That was imprecise, but I was trying to comment on this part of the dialogue using the language that it had established:
Argh! No, damn it, I live in the set theory that really does have all the subsets, with no mysteriously missing subsets or mysterious extra numbers, or denumerable collections of all possible reals that could like totally map onto the integers if the mapping that did it hadn’t gone missing in the Australian outback -
I was also commenting on this part:
Screw set theory. I live in the physical universe where when you run a Turing machine, and keep watching forever in the physical universe, you never experience a time where that Turing machine outputs a proof of the inconsistency of Peano Arithmetic.
The point I was trying to make, and maybe I did not use sensible words to make it, is that This Guy (I don’t know what his name is—who writes a dialogue with unnamed participants, by the way?) doesn’t actually know that, for two reasons: first, Peano arithmetic might actually be inconsistent, and second, even if it were consistent, there might be some mysterious force preventing us from discovering this fact.
I just don’t understand yet what you mean by living in a model in the sense of logic and model theory, because a model is a static thing.
Models being static is a matter of interpretation. It is easy to write down a first-order theory of discrete dynamical systems (sets equipped with an endomap, interpreted as a successor map which describes the state of a dynamical system at time t + 1 given its state at time t). If time is discretized, our own universe could be such a thing, and even if it isn’t, cellular automata are such things. Are these “static” or “dynamic”?
Argh! No, damn it, I live in the set theory that really does have all the subsets, with no mysteriously missing subsets or mysterious extra numbers,
Indeed, I think it’s somewhat unclear what is meant here. The speaker attempts to relate it to physics, referring to the idea that we appear to live in continuous space… but how does the speaker propose to rule out infinitesimals and other nonstandard entities? (The speaker only seems to indicate horror about devils living in the cracks.) Or, for that matter, countable models of the reals, as someone already mentioned. This isn’t directly related to the question of what set theory is true, what set theory we live in, etc… (Perhaps the speaker’s intention in this line was to assume that we live in a Tegmark multiverse, so that we literally do live in some set theory?)
Instead, I think the speaker should have argued that we can refer to this state of affairs, not that it must be the true state of affairs. To give another example, I’m not at all convinced that time must correspond to the standard model of the natural numbers (I’m not even sure it doesn’t loop back upon itself eventually, when it comes down to it, though I agree that causal models disallow this and I find it improbable for that reason). Yet, I’m (relatively) happy to say that we can at least refer to this as a possible state of affairs. (Perhaps with a qualifier: “Unless peano arithmetic turns out to be inconsistent...”)
That was imprecise, but I was trying to comment on this part of the dialogue using the language that it had established
Ah, I was asking you because I thought using that language meant you’d made sense of it ;) The language of us “living in a (model of) set theory” is something I’ve heard before (not just from you and Eliezer), which made me think I was missing something. Us living in a dynamical system makes sense, and a dynamical system can contain a model of set theory, so at least we can “live with” models of set theory… we interact with (parts of) models of set theory when we play with collections of physical objects.
Models being static is a matter of interpretation.
Of course, time has been a fourth dimension for ages ;) My point is that set theory doesn’t seem to have a reasonable dynamical interpretation that we could live in, and I think I’ve concluded it’s confusing to talk like that. I can only make sense of “living with” or “believing in” models.
Set theory doesn’t have a dynamical interpretation because it’s not causal, but finite causal systems have first-order descriptions and infinite causal systems have second-order descriptions. Not everything logical is causal; everything causal is logical.
That was imprecise, but I was trying to comment on this part of the dialogue using the language that it had established:
I was also commenting on this part:
The point I was trying to make, and maybe I did not use sensible words to make it, is that This Guy (I don’t know what his name is—who writes a dialogue with unnamed participants, by the way?) doesn’t actually know that, for two reasons: first, Peano arithmetic might actually be inconsistent, and second, even if it were consistent, there might be some mysterious force preventing us from discovering this fact.
Models being static is a matter of interpretation. It is easy to write down a first-order theory of discrete dynamical systems (sets equipped with an endomap, interpreted as a successor map which describes the state of a dynamical system at time t + 1 given its state at time t). If time is discretized, our own universe could be such a thing, and even if it isn’t, cellular automata are such things. Are these “static” or “dynamic”?
Indeed, I think it’s somewhat unclear what is meant here. The speaker attempts to relate it to physics, referring to the idea that we appear to live in continuous space… but how does the speaker propose to rule out infinitesimals and other nonstandard entities? (The speaker only seems to indicate horror about devils living in the cracks.) Or, for that matter, countable models of the reals, as someone already mentioned. This isn’t directly related to the question of what set theory is true, what set theory we live in, etc… (Perhaps the speaker’s intention in this line was to assume that we live in a Tegmark multiverse, so that we literally do live in some set theory?)
Instead, I think the speaker should have argued that we can refer to this state of affairs, not that it must be the true state of affairs. To give another example, I’m not at all convinced that time must correspond to the standard model of the natural numbers (I’m not even sure it doesn’t loop back upon itself eventually, when it comes down to it, though I agree that causal models disallow this and I find it improbable for that reason). Yet, I’m (relatively) happy to say that we can at least refer to this as a possible state of affairs. (Perhaps with a qualifier: “Unless peano arithmetic turns out to be inconsistent...”)
Ah, I was asking you because I thought using that language meant you’d made sense of it ;) The language of us “living in a (model of) set theory” is something I’ve heard before (not just from you and Eliezer), which made me think I was missing something. Us living in a dynamical system makes sense, and a dynamical system can contain a model of set theory, so at least we can “live with” models of set theory… we interact with (parts of) models of set theory when we play with collections of physical objects.
Of course, time has been a fourth dimension for ages ;) My point is that set theory doesn’t seem to have a reasonable dynamical interpretation that we could live in, and I think I’ve concluded it’s confusing to talk like that. I can only make sense of “living with” or “believing in” models.
Set theory doesn’t have a dynamical interpretation because it’s not causal, but finite causal systems have first-order descriptions and infinite causal systems have second-order descriptions. Not everything logical is causal; everything causal is logical.