papalex

@johncarlosbaez love it, in particular the constant contradictions. First he defines the measure of consciousness via “if, after lengthy conversation, you cannot tell whether it is a human”, and then he ignores that there surely is no single human who can not only write one but like five or so different sonatas in a few minutes? This destroys the whole operational meaning of the test, if you suddenly say “I know it’s not human but this is super human”. No! You (that is he) just proved that Claude strictly speaking fails the Turing test and any “but it’s even better!” just confirms that the Turing test is completely besides the point for actually discussing consciousness, which means his entire argument collapses in the viscous circle of confirming his own beliefs by — tada: his own beliefs.

@johncarlosbaez@mathstodon.xyz @typeswitch@gamedev.lgbt In internal set theory (IST) the idealization principle (I) implements the intuition that we can only test a finite number of objects at a time. A consequence of (I) is that any infinite set must contain a non-standard element and in particular there exists a non-standard number in ℕ (see [1] end of pg. 5). So to get rid of the infinite numbers in ℕ we would have to add further aximos (invoking the external standard predicate), making the standard model of PA ℕˢᵗ a non-standard model of PA within IST. I think the intuition is somewhat natural at first and then makes an interesting twist: We do not tell maths what it may do, but only make axioms about what we can check, this gives us the idealization principle: For any property we can always only check the property on finite elements, which is sort of a pragmatic filter of what we mean by that property. However, the real work of (I) is then that it asserts: If (see LHS of (I)) for any finite standard set 𝑥' there exists a 𝑦(𝑥') such that the property holds (made the 𝑥' dependence explicit), then (I) asserts that there also exists a fixed 𝑦 such that the property holds wrt all standard 𝑥. So the real work of this axiom is in making 𝑦 independent of 𝑥'. I think this is a really nice way of making ∞ and related infinitesimals operational. So the short form intuition could be: IST enforces that the sequence 𝑆ₙ={0,1,...,n} with 𝑆ₙ<𝑆ₙ₊₁ must have a supremum with 𝑆ₙ

@typeswitch@gamedev.lgbt @johncarlosbaez@mathstodon.xyz I think I am a bit confused and likely just miss a key definition somewhere, so I'll just ask a naive question: From N in V you have constructed 'N in *V. You argue that 'N contains N (in a specific sense) but also non-standard numbers (represented by non-constant sequences such as [(0,1,2,3,...)]). Then you observe that 'N is the initial model of PA in *V from which you conclude that ('N,*V) is not the example we want. But as I understand it, you just showed that 'N is richer than N, in particular it contains non-standard elements. And to reduce to only elements of the type [(a,a,a,...)] which correspond to N wihtin 'N would correspond to a non-initial model, right? So why isn't this the example you want?

@typeswitch@gamedev.lgbt @johncarlosbaez@mathstodon.xyz Indeed, with a linguistic glitch: In IST, the standard natural numbers are standard in the sense of the standard predicate. However, the standard predicate is not standard, so the standard natural numbers in IST are non standard in the standard sense 🤯

@johncarlosbaez@mathstodon.xyz @typeswitch@gamedev.lgbt but still, what is wrong with the following reasoning: Since IST is a conservative extension of ZF(C), we have that the reduct of IST models (forgetting the additional structure on these models) gives us ZF(C) models. That means: via IST we get non standard models of ZF(C). In these, we cannot prove via ZF(C) machinery that the nonstandard elements are there, but we know it from our meta view knowing that this is a reduct coming from an IST model.

@typeswitch@gamedev.lgbt @johncarlosbaez@mathstodon.xyz This I understand somewhat as the statement that "one cannot construct the standard (i.e. intended model of) N in *V but any model of PA in *V will have nonstandard elements." Which to me would still be in line with the original quote. That is, to get the standard N one would have to adopt some meta theory (if that is even allowed?) but the standard (i.e. intended) N is definitely not a standard model of PA in *V in any meaningful way. What am I missing?

@johncarlosbaez@mathstodon.xyz @typeswitch@gamedev.lgbt I must admit that I am not an expert and might have misunderstood the assumptions of the quote. In particular, I read the ZFC part in the quote more as an example rather than an assumption. Though I also don't know prior to googling whether "non-standard model of set theory" implicitly excludes IST? Either way, I would enjoy to learn more about any of these things and am happy to accept if my answer was missing the point, if that is the case? Ignoring any nitty gritty interpretations of the original quote. You wrote "in a non-standard model of ZFA". What actually does this include? I would assume anything with ZFA axioms plus whatever other non-standard axioms goes? EDIT: Ah, I think I get it.. Sorry.. IST is a different theory right? Not a different model?