would have existed instead of p3. This is also true for my current approach (and the current specific method might be the last one I try ... since I can only think of few potentially meaningful modifications).
In a way, this does bring down my enthusiasm quite a bit (since I wasn't aware of this implication until a month ago). I have never heard of conception of ω₁ not being a set bringing up a purely syntactic issue. Since I had finally time to look upon the current method in last two months, it does work better than I expected. Since the underlying method is concrete (and fully specified) I probably need to test it more.

Currently I would rate the optimism level regarding the working of this specific method at 5% (part of the reason I am assessing it low is because possibility p3 doesn't seem to exist in this method). In a timeline of one year (if I do keep working on it) I think I can get a very good idea ... either it comes down to 0% (95% chance?) or goes up by around 30--60% (5% chance?).

 

[Frank Peters]

We define the limit as a trend as n gets large but is always finite. We don't talk about the value "at" infinity.

And so the value of that sum is defined in terms of a limit. We define a partial sum as a sum of a finite number of terms, and we analyze what happens as the number of terms in the partial sums gets larger and larger, but remains finite.
...
In an infinite sum, each term is finite. The number of terms is infinite. The sum of the terms is defined as a limit, in terms of finite sums of terms (partial sums). The set of partial sums is infinite, but each individual partial sum is finite.

You seem to be saying two different things here. But mathematic does not admit infinite sums, or an actual infinite series. Such things are always defined in terms of a limit.

However, post-Cantor, it seems that infinite sums are now realizable. We should be able to talk about a one-to-one correspondence between the terms of a convergent series and the natural numbers, which would mean that after aleph-null number of terms the series does actually EQUAL the limit.

Yet conventional mathematics says we can't talk about an actual "infinte" series but only a series that can become "arbitrarily large."

To me, in my admitted ignorance of the complete subject, it seems like an extreme incompatibility.

 

[stevendaryl]

You seem to be saying two different things here. But mathematic does not admit infinite sums, or an actual infinite series. Such things are always defined in terms of a limit.

However, post-Cantor, it seems that infinite sums are now realizable. We should be able to talk about a one-to-one correspondence between the terms of a convergent series and the natural numbers, which would mean that after aleph-null number of terms the series does actually EQUAL the limit.

Yet conventional mathematics says we can't talk about an actual "infinte" series but only a series that can become "arbitrarily large."

To me, in my admitted ignorance of the complete subject, it seems like an extreme incompatibility.

Cantorian set theory does allow us to talk about infinite objects, but it does not give any meaning to the concept of an infinite sum, other than the limit of finite sums. If you have a collection of reals $r_1, r_2, ...$, set theory allows us to group them all into a single infinite set. But it doesn't give any meaning to the sum of that set (except in the sense of limits).

 

[FactChecker]

How could mathematics NOT talk about infinite sets when there are so many of them (integers, real numbers, power sets)? And if one wants to talk about them, there must be some way of discussing infinity and different amounts of infinity. That has all been accomplished with great mathematical rigor. And given infinite sets, the concept of limits and cluster points can be studied. @Frank Peters should accept this and learn about it.

 

[Mark44]

You seem to be saying two different things here. But mathematic does not admit infinite sums, or an actual infinite series. Such things are always defined in terms of a limit.

No, neither of these is true. Please provide a citation of some mathematician who says this.
There are infinite series all over the place, such as this infinite series for $e^x$: $\sum_{n = 0}^\infty \frac {x^n}{n!}$. This series has an infinite number of terms, which makes it a sum with an infinite number of terms. One definition for $e^x$ is this series, not a limit.
There are many more examples such as this one.

However, post-Cantor, it seems that infinite sums are now realizable. We should be able to talk about a one-to-one correspondence between the terms of a convergent series and the natural numbers, which would mean that after aleph-null number of terms the series does actually EQUAL the limit.

This is nonsense.
Having a one-to-one correspondence between the terms of a convergent series and the natural numbers is trivial and irrelevant. The summation notation, $\sum_{k = 1}^\infty a_k = a_1 + a_2 + a_3 + \dots + a_k + \dots$ clearly shows how each term maps to a positive integer.

What is relevant is whether the sequence of partial sums for the series has a limit; if so, the series converges. The series converges if the sequence $S_n = \sum_{k = 1}^n a_k$ has a limit.

Yet conventional mathematics says we can't talk about an actual "infinte" series but only a series that can become "arbitrarily large."

To repeat what @FactChecker said, who says we can't talk about an actual infinite series? I've given one example of an "actual" infinite series, and can provide a ton more of them if you need more convincing.

If you are unable to back up your assertions about what "conventional mathematics" says about infinite series, I will lock this thread.

 

[Stephen Tashi]

Supposedly, infininity has been purged from mathematics.

I hope I am being clear. If not then let me know.

You aren't being mathematically clear. To precisely define what it means for a concept to be "purged from mathematics", you would have to give a mathematically precise definition of the concept.

The words "infinity" and "infinite" do not descibe unique mathematical concepts. They have different meanings in different contexts.

 

[Mark44]

To me, in my admitted ignorance of the complete subject

Don't you think this might be a problem?