I don’t get the connection why the integrals need to be numerated in real numbers.
I do wonder whether surreal non-standard analysis would be a better fit. It seems that people automatically associate integrals with real numbers but there might be cases where real numbers are inadequate.
It might be that having infinite numbers of options trying to turn them into a number system finite numbers are just not enough. But real numbers are already “singly infinite” so I don’t have good intuition which operations eat up that and which ones do not.
Is there such a thing as surreal non-standard analysis? It’s not a subject I follow, but I understood that there still wasn’t a good concept of integration for the surreal numbers.
My general attitude to other systems of non-standard numbers is the same as my attitude to ultrafinitism. Non-standard real numbers are really a way of thinking about the Platonic reals, in which what look like infinite real numbers from within the system are finite real numbers so enormous that in a limited system of reasoning they cannot be reached by any of the available constructions. But I don’t have a formalisation of this idea.
Transfinite induction does feel a bit icky in that finite prooflines you outline a process that has infinitely many steps. But as limits have a similar kind of thing going on I don’t know whether it is any ickier.
Part of my motivation on searching them is that the foundations for surreals are way more elegant than reals. From that perspective it seems very strange how reals would be the “actually existing numbers”.
In particular reals have infinite precision while any human can only determine a scalar to a finite precision. If constructibility would be a concern reals should be concerning but if reals are not concerning there shouldn’t be any additional twists.
Particuarly in surreal thinking ” You might want f(x) = 1/x to have an integral of zero by symmetry, but you can get any value you like depending on how you approach the singularity at 0 ” the multiplicative inverse of the first infinity is epsilon rather than 0 so that would open up as another alternative for the integral value. In real precision any positive number must be finite but for surreals there are positive infinidesimals. Most applications are applied to finite domains so the infinities can be “rounded” to nearest finite. But it wouldn’t be surprising that in a properly infinite application such rounding would introduce non-neglible errors.
In a case where you want to compare a gamble with finite options and a gamble which has infinite outcomes one might sometimes favour one or the other based whether the oods are good or not. In surreals infinidesimals times a first-order infinity yields a finite value so you could write down a formula like (a*x)+w(e*b*y)>(c*z)+w(e*d*v) with w being the first infinite and e=1/w and have it make sense and have it be true or not true based on the weights. With reals you can easily compare finite vs finite and infinite vs infinite but doing a comparison across multiple archimedean fields gets tricky.
With reals you can easily compare finite vs finite and infinite vs infinite but doing a comparison across multiple archimedean fields gets tricky.
Did you mean non-archimedean fields? I regard those as not the real real numbers. For practical purposes in the present context, I don’t think you can beat the Dedekind-complete ordered field (i.e. the real numbers), with nominal ∞ and –∞ symbols added as a shorthand for more verbose statements about infinite integrals and sums.
If you add +1 up from 0 and do −1 from w you never cross because those are part of separate archimedean fields. I think there is also a result that if you have surreals and add the limitation that they need to be archimedean you get the real numbers
Real number have the completeness axiom/property. While for lesser systems the limit fails to exist and thus it is missing a “completing piece” for surreals the limit is not unique. Between the ascending numbers and what would be the real limit there is quaranteed to be a surreal number, so in that sense they are “overcomplete” to follow the standard limit constructions. That is {series|}=U is always going to exist but so will {series|U}=V while for reals V is required to either equal U or equal one of the series.
Surreals are more rich than hyperreals. One could imagine a dart board where some areas scored 10 points, some lines scored 15 points and intersections of lines score 20 points. Treating anything that isn’t area as having literally zero measure the disintions between lines being more probable than line intersections get lost (0=0). With surreals one could try to say that llines are infidesimal in respects to areas and intersection points are infinidesimal with respect to lines. And you could still hold that twice area is twice as likely and twice the line length is twice as likely and twice the amount of points is twice as likely. But areas, lines and points would seem to live in 3 disconnected lands. An abundance of points would not be able to make them reach line probability (unless you provide an actual infinite amount)
However if intersections scored infinite points in respect to lines then it could make expected value sense to aim for intersection points. But if we know that all points are not infinidesimal in respect of each other then we know that any intersection point aiming strategy is inferior.
I guess the connection here is that if you try to divide the dart board into small areas, they are still going to be areas and some mechanism might misvalue an area containing an intersection point that whole area is worth what the point is worth.
I intend not to or atleast the sense is meant to be compatible with the mathematical word. Looking at wikipedia article for “Field” subsection “ordered fields” there is the mention that real are the unique complete field up to isomorphism and there are multiple subsections in surreals that could contend to be isomorphic. Real multiples of w also have an additive inverse and multiplicative inverse making them “another copy” of the reals. I think I might have thougt in error that the other subsection would fullfill field axioms in its own right but rather it is just a subsection that can be mapped to reals.
> Transfinite induction does feel a bit icky in that finite prooflines you outline a process that has infinitely many steps. But as limits have a similar kind of thing going on I don’t know whether it is any ickier.
Well, transfinite induction / recursions is reduced to (at least in ZF set theory) the existence of an infinite set and the Replacement axioms (a class function on a set is a set). I suspect you don’t trust the latter.
The primary first need for transfinite recursion is to go from the successor construction to the natural numbers existing. Going to an approach that assumes an infinite set rather than proves it seems handy but weaker. Althought I guess in reading surreal papers I take set theory as given and while it doesn’t feel like any super advanced features are used there migth be a lot of assumption baggage.
It also feels llike a dirty trick that we don’t need to postulate the existence of zero and that we get surreals from not knowing any surreals. Surreal number definition references sets of surreal numbers? Don’t know any? Worry not there is the set that is of every type. And now that you have read the definition with that knowledge you know a new surreal number which enables you to read the definition again. So we get a lot of finite numbers without positing the existence of a single number and we don’t even need to explicitly define a successor relation.
The base number construction only uses set formation and order and doesn’t touch arithmetic operations, so on that level “the birthday” of mappings has yet to come so it is of limited use. I have seen formulations of surreal theory where it is written in a more axiomatic fashion but a “process” style gives a lto of ground to realise connections betweeen strctures.
The way it’s used in the set theory textbooks I’ve read is usually this:
define a function successor on a set S: S→S∪{S}
assume the existence of an inductive set that contains a set and all its successors. This is a weak and very limited form of infinite induction.
Use Replacement on the inductive set to define a general form of transfinite recursion.
Use transfinite recursion and the union operation to define the step “taking the limit of a sequence”.
So, there is indeed the assumption of a kind of infinite process before the assumption of the existence of an infinite set, but it’s not (necessarily) the ordinal ω. You can’t also use it to deduce anything else, you still need Replacement. The same can be said for the existence and uniqueness of the empty set, which can be deduced from the axioms of Separation.
This approach is not equivalent nor weaker to having fiat transfinite recursion , it’s the only correct way if you want to make the least amount of new assumptions.
Anyway, as far as I can tell, having a well defined theory of sets is crucial to the definitions of surreals, since they are based on set operations and ontology, and use infinite sets of every kind.
On the other hand, I don’t understand your problem with the impredicativity of the definitions of the surreals. These are often resolved into recursive definitions and since ZF-sets are well-founded, you never run into any problem.
I am pretty sure the is not obstacle for applying the successor function to the infinite set. And then there is the construction mirroring ω + ω. If you have the infinite set and it has many successors what limits one to not do the inductive set trick again to this situation?
I kinda know that if you assume a special inductive set that is only one “permitted application” of it and a “second application” would need a separate ad hoc axiom.
Then if we have “full blown” transfinite recursion we just allow that second-level application.
New assumtions assume that there are old assumptions. If we just have non-proof “I have feeling it should be that way” we have a pre-axiomatic system before hand. If we don’t aim to get the same theorems then “minimal change to keep intact” doesn’t make sense. The conneciton here is whether some numbers “fakely exist” where a fake existence could be that some axiom says the thing exist but there is no proof/construction that results in it. A similar kind of stance could be that real numbers are just a fake way to talk about natural numbers and their relations. One could for example note that reals are innumerable but proofs are discrete so almost all reals are undefineable. If most reals are undefinable then unconstructibility by itself doesn’t make transfinites any less real. But if the real field can establish some kind of properness then the same avenues of properness open up to make transfinites “legit”.
I am not that familar how limits connect to the foundamentals but if that route-map checks out then transfinites should not be any ickier than limits.
I don’t get the connection why the integrals need to be numerated in real numbers.
I do wonder whether surreal non-standard analysis would be a better fit. It seems that people automatically associate integrals with real numbers but there might be cases where real numbers are inadequate.
It might be that having infinite numbers of options trying to turn them into a number system finite numbers are just not enough. But real numbers are already “singly infinite” so I don’t have good intuition which operations eat up that and which ones do not.
Is there such a thing as surreal non-standard analysis? It’s not a subject I follow, but I understood that there still wasn’t a good concept of integration for the surreal numbers.
My general attitude to other systems of non-standard numbers is the same as my attitude to ultrafinitism. Non-standard real numbers are really a way of thinking about the Platonic reals, in which what look like infinite real numbers from within the system are finite real numbers so enormous that in a limited system of reasoning they cannot be reached by any of the available constructions. But I don’t have a formalisation of this idea.
Transfinite induction does feel a bit icky in that finite prooflines you outline a process that has infinitely many steps. But as limits have a similar kind of thing going on I don’t know whether it is any ickier.
Part of my motivation on searching them is that the foundations for surreals are way more elegant than reals. From that perspective it seems very strange how reals would be the “actually existing numbers”.
In particular reals have infinite precision while any human can only determine a scalar to a finite precision. If constructibility would be a concern reals should be concerning but if reals are not concerning there shouldn’t be any additional twists.
Particuarly in surreal thinking ” You might want f(x) = 1/x to have an integral of zero by symmetry, but you can get any value you like depending on how you approach the singularity at 0 ” the multiplicative inverse of the first infinity is epsilon rather than 0 so that would open up as another alternative for the integral value. In real precision any positive number must be finite but for surreals there are positive infinidesimals. Most applications are applied to finite domains so the infinities can be “rounded” to nearest finite. But it wouldn’t be surprising that in a properly infinite application such rounding would introduce non-neglible errors.
In a case where you want to compare a gamble with finite options and a gamble which has infinite outcomes one might sometimes favour one or the other based whether the oods are good or not. In surreals infinidesimals times a first-order infinity yields a finite value so you could write down a formula like (a*x)+w(e*b*y)>(c*z)+w(e*d*v) with w being the first infinite and e=1/w and have it make sense and have it be true or not true based on the weights. With reals you can easily compare finite vs finite and infinite vs infinite but doing a comparison across multiple archimedean fields gets tricky.
Did you mean non-archimedean fields? I regard those as not the real real numbers. For practical purposes in the present context, I don’t think you can beat the Dedekind-complete ordered field (i.e. the real numbers), with nominal ∞ and –∞ symbols added as a shorthand for more verbose statements about infinite integrals and sums.
If you add +1 up from 0 and do −1 from w you never cross because those are part of separate archimedean fields. I think there is also a result that if you have surreals and add the limitation that they need to be archimedean you get the real numbers
Real number have the completeness axiom/property. While for lesser systems the limit fails to exist and thus it is missing a “completing piece” for surreals the limit is not unique. Between the ascending numbers and what would be the real limit there is quaranteed to be a surreal number, so in that sense they are “overcomplete” to follow the standard limit constructions. That is {series|}=U is always going to exist but so will {series|U}=V while for reals V is required to either equal U or equal one of the series.
Surreals are more rich than hyperreals. One could imagine a dart board where some areas scored 10 points, some lines scored 15 points and intersections of lines score 20 points. Treating anything that isn’t area as having literally zero measure the disintions between lines being more probable than line intersections get lost (0=0). With surreals one could try to say that llines are infidesimal in respects to areas and intersection points are infinidesimal with respect to lines. And you could still hold that twice area is twice as likely and twice the line length is twice as likely and twice the amount of points is twice as likely. But areas, lines and points would seem to live in 3 disconnected lands. An abundance of points would not be able to make them reach line probability (unless you provide an actual infinite amount)
However if intersections scored infinite points in respect to lines then it could make expected value sense to aim for intersection points. But if we know that all points are not infinidesimal in respect of each other then we know that any intersection point aiming strategy is inferior.
I guess the connection here is that if you try to divide the dart board into small areas, they are still going to be areas and some mechanism might misvalue an area containing an intersection point that whole area is worth what the point is worth.
Ah, you are using “field” in a different sense (than “something with addition and multiplication obeying the usual laws”).
I intend not to or atleast the sense is meant to be compatible with the mathematical word. Looking at wikipedia article for “Field” subsection “ordered fields” there is the mention that real are the unique complete field up to isomorphism and there are multiple subsections in surreals that could contend to be isomorphic. Real multiples of w also have an additive inverse and multiplicative inverse making them “another copy” of the reals. I think I might have thougt in error that the other subsection would fullfill field axioms in its own right but rather it is just a subsection that can be mapped to reals.
> Transfinite induction does feel a bit icky in that finite prooflines you outline a process that has infinitely many steps. But as limits have a similar kind of thing going on I don’t know whether it is any ickier.
Well, transfinite induction / recursions is reduced to (at least in ZF set theory) the existence of an infinite set and the Replacement axioms (a class function on a set is a set). I suspect you don’t trust the latter.
The primary first need for transfinite recursion is to go from the successor construction to the natural numbers existing. Going to an approach that assumes an infinite set rather than proves it seems handy but weaker. Althought I guess in reading surreal papers I take set theory as given and while it doesn’t feel like any super advanced features are used there migth be a lot of assumption baggage.
It also feels llike a dirty trick that we don’t need to postulate the existence of zero and that we get surreals from not knowing any surreals. Surreal number definition references sets of surreal numbers? Don’t know any? Worry not there is the set that is of every type. And now that you have read the definition with that knowledge you know a new surreal number which enables you to read the definition again. So we get a lot of finite numbers without positing the existence of a single number and we don’t even need to explicitly define a successor relation.
The base number construction only uses set formation and order and doesn’t touch arithmetic operations, so on that level “the birthday” of mappings has yet to come so it is of limited use. I have seen formulations of surreal theory where it is written in a more axiomatic fashion but a “process” style gives a lto of ground to realise connections betweeen strctures.
The way it’s used in the set theory textbooks I’ve read is usually this:
define a function successor on a set S: S→S∪{S}
assume the existence of an inductive set that contains a set and all its successors. This is a weak and very limited form of infinite induction.
Use Replacement on the inductive set to define a general form of transfinite recursion.
Use transfinite recursion and the union operation to define the step “taking the limit of a sequence”.
So, there is indeed the assumption of a kind of infinite process before the assumption of the existence of an infinite set, but it’s not (necessarily) the ordinal ω. You can’t also use it to deduce anything else, you still need Replacement. The same can be said for the existence and uniqueness of the empty set, which can be deduced from the axioms of Separation.
This approach is not equivalent nor weaker to having fiat transfinite recursion , it’s the only correct way if you want to make the least amount of new assumptions.
Anyway, as far as I can tell, having a well defined theory of sets is crucial to the definitions of surreals, since they are based on set operations and ontology, and use infinite sets of every kind.
On the other hand, I don’t understand your problem with the impredicativity of the definitions of the surreals. These are often resolved into recursive definitions and since ZF-sets are well-founded, you never run into any problem.
I am pretty sure the is not obstacle for applying the successor function to the infinite set. And then there is the construction mirroring ω + ω. If you have the infinite set and it has many successors what limits one to not do the inductive set trick again to this situation?
I kinda know that if you assume a special inductive set that is only one “permitted application” of it and a “second application” would need a separate ad hoc axiom.
Then if we have “full blown” transfinite recursion we just allow that second-level application.
New assumtions assume that there are old assumptions. If we just have non-proof “I have feeling it should be that way” we have a pre-axiomatic system before hand. If we don’t aim to get the same theorems then “minimal change to keep intact” doesn’t make sense. The conneciton here is whether some numbers “fakely exist” where a fake existence could be that some axiom says the thing exist but there is no proof/construction that results in it. A similar kind of stance could be that real numbers are just a fake way to talk about natural numbers and their relations. One could for example note that reals are innumerable but proofs are discrete so almost all reals are undefineable. If most reals are undefinable then unconstructibility by itself doesn’t make transfinites any less real. But if the real field can establish some kind of properness then the same avenues of properness open up to make transfinites “legit”.
I am not that familar how limits connect to the foundamentals but if that route-map checks out then transfinites should not be any ickier than limits.