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srn347 comments on Second-Order Logic: The Controversy - LessWrong
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Alternatively, one could start by asking whether 2^50 is a real number or not, if yes go up to 2^75, if no go to 2^25, and in up to 7 steps find a real number that, when doubled, ceases to be a real number. There may be impractical or even noncomputable numbers, but continuity holds that doubling a real number always yields a real number.
I think the point of the fable is that Yesenin-Volpin was counting to each number in his head before declaring whether it was 'real' or not, so if you asked him whether 2^50 was 'real' he'd just be quiet for a really really long time.
But wouldn't that disprove ultrafinitism? All finite numbers, even 3^^^3, can be counted to (in the absence of any time limit, such as a lifespan), there's just no human who really wants to.
As I understand it, this is precisely the kind of statement that ultrafinitists do not believe.
If that's true, then Yesenin-Volpin was just playin the role of a Turing machine trying to determine if a certain program halts (the program that counts from 0 to the input). If 3^^^3 (say) for a finitist doesn't exists, then s/he really has a different concept of number than you have (for example, they are not axiomatized by the Peano Arithmetic). It's fun to observe that ultrafinitism is axiomatic: if it's a coherent point of view, it cannot prove that a certain number doesn't exists, only assume it. I also suspect (but finite model theory is not my field at all) that they have an 'inner' model that mimics standard natural numbers...
Well, that's what the anti-ultrafinitists say. It is precisely the contention of the ultrafinitists that you couldn't "count to 3^^^3", whatever that might mean.
Hmm.
So, it's not sufficient to define a set of steps that determine a number... it must be possible to execute them? That's a rather pragmatic approach. Albeit it one you'd have to keep updating if our power to compute and comprehend lengthier series of steps grows faster than you predict.
No, ultrafinitism is not a doctrine about our practical counting capacities. Ultrafinitism holds that you may not have actually denoted a number by '3^^^3', because there is no such number.
Utlrafrinitists tend no to specfify the highest number, to prevent people adding one to it.
Hence "may not"
I would have done the following if I had been asked that: calculate which numbers I would have time to count up to before I was thrown out/got bored/died/earth ended/universe ran out of negentropy. I would probably have to answer I don't know, or I think X is a number for some of them, but it's still an answer, and until recently people could not say wether "the smallest n>2 such that there are integers a,b,c satisfying a^n + b^n = c^n" was a number or not.
I'm not advocating any kind of finitism, but I agree that the position should be taken seriously.