User blog:Deedlit11/Keith Ramsay on the largest definable number

Just thought I would share this post by Keith Ramsay on various methods of defining incredibly large numbers. It's from way back in 2004, but it's very informative.

John Tapper: |What is the largest (finite) number ever written down?

I don't know.

Probably the contestants need to be divided into categories here: numbers given in decimal, numbers for which a recursive algorithm has been given, and numbers given by mathematical definitions. The "recursively given" category is perhaps the meatiest, but I have a few things to say about the (nonconstructive) "arbtrary definition" category.

Defining big numbers in any arbitrary way becomes a matter of defining strong languages. One greater than the largest number which can be defined using a given number of symbols (a million, say) in a well-defined language ordinarily requires just over the given number of symbols to define in that language, but it's relatively easy to define in a slightly more expressive language, which can define the <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">original language.

<span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">So we have a kind of loose hierarchy. Values of the busy <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">beaver function are good for getting above the contestants <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">in the recursively-exhibited number category. But the <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">first-order language of arithmetic is a stronger means of <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">defining numbers, so we can beat those by referring to the <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">largest natural number N which is the unique natural number <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">satisfying a formula P(n) with one free variable in the <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">first-order language of the natural numbers with + and *, <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">but with some carefully formulated bound on the complexity <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">of the formula P.

<span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">Second order arithmetic is stronger than first order <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">arithmetic. Once on sci.math a "name a big number" contest <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">was held, and I submitted an entry based on second order <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">arithmetic, although nobody made any reply to it. As far as <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">I could tell, it was the biggest number named then.

<span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">The first-order language of set theory (for the cumulative <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">hierarchy, the usual intended model of ZFC) is stronger <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">still. At this point, one runs into the sticky problem <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">that some (more) people at least would stop considering the <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">number definitions to be well-defined. Some people would <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">stop before we got here, but my impression is that a <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">considerably larger group would want to stop here.

<span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">For instance, "the n for which aleph-n is the continuum, or <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">0 if the continuum is not aleph-n for any natural number n" <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">might define a natural number bigger than all the ones <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">mentioned so far! I mean "might" in the sense that for <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">each natural number n>1, it's consistent with the usual <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">axioms of set theory that the continuum is aleph-n. (It's <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">also consistent that the continuum is greater than aleph-n <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">for each natural number n.) If we defined a number to be <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">the largest natural number definable in set theory using <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">some substantial-sized expression, we would be taking a <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">maximum of a set which implicitly includes that n, whereas <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">there are plenty of people who are not so convinced that <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">whether the continuum hypothesis (which says that <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">the continuum is aleph-1) is true has a well-defined answer, <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">which implies that this is not genuinely a definition.

<span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">If one "believes in" set theory, one can take things <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">further, although this starts to get messier. One way to <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">extend set theory is with truth-predicates. First-order <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">set theory extended with a predicate T(n) which is true <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">when n is the code number of a true sentence in set <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">theory is a stronger language. (We could strengthen it <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">a bit more by allowing parameters: T(n, x1,...,xn) defined <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">to mean that n is the code of a predicate P such that <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">P(x1,...,xn) holds.)

<span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">Once we've extended the language that way, the next natural <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">step is to iterate the extension. We define a sequence of <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">truth-predicates T0, T1, T2, ... where T0 is the truth <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">predicate for first-order set theory, and for each n, <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">T_{n+1} is the truth-predicate for set theory extended by <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">T-n.

<span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">One doesn't need to restrict oneself to only natural <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">numbers n. We can define T_alpha for ordinal notations <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">alpha. T_omega for instance can be defined as the truth <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">predicate for a language consisting of set theory augmented <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">by the T_n for all natural numbers n. A little caution is <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">needed because the encoding of formulas into natural numbers <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">depends on our having a code for each of the symbols being <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">added to the language. That's why I wrote "ordinal notations" <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">instead of ordinals. For the language with {T_x : x < alpha} <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">to be encoded as a natural number, we need essentially to <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">have a countable enumeration of {x: x< alpha}.

<span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">On the other hand, there's no reason why formulas can't be <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">taken to have ordinals embedded in them, so long as when <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">all is said and done we have a language with an enumerated <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">list of symbols. We could define T_x(n, x1,...,xn) where <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">n is no longer a natural number, but a structure with <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">T_y's in it, where y is simply an ordinal itself. I'm not <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">sure whether this actually enables one to define a language <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">stronger than what one can define using notations for <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">countable ordinals.

<span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">I believe all of these methods of defining language (and <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">consequently big natural numbers) are trumped by extending <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">set theory with a hierarchy of "superclasses" above the <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">usual hierarchy of ordinary sets. The inductive definitions <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">of these truth predicates fail to work in ordinary set <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">theory because they require dealing with proper-class-sized <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">constructions. One wants to define the relationship of a <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">formula P(x1,...,xn) to an n-tuple of sets (a1,...an) which <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">says P(a1,...,an) holds. As long as the number of variables <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">and quantifications is fixed, one can define the relationship <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">inside ordinary set theory. Each relationship can be <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">represented as a single formula in that language. The formula <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">can be considered as in Goedel-Bernays set theory to <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">correspond to a proper class of n+1 tuples (P,a1,...,an). <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">But in order to define the general relationship between <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">arbitrary formulas P and substitutions into its free <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">variables, one needs an inductive definition, which <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">implicitly involves considering a _sequence_ of such <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">relationships, which is an object of one higher rank than <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">the proper classes in it.

<span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">But if we soup up the language by permitting references to <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">proper class and "superclasses" whose elements are proper <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">classes, then the truth predicate for the original language <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">of set theory can be defined. Likewise, iterated truth <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">predicates can be defined. So it seems to me that this is a <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">stronger method of making the language more expressive.

<span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">One can conceivably keep going in this same direction by <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">adding more ranks beyond the point where the objects stop <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">being sets. I remember once seeing a reference to an imagined <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">hierarchy which has "as many" additional ranks as the original <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">hierarchy has, i.e., one for each ordinal. If the class of all <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">ordinals is denoted On, then the ranks of this realm are <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">ordered like On+On.

<span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">Perhaps this allows us to define almost inconceivably large <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">natural numbers! At this point, however, I have a harder time <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">seeing how a language which refers to such a hierarchy should <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">be interpreted, and whether it makes well-defined sense at all. <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">If the hierarchy keeps going, then in what sense do the <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">elements stop being sets when we get to the proper classes?

<span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">Looking out at the big natural numbers, we see on the horizon <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">flickering images which look sort of like numbers definable <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">only using the most tenuous of notions, but it's hard to say <span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">whether they are there, or just mirages.

<span style="color:rgb(0,0,0);font-family:tahoma,arial,helvetica,sans-serif;font-size:15px;line-height:normal;">Keith Ramsay

So we get methods for going beyond Rayo's number, using truth predicates and ultimately superclasses. I'm not sure what he means by "there's no reason why formulas can't be taken to have ordinals embedded in them", but it looks intriguing. Here's a question:  how does LittlePeng9's idea of going to higher order logic compare with taking superclasses? Are they equivalent? How high can you go in higher-order logic anyway?