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#math
<futurist> N
<futurist> err, and ordered tuple from N
<Olathe> The thing is that you can write each in some way so that you have a finite number of columns. Each column has a countably infinite number of possible items.
<Olathe> So, you have a finite-size vector of naturals.
<futurist> write each *closed term* so that you have a finite number of colums?
<Olathe> Yes.
<futurist> but they can have arbitrariliy large arity..
<futurist> i mean, the highest function
<futurist> i must be thinking of the wrong way of writing them
<Olathe> Right, but it is finite, right ?
<Olathe> I mean, the naturals are unbounded, but each case is finite.
<Olathe> So is the arity.
<futurist> for any given one, but for any k in N, k is not enough columns if each colum corresponds to a character
<Olathe> It's unbounded, but each arity is finite.
<futurist> yeah, each arity is finite
<Olathe> OK, so the entire thing is unbounded, but each closed term is finite in length.
<futurist> oh, ok
<futurist> so we have infintie columns, but it's ok, i think
<Olathe> So, with N^finite, you can interleave the bits (with perhaps a length of it all encoded somehow) and get a natural.
<futurist> that last sentence i totally don't understand
<Olathe> Do you see how each closed form is a finite vector of naturals ?
<futurist> i think so, lemme think about that for one sec
<futurist> arity, function_number, argument1, argument2, ??
<Olathe> Yes, for one function, that could be a format.
<futurist> what do you mean 'for one function'?
<Olathe> That's like f(a, b, c, ...): it can only give information about one function application.
<futurist> oh
<futurist> right
<futurist> then, i don't see how it works in general
<Olathe> Alright, let's assume you can take a finite vector of naturals and compress it as one natural.
<futurist> wait, we should just use the numbering of characters, right?
<Olathe> What do you mean ?
<futurist> because the language as a whole must be countable, just assign a natural to every function sign and constant
<Olathe> Yes.
<futurist> then f(g(a,b), c) would be something like (2,4,1,3,5)
<futurist> if constants are odd, functions even, or something
<Olathe> Yes, you could do something like that, too.
<Olathe> (I was going to do it in a more complex recursive way)
<joblot> futurist: it would be RPN or some other encoding
<futurist> yeah i don't need to prove anything using this encoding other than that it's countable ;-)
<futurist> so then set up a matrix where the 2nd colum is ordered pairs, 3rd colum is ordered triples, etc. and zig-zag, voila, right?
<Olathe> That would be weirdly multidimensional, I think.
<futurist> hmm?
<Olathe> I'm not sure what picture you're making.
<futurist> well i know i can list ordered pairs
<futurist> and i know i can list ordered triples (thinking of them as pairs, the first entry of which is a pair)
<futurist> etc.
<futurist> so i have a matrix like:
<futurist> a (a,a) (a,a,a) ...
<futurist> b (a,b) (a,a,b) ...
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