The "elements" of a real number

Question

This question is essentially a paraphrase of a separate (deleted) question, which talks about a comment by Asaf Karagila about the "elements of $\pi$".

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I'm aware of how natural numbers can be viewed as sets, so for example $3$ may be viewed as the set $\{\:\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\:\}$, so has elements $\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}$. This seems pretty concrete and natural.

I can see that we can adapt this to deal with the integers, by for example adding in a second empty set as a "marker" element (so $-3$ corresponds to $\{\:\emptyset, \emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\:\}$). I can also see that we can adapt this "marker" idea to deal with the rational numbers (which are pairs of integers, and we "mark" the top one and the bottom one in a certain way). However, I am already getting nervous here as this seems much more synthetic than how we viewed the natural numbers.

Anyway. It not clear to me how a number like $\pi$ or $e$ can have elements. One way might be to view these numbers as limits of sequences, and so as lists of rational numbers. However, this seems suspicious as these numbers are limits of multiples sequences, so this does not give me a canonical set which represents these numbers, but instead a family of sets. Is this OK, or is my reasoning broken?

So what I want to ask is:

What are the elements of $e$?

Or, more subtly, does this question make sense, or should we remove the word "the" from it?

Answer 1

What are the exact bits in the string that is my answer? Well, that depends on how you code it. Is it a UTF-8, or maybe UTF-16, or maybe it's ASCII. Maybe you're using a browser that represents strings as null-terminating sequences, or maybe it's a more elaborate type of object.

The point is, that this string of text that you're reading right now has a myriad of ways, all valid and all useful in their own way, to become a sequence of bits in the memory of your computer.

Likewise, real numbers are, as most people usually think of them, just an abstract entity. Like a string of text. Set theory, and indeed any foundation of mathematics, implements these abstract entities as sets (or otherwise in other foundations). Which are the exact sets which are the real numbers? That depends on how you implement them.

The so-called standard route in the case of the real numbers and $\sf ZFC$ (and its related set theories), would be:

0. Fix an encoding of ordered pairs, usually the Kuratowski pairing.
1. Use $\omega$, the least infinite ordinal, to model the natural numbers.
2. Define $\Bbb Z$ as the quotient of $\omega\times\omega$ in the algebraic way.
3. Define $\Bbb Q$ as the quotient of $\Bbb{Z\times Z}$ in the algebraic way.
4. Define $\Bbb R$ as a completion of $\Bbb Q$, which in the case of set theory is somewhat more natural via Dedekind cuts.

In that case the elements of $e$, or indeed any real number, are rational numbers smaller than $e$. But what are the rational numbers? Well, those are sets of pairs of integers, which themselves are sets of pairs of finite ordinals, which themselves have a fairly well-understood structure.

However, that is not the only way to encode the real numbers. We can choose a different way to encode ordered pairs, or we can use the Cauchy completion using equivalence classes of Cauchy sequences of rational numbers. We can encode the integers differently, or we can decide to move from $\omega$ to the non-negative rational numbers, and only then introduce the negative ones.

There are many, many ways of encoding a real number into sets. In fact, we can just take any set of size $2^{\aleph_0}$, run "the standard route", and then use a bijection to make this encoding. In set theory real numbers are often regarded as:

1. Subsets of $\omega$.
2. Functions from $\omega$ to $\omega$.
3. Functions from $\omega$ to $2$.
4. Some combination of the sets above.

This is context dependent. Just like the question of whether or not the string you're reading is represented one way or another depends on what browser you're using, which operating system you're using, etc.

So to your question, are the elements of $e$? Well, that depends on which set is $e$.

Answer 2

Following Dedekind, let's model every real number as a downward-closed proper subset of the rationals without a greatest element. Then $$e=\{x\in\mathbb Q:x<2.7\dots\}=\left\{\frac01,\frac11,-\frac11,\frac21,-\frac21,\frac12,-\frac12,-\frac31,\frac13,-\frac13,\frac23,-\frac23,\frac32,-\frac32,-\frac41,\dots\right\}$$

Answer 3

It depends on how you define the real numbers.

The Cauchy sequence method for defining the reals is a little more complicated than the Dedekind approach. A real number in this definition is an equivalence class of Cauchy sequences of rational numbers.

This means that an element is a single Cauchy sequence $(x_n)_{n=1}^{\infty}$ of rational numbers which converges to $e,$ or, less cyclically, given:

$$e_n=\sum_{k=1}^n \frac{1}{k!}$$ we have $(x_n)_{n=1}^{\infty}\in e$ if $x_n-e_n\to 0.$

Answer 4

The standard construction already of $\Bbb Z$ from $\Bbb N$ is different from your ad hoc idea (which is but perhaps a lot leaner). We will heavily use pairs and equivalence relations along the way. So remember that for pairs may use Kuratowski's definition $$ (a,b):=\{\{a\},\{a,b\}\}$$ We define as set of equivalence classes of pairs of natural numbers, $$\Bbb Z:=\Bbb N^2/{\sim}$$ where $$ (a,b)\sim(c,d)\iff a+d=b+c.$$ And we inject $\Bbb N\to \Bbb Z$ via $n\mapsto \overline{(n,0)}$. With this, $0\in\Bbb Z$ is the set $$\tag1\{\,(n,n)\mid n\in\Bbb N\,\}=\{\,\{\{n\}\}\mid n\in\Bbb N\,\}=\{ \{\{\emptyset\}\}, \{\{ \{\emptyset\} \}\}, \{\{\{ \{\emptyset,\{\emptyset\}\} \}\}\},\ldots\} $$ and already a lot more complex than the $0=\emptyset\in\Bbb N$ we had before.

Next, we would typically define $\Bbb Q$ as equivalence classes of pairs of integers, $$ \Bbb Q:=\Bbb Z\times(\Bbb Z\setminus 0_{\Bbb Z})/{\sim}$$ where this time $$(a,b)\sim (c,d)\iff ad=bc. $$ And we inject $\Bbb Z\to \Bbb Q$ via $k\mapsto \overline{(k,1)}$.

Several ways to get to the reals are possible. I'd suggest Dedekind cuts, but other than in the original work of Dedekind (pairs of sets of rationals), I would suggest to use only single sets of rationals (with specific properties) $$\Bbb R:=\{\,A\in\mathcal P( \Bbb Q)\mid A\ne\emptyset\land A\ne\Bbb Q\land \forall x\in A,\exists y\in A, y>x\land \forall x\in A,\forall y\in \Bbb Q,y<x\to y\in A\,\}.$$

With this, the elements of $\pi$ are simply all rational numbers $<\pi$ (for example $3$ or $\frac{333}{106}$ or $-42$). In particular $0$ is such an element, but it is not $0=0_{\Bbb N}=\emptyset$, nor the $0_{\Bbb Z}$ described in $(1)$ above, but the equivalence class of fractions $\{\,(0_{\Bbb Z},n)\mid n\in\Bbb Z, n\ne0_{\Bbb Z}\,\}$.