David fucking Hilbert wrote 'after infinity, counting continues naturally, infinity plus one, infinity plus two, and so on' (I'm quoting from memory, so could be slightly wrong).
I consider this anathema. I'm proposing that any computation which adds any number to infinity, or multiplies any number by infinity, should return infinity.
This is, as I understand it, the intuitionist heresy.
Is there anyone prepared to argue I shouldn't do this?
@semit0ne that's expected behaviour when dealing with infinity. Infinity is bloody horrible to deal with.
I'm broadly trying to implement intuitionist arithmetic.
See e.g.:
@simon_brooke I don't see the connection. My (admittedly limited) understanding ist that in an intuitionistic setting you don't define the natural numbers differently but you use intuitionistic logic to prove propositions in arithmetic.
Let's pretend we had the natural numbers augmented with the symbol infinity. What is the element of this set that when added to "infinity" gives 0, i.e. the additive inverse of "infinity"?
@semit0ne I am not certain that that question makes sense. A lot of questions about the nature of the infinite do not. We can trivially show that some infinities are very much larger than others, so they have ordinality; yet it seems to me they all taste the same. I think it may be wrong to think of infinity as a quantity, rather than as some weird monstrosity that lurks in the darkness beyond the end of number.
If you subtract aleph one from aleph null or vice versa, what do you have?
Simon Brooke
semitone