The axiom of choice (2)

After reading what I wrote about the axiom of choice about two years ago, it appears to me that I forgot to mention something important. I claimed that there are numbers which we are not able to tell. But it all depends on the language. It can be proved that for each real or complex number, there exists a language (or actually, there is an infinite number of languages) in which this number can be told. It’s very easy to prove – we can just define a language in which this number has a symbol, such as 1, 0 or o. We tend to look at some symbols as universal, but they are not. For example, the digit 0 means zero in English, but five in Arabic. The dot is used for zero in Arabic. So defining numbers depends on the language we use.

But, since there are infinite possible languages, the language itself has to be defined, or told, in order for other people to understand. We come to a conclusion that the ability to tell numbers depends on some universal language, in which we can tell the number either directly, or by defining a language and then tell the number in this language (any finite number of languages can be used to define each other). But in order to communicate and understand each other, we need a universal language in which we can start our communication.

It still means we are not able to communicate more than a countable number of numbers, or a finite number of numbers in any given finite binary digits or time, but the set of the countable (or finite) number of numbers that we can communicate depends on the language we use. For example, if we represent numbers as rational numbers (as a ratio of two integers) then we can represent any rational number, but we can’t represent irrational numbers such as the square root of 2. But if we include the option of writing “square root of (a rational number)” then we can represent also numbers which are square roots of rational numbers. In this way we can extend our language, but it’s hard to define which numbers we are able to define in non-ambiguous definitions. An example of a set of numbers we can define in such a way are the computable numbers.

In any case, for any language the number of numbers that can be defined in it is countable, and we can conclude that any uncountable set has an (uncountable) subset of numbers which can’t be defined in this language. If we subtract the set of numbers that can be defined from the original uncountable set, we can define an infinite set of numbers, none of which we are able to define or express. If there are languages in which these numbers can be expressed – these languages too can not be expressed in the original language.

It’s similar to what we have in natural languages. Some expressions (or maybe even any expression) can’t be translated from one language to another. For example, in Hebrew there is no word for tact. The word tact is sometimes used as it is literally, but this is not Hebrew. There are many words in Hebrew, and any language, from other languages. But the Hebrew language itself does not have a word for tact.