I don't think people really understood what I was talking about in that thread. I would have to write a sequence about

the difference between first-order and second-order logic

why the Lowenheim-Skolem theorems show that you can talk about integers or reals in higher-order logic but not first-order logic

why third-order logic isn't qualitatively different from second-order logic in the same way that second-order logic is qualitatively above first-order logic

the generalization of Solomonoff induction to anthropic reasoning about agents resembling yourself who appear embedded in models of second-order theories, with more compact axiom sets being more probable a priori

how that addresses some points Wei Dai has made about hypercomputation not being conceivable to agents using Solomonoff induction on computable Cartesian environments, as well as formalizing some of the questions we argue about in anthropic theory

why seeing apparently infinite time and apparently continuous space suggests, to an agent using second-order anthropic induction, that we might be living within a model of axioms that imply infinity and continuity

why believing that things like a first uncountable ordinal can contain reality-fluid in the same way as the wavefunction, or even be uniquely specified by second-order axioms that pin down a single model up to isomorphism the way that second-order axioms can pin down integerness and realness, is something we have rather less evidence for, on the surface of things, than we have evidence favoring the physical existability of models of infinity and continuity, or the mathematical sensibility of talking about the integers or real numbers.

Comment author:[deleted]
14 September 2011 06:33:15AM
3 points
[-]

Lowenheim-Skolem, maybe?

But that does not imply that you can't talk about integers or reals in first order logic. And indeed you can talk about integers and real numbers using first-order logic, people do so all the time.

Only in the same sense that you can talk about kittens by saying "Those furry things!" There'll always be some ambiguity over whether you're talking about kittens or lions, even though kittens are in fact furry and have all the properties that you can deduce to hold true of furry things.

Comment author:[deleted]
14 September 2011 07:02:33AM
3 points
[-]

Not in the same sense at all. All of the numbers that you have ever physically encountered were nameable, definable, computable. Moreover they came to you with algorithms for verifying that one of them was equal to another.

Yes, and that's OK. I suspect you can't do qualitatively better than that (viz ambient set-theoretic universe for second-order logic), but it's still possible (necessary?) to work under this apparent lack of absolute control over what it is you are dealing with. Even though (first order) PA doesn't know what "integers" are, it's still true that the statements it believes valid are true for "integers", it's useful that way (just as AIs or humans are useful for making the world better). It is a device that perceives some of the properties of the object we study, but not all, not enough to rebuild it completely. (Other devices can form similarly imperfect pictures of the object of study and its relationship with the device perceiving it, or of themselves perceiving this process, or of the object of study being affected by behavior of some of these devices.)

Likewise, we may fail to account for all worlds that we might be affecting by our decisions, but we mostly care about (or maybe rather have non-negligible consequentialist control over) "real world" (or worlds), whatever this is, and it's true that our conclusions capture some truth about this "real world", even if it's genuinely impossible for us to ever know completely what it is. (We of course "know" plenty more than was ever understood, and it's a big question how to communicate to a FAI what we do know.)

I don't believe it's good math until it becomes possible to talk about the first uncountable ordinal, in the way that you can talk about the integers. Any first-order theory of the integers, like first-order PA, will have some models containing supernatural numbers, but there are many different sorts of models of supernatural numbers, you couldn't talk about the supernaturals the way you can talk about 3 or the natural numbers. My skepticism about "the first uncountable ordinal" is that there would not exist any canonicalizable mathematical object - nothing you could ever pin down uniquely - that would ever contain the first uncountable ordinal inside it, because of the indefinitely extensible character of well-ordering. This is a sort of skepticism of Platonic existence - when that which you thought you wanted to talk about can never be pinned down even in second-order logic, nor in any other language which does not permit of paradox.

Comment author:Sniffnoy
14 September 2011 07:00:38AM
6 points
[-]

You seem to keep forgetting that the whole notion of "second-order logic" does not make sense without some ambient set theory. (Unless I am greatly misunderstanding how second-order logic works?) And if you have that, then you can pin down the natural numbers (and the first uncountable ordinal) in first-order terms in this larger theory.

Only to the same degree that first-order logic requires an ambient group of models (not necessarily sets) to make sense. It's just that the ambient models in the second-order theory include collections of possible predicates of any objects that get predicates attached, or if you prefer, people who speak in second-order logic think that it makes as much sense to say "all possible collections that include some objects and exclude others, but still include and exclude only individual objects" as "all objects".

Comment author:[deleted]
14 September 2011 07:24:36AM
6 points
[-]

Only to the same degree that first-order logic requires an ambient group of models (not necessarily sets) to make sense.

Well, it makes sense to me without any models. I can compute, prove theorems, verify proofs of theorems and so on happily without ever producing a "model" for the natural numbers in toto, whatever that could mean.

Comment author:CronoDAS
14 September 2011 07:55:29AM
0 points
[-]

Hmmm...

::goes and learns some more math from Wikipedia::

Okay... I now know what an ordinal number actually is. And I'm trying to make more sense out of your comment...

So, re-reading this:

or even be uniquely specified by second-order axioms that pin down a single model up to isomorphism the way that second-order axioms can pin down integerness and realness, is something we have rather less evidence for

So if I understand you correctly, you don't trust anything that can't be defined up to isomorphism in second-order logic, and "the set of all countable ordinals" is one of those things?

(I never learned second order logic in college...)

Comment author:komponisto
14 September 2011 07:37:31AM
6 points
[-]

Everything sounded perfectly good until the last bullet:

why believing that things like a first uncountable ordinal can contain reality-fluid in the same way as the wavefunction

ERROR: CATEGORY. "Wavefunction" is not a mathematical term, it is a physical term. It's a name you give to a mathematical object when it is being used to model the physical world in a particular way, in the specific context of that modeling-task. The actual mathematical object being used as the wavefunction has a mathematical existence totally apart from its physical application, and that mathematical existence is of the exact same nature as that of the first uncountable ordinal; the (mathematical) wavefunction does not gain any "ontological bonus points" for its role in physics.

or even be uniquely specified by second-order axioms that pin down a single model up to isomorphism the way that second-order axioms can pin down integerness and realness

Pinning down a single model up to isomorphism might be a nice property for a set of axioms to have, but it is not "reality-conferring": there are two groups of order 4 up to isomorphism, while there is only one of order 3; yet that does not make "group of order 3" a "more real" mathematical object than "group of order 4".

## Comments (390)

Old*10 points [-]I don't think people really understood what I was talking about in that thread. I would have to write a sequence about

lessevidence for, on the surface of things, than we have evidence favoring the physical existability of models of infinity and continuity, or the mathematical sensibility of talking abouttheintegers or real numbers.Lowenheim-Skolem, maybe?

But that does not imply that you can't talk about integers or reals in first order logic. And indeed you can talk about integers and real numbers using first-order logic, people do so all the time.

Only in the same sense that you can talk about kittens by saying "Those furry things!" There'll always be some ambiguity over whether you're talking about kittens or lions, even though kittens are in fact furry and have all the properties that you can deduce to hold true of furry things.

Not in the same sense at all. All of the numbers that you have ever physically encountered were nameable, definable, computable. Moreover they came to you with algorithms for verifying that one of them was equal to another.

*2 points [-]Yes, and that's OK. I suspect you can't do qualitatively better than that (viz ambient set-theoretic universe for second-order logic), but it's still possible (necessary?) to work under this apparent lack of

absolute controlover what it is you are dealing with. Even though (first order) PA doesn't know what "integers" are, it's still true that the statements it believes valid are true for "integers", it's useful that way (just as AIs or humans are useful for making the world better). It is a device that perceives some of the properties of the object we study, but not all, not enough to rebuild it completely. (Other devices can form similarly imperfect pictures of the object of study and its relationship with the device perceiving it, or of themselves perceiving this process, or of the object of study being affected by behavior of some of these devices.)Likewise, we may fail to account for all worlds that we might be affecting by our decisions, but we mostly care about (or maybe rather have non-negligible consequentialist control over) "real world" (or worlds), whatever this is, and it's true that our conclusions capture some truth about this "real world", even if it's genuinely impossible for us to ever know completely what it is. (We of course "know" plenty more than was ever understood, and it's a big question how to communicate to a FAI what we do know.)

In other words, a first uncountable ordinal may be perfectly good

math, but it's notphysics?I don't believe it's good

mathuntil it becomes possible to talk aboutthefirst uncountable ordinal, in the way that you can talk abouttheintegers. Any first-order theory of the integers, like first-order PA, will have some models containing supernatural numbers, but there are manydifferentsorts of models of supernatural numbers, you couldn't talk aboutthesupernaturals the way you can talk about 3 or the natural numbers. My skepticism about "the first uncountable ordinal" is that there would not exist any canonicalizable mathematical object - nothing you could ever pin down uniquely - that would ever contain the first uncountable ordinal inside it, because of the indefinitely extensible character of well-ordering. This is a sort of skepticism of Platonic existence - when that which you thought you wanted to talk about can never be pinned down even in second-order logic, nor in any other language which does not permit of paradox.You seem to keep forgetting that the whole notion of "second-order logic" does not make sense without some ambient set theory. (Unless I am greatly misunderstanding how second-order logic works?) And if you have that, then you can pin down the natural numbers (and the first uncountable ordinal) in first-order terms in this larger theory.

Only to the same degree that first-order logic requires an ambient group of models (not necessarily

sets) to make sense. It's just that the ambient models in the second-order theory include collections of possible predicates of any objects that get predicates attached, or if you prefer, people who speak in second-order logic think that it makes as much sense to say "all possible collections that include some objects and exclude others, but still include and exclude only individual objects" as "all objects".Well, it makes sense to me without any models. I can compute, prove theorems, verify proofs of theorems and so on happily without ever producing a "model" for the natural numbers in toto, whatever that could mean.

Hmmm...

::goes and learns some more math from Wikipedia::

Okay... I now know what an ordinal number actually is. And I'm trying to make more sense out of your comment...

So, re-reading this:

So if I understand you correctly, you don't trust anything that can't be defined up to isomorphism in second-order logic, and "the set of all countable ordinals" is one of those things?

(I never learned second order logic in college...)

Everything sounded perfectly good until the last bullet:

ERROR: CATEGORY. "Wavefunction" is not a mathematical term, it is a physical term. It's a name you give to a mathematical object

when it is being used to model the physical world in a particular way, in the specific context of that modeling-task. The actual mathematical object being used as the wavefunction has a mathematical existence totally apart from its physical application, and that mathematical existence is of the exact same nature as that of the first uncountable ordinal; the (mathematical) wavefunction does not gain any "ontological bonus points" for its role in physics.Pinning down a single model up to isomorphism might be a nice property for a set of axioms to have, but it is not "reality-conferring": there are two groups of order 4 up to isomorphism, while there is only one of order 3; yet that does not make "group of order 3" a "more real" mathematical object than "group of order 4".

*6 points [-]I would like very very much to read that sequence. Might it be written at some point?