hey lemme squeeze this in during the interim…
taken from this excellent article: academia.edu/1455334/WITTGE … SET_THEORY
for all this technical talk, the moral of the story is really rather simple. wittgenstein is suggesting that we are bewitched ‘metamathematically’ when we take an otherwise perfectly sensible language of rule governed abstract symbols - in this case numbers - to use for quantifiying our experiences of things, processes, events, durations, etc., … all of which are finite and limited experiences - and naturally mistake the intensional use of the language as a proper representation (or i should say evidence for) of an actual, extensional instance of an ‘endless’ experience (e.g., counting infinitely). we extend the rule beyond our experience of what the rule can yield as extension in normal experience, and imagine that to simply continue to follow the rule would necessarily result in an extension of the infinite. but he points out that endlessness can’t be done, and is therefore a senseless notion despite it being perfectly logical that the rule (intension) should produce an infinity if it is simply followed through endlessly. it’s that concept of ‘endlessly’ that gets us all befuddled. we are writing a list… and we stop. we’ve created a set. but why should’t we be able to list without ever stopping? it’s here that the intensional and the extensional intersect and create the confusion. the extension of the list is completed whenever the listing stops. it reaches an extensional terminus, so to speak, while its still possible to continue with the rule indefinitely. therein lies the bewitchment.
so think about that quote above, again. he says ‘no such thing as all numbers’. you will never complete a finished set of all numbers… but what you can experience directly is applying the rule. following the rule in a direction toward infinity is only ever an ‘approach’, as sil put it. if one insists that an ‘actual’ infinity can exist, well this is some kind of quasi-platonic realism so far divorced from empiricism that it’s… well it’s just fucked up, man. wtf.