it's not possible to program infinity, is it?

[PaddyD]

"But wouldn't that still qualify for the "there are as many even numbers as there are numbers" thing, i.e. there are as many sensical sentences as there are sensical + gibberish sentences? "<?xml:namespace prefix = o ns = "urn:schemas-microsoft-comfficeffice" /><o></o>
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Hmmm. That one's a bit more complicated. Thinking out loud:<o></o>
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- let us assume a finite dictionary of meaningful words (he, she, said, it, that, was raining) and a defined grammar that allows recursion (i.e. you can only generate gibberish by grammatical errors, not meaningless words, and sentences of the form "He said that she said that he said...." are OK). <o></o>
- you will exhaust the list of grammatically well-formed, non-recursive sentences pretty quickly (let's call this set the set of first order sentences – things like ‘He said it was raining.’)<o></o>
- add the first layer of recursion to each of the sentences in the first set to get second order set and repeat indefinitely (‘He said she said it was raining’). <o></o>
- the recursive process for generating new sentences is exactly the same as the recursive process for generating new numbers (n+1). So the size of the set of sentences in this language will be the same as the size as the set of the natural numbers (or odd numbers, or even numbers, or numbers ending in 7 or whatever). <o></o>
- it feels like the non-grammatical sentence set should also be countably infinite, because the sentences are generated either (a) by a one-off breach of the grammar (an atomic or first order item again – e.g. ‘He it was raining’), or (b) wrapping an atomic error in an otherwise grammatically well-formed recursive structure that is not well formed as a whole (‘She said he it was raining’), or variations on those two approaches <o></o>
- so the question is then (a) does expanding the dictionary make any difference and (b) can you run a version of Cantor’s diagonalisation argument across this?<o></o>
- and I don’t think either would change the outcome, because (a) dictionary expansion would only (countably) increase the size of first order sentences (or that sort of thing) and (b) I can only imagine diagonalisation for natural language as, effectively, changing the spelling of words in the dictionary, in which case back to (a).<o></o>
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But probably not <o></o>

 

[SuperFerret]

You'd also have to account for the non-grammatically correct, yet frequently used sentences such as "Going pub" (instead of going to the pub) that I frequently hear uttered by family and friends.

They could potentially add to the finite/infinite sentences utterable

 

[Lewiy]

someone mentioned the anthropic principle. they're misunderstanding it. you could say the same thing about stonehenge:

stonehenge exists, therefore it could come about at random.

just because something exists, does not follow that it is random

I think you’ve missed the point of the anthropic principle.
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Stonehenge was created for a purpose by human beings. The concept of purpose is unique to things that are created by or used (in ways other than for their original “design” – for want of a better phrase) by living organisms. As such, Stonehenge almost certainly wasn’t created at random (although the argument that it could have been created at random still holds to some degree).
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The anthropic principle covers the argument that the conditions for life MUST exist on Earth in our Universe because here we are pondering its existence. This in turn suggests a much higher possibility that the properties of the Universe are random than that Stonehenge came into being at random. I don’t believe that the anthropic principle has any connection at all with the concept of “man-made” objects.