How to write an individual real number?

Question

I just read an interesting book: "Classical and nonclassical logics", Princeton Univ. Press (2005) by Eric Schechter. On p. 208 he writes:

Also for simplicity of notation, we have chosen an alphabet that is only countably infinite. That alphabet is adequate for most applications of logic, but some logicians prefer to allow uncountable alphabets as well. (Imagine an even larger infinite computer keyboard, with real numbers written on the key caps!)

My question: How could the manufacturer write a real number except the few which have their own names like $2$ or $\sqrt3$ or $1/4$ or $\pi$?

If a student would ask me, I really could not answer since the real numbers written on the key caps have to be individuals, i.e., it is not sufficient to distinguish each one from some "given" real numbers but each one must differ from all other real numbers. How can that be possible by finite strings of symbols on the key caps? I assume consent that infinite strings of symbols don't carry any information that could be with sufficient completeness conveyed to the typist.

Answer 1

Schechter is talking about infinitary logic (see the Wikipedia and Stanford Encyclopedia of Philosophy entries), although Schechter's "infinite computer keyboard" comment is meant to be picturesque and not mathematically precise.

See also Keisler's 1970 paper Logic with the quantifier "there exist uncountably many" and David Marker's Fall 2007 A Primer on Infinitary Logic and Dickmann's 1975 book Large Infinitary Languages (book reviewed here). Incidentally, Dickmann's book deals with large cardinal infinitary languages, in case you were wondering just how far down the uncountable rabbit hole logicians have gone with this stuff. (Answer: Further than you can probably imagine.)

Finally, the google search infinitary languages brings up a lot of relevant hits.

Answer 2

I assume you mention the real numbers because you are considering the case where the set of real numbers is taken to be your alphabet.

In this case, the "keyboard manufacturer's" job is simple; every key is labeled with a single symbol.

The point you're missing is that, if you take you take the real numbers to be your alphabet, they are the very symbols you use to write with.

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It may help to understand why one might want to use an uncountable alphabet. One might get the impression that strings over an alphabet are meant to model human writing, and while that's one application, it is by no means the only one.

A different example would be to axiomatically define the notion of a "real vector space". A typical method would be to define a language consisting of:

• The usual logical symbols
• The constant symbol $\vec{0}$ (the "zero vector")
• The binary function symbol $+$ (the operation of "vector addition")
• For every real number $r$, the unary function symbol "$r \cdot {}$" (the operation of "scalar multiplication by $r$")

and formally write out the axiom schema for the real vector spaces in this language:

• The axiom $a+(b+c) = (a+b)+c$
• For every real numbers $r,s,t$ with $r+s=t$, include the axiom $t \cdot a = r \cdot a + s \cdot a$
• ... and so forth

One's first tendencies might be to try and include in the axioms a minimalist axiomatization of the real numbers, but that's missing the point — the real numbers here are a given, and the point of this theory is to define what a "real vector space" is, not what a "real number" is.

Also, there are important mathematical reasons for having the theory presented in this particular form; e.g. it shows that "real vector space" is an example of a thing called a universal algebra.

Answer 3

So, here are two assumptions that I think are built into your question.

1. The manufacturer is writing on the keys with only a finite, or perhaps countable alphabet of symbols.

2. The manufacturer can only fit a finite number (maybe unboundedly large) of symbols on each key.

If assumption one fails, then you could just give each real number a symbol, and be done with it.

If assumption two fails, you could name each real number by its perhaps infinitely long decimal expansion.

If both assumptions hold, then the answer is "They can't". This is because, as Cantor proved a while back, there are uncountably many real numbers. But there are only countably many finite strings of symbols over a given finite (or countable) alphabet.

The manufacturer's best bet, if they want to do this systematically, might be to write on each key a computer program that prints out the decimal representation of the real number that key is supposed to produce. This would get you the computable numbers, a larger set of reals than just algebraic numbers plus common transcendentals like π. But, there remain some reals with names that can't be represented this way, like Chaitin's Ω, so this still isn't optimal.

However, there is no optimal solution. Suppose there were. Then we could add a key which said "the real whose first digit is the first digit of the alphabetically least real from the old keyboard, plus one (or minus one, if the first digit is 9), whose second digit is the same as the second digit of the alphabetically second-least real from the old keyboard, and so on", and in that way add a key which denotes a real not denoted by any key on the first keyboard (it can't be denoted by the first key, because it differs in the first digit from that real, it can't be denoted by the second, because it differs in the second digit...). So, since we've added a key, the original keyboard was not optimal. Contradiction, and our assumption must have been false.

I'm slightly fudging things for simplicity in the previous paragraph by pretending that decimal representations are unique. But you could run the same argument but on a scheme where they are unique by using e.g. continued fractions.

Answer 4

Well if you are imagining a keyboard with an infinite amount of keys, you would want to imagine an infinite amount of space to write the number on each key. So the unnamed real numbers would contain infinite space to write.

But how would the typist distinguish them? Well how does the mathematician distinguish them?

By their position. We don't read off the digits but locate them in the place where they belong.

So each key and its neighbor would have the ability to have a key between them. And likewise for those keys.

If we are imagining; here this is what this means. If each key were a distinct entity holding position to another key with nothing between them, it would be countable.

So in our imagining we have to come up with a mechanism to make the numbers infinitely dense. So using the definition of this in real numbers we will place a key inbetween two other keys as needed.

Not sure if this is the point of the analogy or the imagining exercise, but as analogy it breaks down; if we don't add room for that infinite density of real numbers.

And so in the imagining we create a contraption that looks like a keyboard from afar but if we stare at the spaces between the keys is different. Much like real numbers look like integers, until we look more closely and find we have to squeeze more and more in the spaces between each named thing.