Is Language Infinite?

Sir Bedevere on the Infinity of Joseph's Ancestry

The creed in some linguistic circles is that there is a potentially infinite number of different sentences in any language. The calculation of that infinite number is the topic of a book by Langendoen and Postal entitled "The Vastness of Language". This appears to have been the last of Frogguy's contribution to a long discussion on Linguist about the infinity of language, which started in late 1991.. I have retained of his post only the part where he applies the linguists' formal argument to genealogies out of Matthew's Gospel and reaches an absurd conclusion: that there are infinitely many ways in which Joseph, Jesus's father, could have descended from Abraham. Using the same sort of argument, you can prove that the population of the Earth is infinite. Proof ab absurdo, proof a ridiculo of the inanity of that linguistic creed. Jacques Guy had presented Sir Bedevere's generative rules in Backus-Naur Form, which I cannot get to look even half right in HTML, so I have taken the liberty to edit that part of his demonstration to make it more legible.

ARTHUR: This new learning amazes me, Sir Bedevere. Explain again how there are infinitely many ways in which Joseph could have descended from Abraham.

BEDEVERE: Of course, my Liege. Matthew has written that Abraham begat Isaac who begat Jacob who begat Juda, and so on, who begat Joseph.

ARTHUR: Yes.

BEDEVERE: Behold then, my Liege, these generative rules:

  1. Genealogy of Joseph from Abraham == "Abraham begat" some Hebrews

  2. some Hebrews == one Hebrew (end of story).
    or:
    some Hebrews == one Hebrew "who begat" some Hebrews

  3. one Hebrew == "David" or "Joshua" or "Solomon" or ... ... ... "Brian" :-)
They are truly wondrous, my Liege, for they account for all the possible filiations through which Joseph could have descended from Abraham, including that reported by Matthew: just replace (one-Hebrew) with a Hebrew and have him beget.

ARTHUR: Uh, yeah, I can see that. Randy lot those Hebrews, eh?

BEDEVERE: And those alternative genealogies, my Liege, are infinite in number because (some-Hebrews) is recursive.

Have you proved that the population of the Earth is infinite yet? It's easy... Want a peek at one possible solution? There it is. Now you may wonder "what is the point? what does it matter?". Weeell...

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Proof that the Population of the Earth is Potentially Infinite

  1. Population-of-the-Earth = Someone and (some-more-people) someone else.
  2. (some-more-people) =
Easy, wasn't it? Those rules generate: Someone and someone else and someone else and... ad infinitum. There is no limit to it, you can stop when you like, or continue as long as you like, therefore... the population of the Earth is potentially infinite! QED.
Note that the minimal population of the earth is three. Here is another definition that gives a minimal population of two:
  1. Population-of-the-Earth = Someone (and-some-more-people).
  2. (and-some-more-people) =
Would you now like to prove that the number of people who can fit in a phone booth is infinite? I leave you to it. Or you may prefer to prove that the number of different sentences in a language, any language, is infinite too. As another exercise, have a look at Langendoen's very own proof that the numbers of sentences in a language is infinite, and apply it to phonebooths to see how many people fit in one.