Why can't consecutive irrational numbers be treated mathematically as limits?

Question

I'm a relative newcomer to these stackexchange websites, and this post will serve as my introduction to the Mathematics stackexchange site. After perusing some of the related questions, I found these to be the most relevant to my question: 1) Why can't you count real numbers this way?, 2) Why there is not the next real number?, 3) Proof that the real numbers are countable: Help with why this is wrong, and 4) Why does the Dedekind Cut work well enough to define the Reals? .

In related question #1), the concept of an "index" with regard to the un/countability of the set of real numbers was mentioned. I have encountered that term before in past discussions regarding the countability of transfinite ordinal sets (in other math forum websites), but I'm not sure what it means.

In any case, I understand the "real-number line" to be composed of the set of rational numbers (whose cardinality is $\aleph_0$) which have sets of "gaps" between them, each of which is filled in by a set of irrational numbers (whose cardinality is $\aleph_1$). My "understanding" is that there are no "gaps" between any two consecutive irrational numbers, and likewise, no "gaps" between a rational number and the irrational numbers that immediately precede and follow it. If all of this is true, then given some number $n$ (rational or irrational), why can't I write the next consecutive irrational number that immediately follows n as $\lim_{t\to\infty}n+10^{-t}$.

Answer 1

You can't have two 'adjacent' real numbers like that. Suppose $\alpha$ and $\beta$ were two such numbers, then $\frac{\alpha+\beta}{2}$ would be between them.

As already pointed out in the comments, $$\lim_{t\to\infty}n+10^{-t}=n.$$

Answer 2

You asked specifically

why can't I write the next consecutive irrational number that immediately follows n as $\lim_{t\to\infty}n+10^{-t}$.

This $\lim_{t\to \infty}$ notation is a mathematical term of art; it has a specific meaning, and it simply doesn't mean what you think it does. You have the idea that it means $n$ plus some tiny tiny number that is smaller than every other number. But it doesn't mean that, because there is no such thing as a tiny tiny number that is smaller than every other number. What you have written is actually a complicated way of writing the number $n$, nothing more.

I sympathize with your desire to invent a tiny tiny number that is smaller than every other number, because it sounds plausible, and almost everyone tries to do it at some point. But on closer investigation, the idea turns out to be incoherent, and this is why there is no notation for that tiny tiny number.

Consider this analogy: the phrase “the biggest purple hat in my closet” sounds reasonable, and there is no obvious reason why such a thing doesn't exist. But in fact there is no such thing as the biggest purple hat in my closet, although you have to know a bit about the contents of my closet to know this. (I don't have any purple hats.)

Similarly, there is no such thing as the number that is the next bigger real number after $n$, although you have to know a little bit about real numbers to realize this. For suppose there were such a number; let us call it $y$. Then let $\epsilon = y - n$. Since $y$ is bigger than $n$, the number $\epsilon$ is positive. But then $\frac\epsilon2$ is positive also, although smaller than $\epsilon$, so we have $0 < \frac\epsilon2 < \epsilon$. Then adding $n$ to each part of that inequality, we get $$n < n+\frac\epsilon2 < n+\epsilon = y$$ which shows that $n+\frac\epsilon2$ is between $n$ and $y$, and therefore that $y$ was not the next number bigger than $n$. But this argument works for any possible number $y$ that you could invent, so the whole idea of a number $y$ that is the next bigger number after $n$ has to be scrapped.

I'm not sure this answers your question, but your question is based on several basic (but very common) misunderstandings of how numbers work, so I hope this helps clear up some of it. But the question posed by the title of your question, “Why can't consecutive irrational numbers…” can be answered without even reading the whole thing; the answer is “because there is no such thing as consecutive irrational numbers.”