I heard that there are different ranks of infinity, like $\aleph_0, \aleph_1, \aleph_2$, etc, my question is, the base of natural log, i.e. '$e$' is defined by a limit of taking $n\rightarrow$infinity , will the limit for the value of '$e$' be different if $n\rightarrow\aleph_1$ or $n\rightarrow\aleph_2$ or $n\rightarrow\aleph_3$ etc? That is if $n$ tends to different ranks of infinity, will the limit be different?

  • $\begingroup$ DIFFERENT RANKS of infinity??? Do you really know what INFINITY is? $\endgroup$ – SchrodingersCat Jan 26 '16 at 6:23
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    $\begingroup$ Easy, easy. ¶ @Henry: The notation $\lim_{x \to \infty} f(x) = L$ is shorthand for a mathematical statement that doesn't make direct reference to any infinity, so the differing orders of infinity don't come into play. What the above equation means, symbolically, is that for any positive value $\varepsilon$, $f(x)$ is within $\varepsilon$ of $L$ for all $x$ beyond some point that may depend on $\varepsilon$. For instance, we say that $\lim_{x \to \infty} 1/x = 0$ because for any $\varepsilon > 0$, $|1/x-0| = 1/x < \varepsilon$ for any $x > 1/\varepsilon$. Note: No $\infty$. $\endgroup$ – Brian Tung Jan 26 '16 at 6:44
  • $\begingroup$ I'm referring only to standard analysis in the foregoing, incidentally. $\endgroup$ – Brian Tung Jan 26 '16 at 6:46
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    $\begingroup$ $\lim_{x\to\infty}$ is just shorthand for $x$ growing without bound within the reals, it doesn't deal with any real infinity. $\endgroup$ – Zelos Malum Jan 26 '16 at 7:00
  • $\begingroup$ I don't think the downvotes on this are fully justified (at least not if they are based merely on the OP mixing up the various different notions comprised under the name "infinity"). It a very important question and it should especially be addressed that in symbolisms such as $\lim_{x\to\infty}$ or $\sum_{n=0}^\infty$, there isn't really an object/entity "infinity" involved and this contrasts a lot with the infinite cardinalities ... $\endgroup$ – Hagen von Eitzen Jan 26 '16 at 7:36

I'm going to assume that by the limit definition of $e$, you mean

$$ e=\lim_{n \rightarrow \infty} {\left(1+\frac{1}{n}\right)^n} \quad \text{where } n\in \mathbb{N} $$

In a way (but not all ways!), the $\infty$ can be thought of as the number of objects $n$ will pass by on the way to $e$.

The number of elements in $\mathbb{N}$, called the cardinality of $\mathbb{N}$, is $\aleph_0$.

The following definition of $e$ is also valid:

$$ e=\lim_{x \rightarrow \infty} {\left(1+\frac{1}{x}\right)^x} \quad \text{where } x\in \mathbb{R} $$

The cardinality of $\mathbb{R}$ is $\mathfrak{c}$, where it is known that $\mathfrak{c} > \aleph_0$ (and if the continuum hypothesis is true then $\mathfrak{c} =\aleph_1$).

So in this case it seems that passing $\aleph_0$ objects in $\mathbb{N}$ is the same as passing $\mathfrak{c}$ objects in $\mathbb{R}$, which isn't surprising since $\mathbb{N} \subset \mathbb{R}$.

Consider the following description of these limits using their definitions: $$ \forall \epsilon > 0 \ \exists N \in \mathbb{N}: n\geq N \Rightarrow |a_n - e| <\epsilon \text{ where } (a_n):\mathbb{N} \rightarrow \mathbb{R}\text{ is defined by }a_n = \left(1+\frac{1}{n}\right)^n $$ and $$ \forall \epsilon > 0 \ \exists \delta \in \mathbb{R}: x\geq \delta \Rightarrow |f(x) - e| <\epsilon \text{ where } f:\mathbb{R} \rightarrow \mathbb{R}\text{ is defined by }f(x) = \left(1+\frac{1}{x}\right)^x $$

Is there anything that can be said about some type of limit that involves more than $\mathfrak{c}$ objects?

In topology the notion of the limit of a sequence is generalized by the limit of a topological net. For a topological space $(X,\tau)$ and a directed set $\Lambda$, a net is some function $(x_\lambda): \Lambda \rightarrow X$.

We can talk about convergence, that is the net $(x_\lambda)_{\lambda \in \Lambda}$ converges to $x \in X$ and we write:

$$ \lim_{\lambda \in \Lambda} x_\lambda =x $$

which has a similar definition to the ones above:

$$ \text{For every neighbourhood } U \text{ of } x \text{, there exists } \lambda_0 \in \Lambda \text{ such that } \lambda \geq \lambda_0 \Rightarrow x_\lambda \in U $$

So think about the notions presented here $$ n \rightarrow \infty \text{ where } n \in \mathbb{N} \text{ and } |\mathbb{N}| = \aleph_0 $$ $$ x \rightarrow \infty \text{ where } x \in \mathbb{R} \text{ and } |\mathbb{R}| = \mathfrak{c} $$

With nets we can talk about the situation where $\lambda \in \Lambda$ and $|\Lambda| > \mathfrak{c}$.

Is it possible to define some net $(x_\lambda)_{\lambda \in \Lambda}$ for some topological space $(X,\tau)$ and some directed set $\Lambda$ where $|\Lambda|>\mathfrak{c}$ that behaves like $(a_n)^{\infty}_{n=1}$ and $f$ above? It's gotten too abstract to make that easy to talk about.

Although certainly, since the limit $x\in X$, then $x\neq e$ if $e \notin X$.

  • $\begingroup$ Do I understand you right? You're suggesting that limits of sequences are limits up to $\aleph_0$ and limits over $\Bbb R$ are limits up to $\frak c$ and so on? $\endgroup$ – Asaf Karagila Jan 28 '16 at 5:32
  • $\begingroup$ No, not really, just that one limit is defined on a sequence with indices in $\mathbb{N}$ (a set with cardinality $\aleph_0$), and one limit is defined on a function with a domain in $\mathbb{R}$ (a set with cardinality $\mathfrak{c})$. $\endgroup$ – Adam Francey Jan 28 '16 at 5:37

There are different kinds of infinity.

When talking about limits you see if you consult the definition of $\lim_{x\to\infty}$ you'll see that there's actually no infinity involved. This is because the concept of infinity is a bit confusing and here's a occasion where you can easily avoid an actual infinity.

This is a case of infinity that could be called potential infinity. While there's no actual infinity involved. The quantities are all finite, but we don't restrict the size of them.

Another concept of the theme is Bachmann-Landau notation (or ordo notation). It's a notation describing the assymptotic behavior of functions (ie how it grows or diminishes for large $x$). In some sense this is a description of potential infinities, in addition these can be compared in some cases (for example $\Theta(x^2) > \Theta(x)$).

Then of course there are actual infinities, but these should not (IMHO) be studied before you actually learned the concept of limits, by it's definition. This is because at first one think may think that actual infinities would simplify the concepts instead of relying on the $\epsilon$-$\delta$ definition of limits, but in reality I don't think that's the case.

There's not only one kind of actual infinities (but keep in mind none of them are the infinity sign in $\lim_{x\to\infty}$) that is used in different fields. There's for example cardinal numbers which can be thought as the count of elements in a set - for these some are larger than others, but this kind of infinity is normally not used in mathematical analysis. The one's that's used there are used there normally doesn't come in different sizes.


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