User blog:P進大好きbot/List of Common Failures in Googology

I listed common failures appearing in definitions of large numbers. See also the list of common mistakes and questions on formal logic appearing in googology.

= Infinite Loop =

When you define a new term \(t\) using a given recursive system \(T\), the recursion should actually terminate. If it does not terminate, then the computation process is roughly said to include an infinite loop. One of the most elementary infinite loops is an actual loop, i.e. the recursive function \(f(x)\) called in the system \(T\) calls \(f(x)\) with the same input \(x\). Even if the non-termination is not due to an actual loop, it is often also called an infinite loop in googology.

Here are examples of infinite loops.

Actual Loop
"I define a function \(f(x)\) in the following recursive way: If \(x\) is presentable as \(2^n\) for some natural number \(n\), then \(f(x) = f(1)^x\). Otherwise \(f(x\) = \sum_{i=0}^{x} f(2^x)\). Put \(N = f(10^{100}).\)"

This sentence does not define \(f\) or \(N\), because the computation of \(f(x)\) includes an actual loop for the case \(x = 1\). This ill-definedness sometimes occurs when a googologist simply forgets to check the behaviour for the easiest cases or the system is too complicated to discover loops.

Divergence
"I define a function \(f(x)\) in the following recursive way: \(f(x) = 2^{f(x-1)}\). Put \(N = f(10^{100}).\)"

This sentence does not define \(f\) or \(N\), because the computation of \(f(x)\) yields the divergence of the inputs. This ill-definedness sometimes occurs when a googologist simply forgets to set the start point of the recursion or the system is too complicated to check the behaviour.

Complicated Loop
"In the following, \(x\) denotes a natural numer, and \(s\) denotes an array of natural numbers, which can be empty. I define a function \(f\) in the following way: \(f = x+1, f(s,0) = f(s)+1, f(s,x+1) = f(\underbrace{s,x,\ldots,s,x}_{x+2})\). Put \(N = f(10^{100})\)."

This sentence does not define \(f\) or \(N\), because the computation of \(f(10^{100})\) calls \(f(s,1)\) infinitely many times. This ill-definedness often occurs when a googologist tries to imitate or extend an existing notation by the wrong belief that replacing the rule sets by similar ones would not change the termination.

= Undefined Reference =

When you use a new term \(t\), then you need to do either one of the following: Otherwise, \(t\) is ill-defined. In addition, if \(t\) depends on another term \(s\) which is non-constant in the context, then it is better to clarify the dependence in order to avoid the ambiguity. For example, it is better to denote \(t\) by something like \(t_s\) or \(t(s)\).
 * 1) Define \(t\) only using other terms which have already been defined or quantified and other free variables in the current scope.
 * 2) Quantify \(t\), and declare the range in which \(t\) runs through.
 * 3) Declare \(t\) as a free variable in the current scope.

Here are examples of undefined references.

Undefined Value
"Let \(x\) be a natural number. Put \(f(x) = x^s)."

This sentence does not define a function \(f\) unless \(s\) is a term which has already been defined or quantified and is a free variable in the current scope. This ill-definedness sometimes occurs by forgetting to write down the definition which is regarded as a "trivial" one by the author.

Undefined Case Classification
"Let \(x\) be a natural number. If \(x \neq s\), then put \(f(x) = x\). Otherwise, put \(f(x) = 0\)."

This sentence does not define a function \(f\) unless \(s\) is a term which has already been defined or quantified and is a free variable in the current scope. This ill-definedness sometimes occurs because terms appearing only in case classifications are unobtrusive.

Incomplete Case Classification
"Let \(x\) and \(y\) be a natural number. If \(x < y\), then put \(f(x,y) = x^y\). If \(x^3 > y^4\), then put \(f(x,y) =(x^3-y^4)^x\). Put \(N = f(2,2)\)."

This sentence does not define \(N\) unless you add the rest case classification, i.e. the definition of \(f(x,y)\) for the case where \(x \geq y\) and \(x^2 \leq y^4\). This ill-definedness sometimes occurs when a googologist simply does not check the completeness of case classifications or the case classification itself is too complicated to check because it contains massive appearance of "\(\ldots\)"-type abbreviations.

Undefined Comparison
"I define a notation \(T\) as …. I define a cofinality as …. I define a recursive system of fundamental sequences. For any \((t,s) \in T^2\), if \(t < \textrm{cof}(s)\), then \(\psi_t(s)[0] = \psi_t(s[t])\) and \(\psi_t(s)[n+1] = \psi_t(s[\psi_t(s)[n])]\)."

This sentence does not define a recursive system of fundamental sequences unless you explicitly define the inequality \(<\). I note that the use of fundamental sequences in the definition of \(<\) often causes a circular logic because of the lack of the well-foundedness. This ill-definedness sometimes occurs when a googologist tries to imitate an ordinal notation without understanding the definition of the notion of an ordinal notation.

Ill-defined Reference
"I denote Meameamealokkapoowa oompa by \(M\), and BIG FOOT by \(BF\). Put \(N = M^{BF}\)."

This sentence never defines \(N\) because \(M\) and \(BF\) refer to ill-defined natural numbers. This ill-definedness sometimes occurs when a googologist uses googological stuffs without understanding their definitions.

Undefined Substitution
"I define \(f(x)\) as \(x \uparrow^x x\) and \(g(x)\) as \(f^x(x)\). Put \(\alpha = g(\omega_1)\)."

This sentence does not define \(\alpha\) unless you add the definition of the domains and the values of \(f\) and \(g\). In mathematics, the notion of a map is defined as a tuple including the domain and the graph, and hence you need to clarify the domain. Traditionally in googology, a function is regarded as a map between suitable sets of numbers, and hence you can skip the declaration as long as the domain is a reasonable of numbers. On the other hand, if you define a function with wider domain, you need to explicitly declare the domain. If you want to substitute an ordinal for a variable of a function defined as a map between suitable sets of numbers, then you need to specify an explicit extension of the original functions. This ill-definedness sometimes occurs when a googologist applied unspecified hyper operators to ordinals.

Undefined Truth Predicate
"Let \(L\) denote the formal language define in…. Let \(T\) denote the truth predicate for \(L\)-formulae."

This sentence does not define \(T\) unless you give an explicit formalisation of the truth of \(L\)-formulae in the domain of the base theory. For example, if you are working in \(\textrm{ZFC}\) set theory, you need to define the truth of \(L\)-formulae in \(V\). This ill-definedness frequently occurs when a googologist without the knowledge of set theory tries to imitate Rayo's number.

= Multiple Reference =

When you introduce a new term \(t\), then the definition of \(t\) should be precisely one formula. More precisely speaking, it is required that there exists at least one defining formula applicable to \(t\) and for any defining formulae \(F_1\) and \(F_2\) applicable to \(t\), \(F_1\) and \(F_2\) defines the same term, i.e. the equality between. those two terms can be provable under the base theory. Otherwise, \(t\) is ill-defined.

Here are examples of multiple references.

Overlapping Case Classification
"Let \(x\) and \(y\) be a natural number. If \(x < y\), then put \(f(x,y) = \frac{1}{y-x}\). If \(x^3 > y\), then put \(f(x,y) = \frac{1}{x^3-y}\). Put \(N = f(2,3)\)."

This sentence does not define \(N\) unless you set an order of the priority to apply the case classification. When you write expansion rules of a given computable notation without clarifying the order, you are automatically supposed to set the order following the numbering of the lines. Namely, the first line should be applied if possible. Otherwise, the second line should be applied if possible. Therefore this ill-definedness rarely occurs. On the other hand, if you implicitly intend another order, then other googologists will not follow what you intend. You might claim that the order should be "trivially" the same as what you intend, but it is wrong. In order to avoid such troubles, you need to explicitly clarify the order if it does not follow the numbering of lines.

Non-Unique Substring Searching
"Let \(s\) be a string. If there is a substring \(t\) of the form \((\#)\), where \(\#\) is any string, replace \(t\) in \(s\) by \((\#,t)\). Denote by \(X\) the result of the application of this rule to \((,(,))\)."

This sentence does not define the resulting string \(X\) unless you set an order of the priority of \(t\) to apply this replacement. This ill-definedness frequently occurs when a googologist tries to write rule sets applicable to any substrings without setting a reasonable syntax helpful to determine an explicit substring to apply those rules to.

Non-Unique Division into Substrings
"Let \(s\) be a string. If it is of the form \((\#_1,(\#_2))\), where \(\#_1\) and \(\#_2\) are any strings, replace \(s\) by \((\#_1,(\#_1,\#_2))\). Denote by \(X\) the result of the application of this rule to \((,(,))\)."

This sentence does not define the resulting string \(X\) unless you set an order of the priority of \((\#_1,\#_2)\) to apply this replacement. This ill-definedness frequently occurs when a googologist tries to write rule sets using explicit expressions with substrings without setting a reasonable syntax helpful to determine explicit substrings to apply those rules to.

Multiple Comparison
"I define a notation \(T\) as …. I defined a recursive system of fundamental sequences. For any \(t,s \in T\) satisfying \(t = (s)\), \(t[n] = (s[n])\). … I define an equality between terms in \(T\). For any \(t,s \in T\), the equality \(t = s\) holds if …. I define a term rewriting in the following way: \(t + 0 = t\), \(t \times 1 = t\), …"

This sentence does not work as you hope, because you have the equality "\(=\)" play distinct roles, i.e. the strict equality as strings, the overloaded equation, and the term rewriting. This ill-definedness sometimes occurs because the difference of the three roles is not easy to realise before pointed out.

= Intuition-Based Description =

When you define a term \(t\), then you need to describe its definition. Since a definition is a formula, no occurrence of an intuition-based stuff which is not formalised in mathematics is allowed.

Here are examples of intuition-based description.

Intuitive Pattern Matching
"I define \(f(0) = \epsilon_0\), \(f(1) = \zeta_0\), and \(f(2) = \psi_{\chi_0(0)}(\psi_{\chi_{\varpepsilon_{M+1}}(0)}(0))\). Repeat this pattern. Then I put \(N = f^{100}(100)\)."

This sentence does not define \(f\) unless the "pattern" is clearly formulated in terms of a mathematical formula. This ill-definedness frequently occurs when a googologist who likes "\(\ldots\)"-type abbreviations tries to imitate the use of ordinals without understanding the precise definition.

Intuitive Translation
"I verify the termination of my notation \(T\) with the system of fundamental sequences defined as …. This expands in this way, and hence very similar to another system \(T'\) which is known to terminate. Therefore it terminates."

This sentence does not define a conversion from \(T\) to \(T'\) satisfying several compatibility which ensures the termination of \(T\). This ill-definedness frequently occurs when a googologist without the knowledge of mathematical proofs tries to imitate proofs.

Intuitive Oracle
"Let \(T\) denote the \(\tetxtrm{ZFC}\) set theory augmented by an oracle which can solve any problem in \(\textrm{ZFC}\) set theory."

This sentence is non-sense, because \(\textrm{ZFC}\) set theory is not a computation model. This ill-definedness frequently occurs when a googologist without the knowledge of an oracle or set theory tries to imitate Busy beaver function.

Intuitive Truth Predicate
"Let \(\Sigma\) denote the set of true sentences in FOST of length \(< 10^{100})."

This sentence does not define \(\Sigma\) unless the truth predicate is defined. This ill-definedness frequently occurs when a googologist without the knowledge of set theory tries to imitate Rayo's number.

Intuitive Definability
"Let \(N\) denote the least natural number greater than any natural numbers definable (resp. namable) by formulae of length \(< 10^{100}\)."

This sentence does not define \(N\) unless a formal language \(L\) is explicitly declared and the definability of a natural number in the base theory by an \(L\)-formula is formalised. This ill-definedness frequently occurs when a googologist without the knowledge of the difference of the meta theory and the base theory tries to imitate Rayo's number.

Intuitive Formal Theory
"I use a new set theory. The first theory is \(\textrm{ZFC}\) set theory. The second theory is the strongest one among the first theory augmented by independent formulae of length \(10^{100}\). Similarly, the \(n+1\)-st theory is the strongest one among the \(n\)-th theory augmented by independent formulae of length \(10^{100}\). I define \(N\) as the least natural number in \(10^{100}\)-th theory greater than any natural numbers definable by formulae of length \(< 10^{100}\)."

This sentence does not define a sequence of formal theories and \(N\), because the intuitive comparison of the "strength" does not make sense. This ill-definedness sometimes occurs when a googologist without the knowledge of formal logic successfully imitates Utter Oblivion.

= Lack of the Unique Existence =

When you define a term \(t\) by a certain property \(P\), then \(P\) should characterise precisely one term. Otherwise, \(t\) is ill-defined. There could be a multiple candidates of \(t\), or there could be no candidate of \(t\).

Here are examples of lacks of the unique existence.

FGH without Fundamental Sequences
"Let \(\alpha\) denote the countable ordinal defined as …. Put \(N = f_{\alpha}^{100}(10^{100})\)."

This sentence does not define \(T\) unless you give an explicit system of fundamental sequences below \(\alpha + 1\). This ill-definedness sometimes occurs when a googologist who does not understand ordinals but just heard names of large ordinals tries to use large ordinals such as \(\textrm{PTO}(\textrm{ZFC} + \textrm{I}0)\).

OCF Based on Unspecified Large Ordinals
"Let \(X\) denote a sufficiently large ordinal. For an ordinal \(\alpha\), I define \(\psi(\apha)\) as the least ordinal which does not belong to the closure of \(\{0,X\}\) with respect to …."

This sentence does not define \(\psi\) unless you explicitly fix \(X\). Stating something like "\(X\) can be taken as the least ordinal greater than any value of \(\psi\)" is a circular logic. This ill-definedness sometimes occurs when a googologist wants to remove the occurrence of large cardinals in the definition of an OCF so that it becomes definable in \(\textrm{ZFC}\) set theory.

= Circular Logic =

When you define several terms, then there should be a well-founded order of the definitions so that transcendental induction works. In particular, the definition of a term \(t\) should not refer to a term \(s\) whose definition refers to \(t\).

Here are examples of circular logics.

Circular Recursion
"For a natural number \(n\), I define \(f(n)\) and \(g(n)\) in the following way: \(f(n) = g(n+1), g(n) = f(n-1)\)."

This sentence does not define \(f\) or \(g\), unless either one of them is defined by another formula. This ill-definedness sometimes occurs when a googologist without knowledge of transcendental induction tries a mutual recursion.

Circular Proof
"I verify the termination of my notation \(T\) with the system of fundamental sequences defined as …. I define a map \(o \colon T \to \textrm{Ord}\) in the following way: Put \(o(0) = 0). For any successor explression \(t \in T\)), put \(o(t) = o(t[0]) + 1\). For any non-zero limit expression \(t \in T\), put \(o(t) = \sup_{n \in \mathbb{N}} o(t[n])\). Then we have \(o(t[n]) < o(t)[n]\) for any non-zero expression \(t\). Therefore by the well-foundedness of ordinals, \(T\) terminates."

This sentence does not ensure neither the well-definedness of \(o\) nor the termination of \(T\), because the proof of the well-foundedness of \(T\) explicitly relies on the well-definedness of \(o\) and the well-definedness of \(o\) implicitly relies on the well-foundedness of \(T\). This circular logic sometimes occurs when a googologist without knowledge of transcendental induction tries to prove the termination of a system whose termination is strongly believed.

Circular Meta-Proof
"I verified the termination of my notation \(T\) with the system of fundamental sequences defined as … under \(\textrm{ZFC} + \textrm{I}0\). Then the termination can be verified also in \(\textrm{ZFC}\) set theory in the following way: The termination of \(T\) under \(\textrm{ZFC} + \textrm{I}0\) ensures the existence of a finite sequence \(a\) which displays all the computation process of \(T\). For each entry \(e\) of \(a\), the sentence that \(e\) is the rightmost entry or the next entry of \(e\) in \(a\) is given by applying the computation step to \(e\) is verifiable under arithmetic, and hence under \(\textrm{ZFC}\). Therefore combining the proof for each \(e\), I obtain a proof under \(\textrm{ZFC}\) of the sentence that \(a\) displays all the computation process of \(T\). It gives a proof of the termination of \(T\) under \(\textrm{ZFC}\)."

This sentence does not ensure the provability of the termination of \(T\), because the finiteness of \(a\) is just provable under \(\textrm{ZFC} + \textrm{I}0\) and combining the proof for each \(e\) itself does not necessarily terminates unless the termination of \(T\) is provable under the meta-theory. This circular logic occurs when a googologist without knowledge of \(\Sigma_1\)-soundedness tries prove the termination of a system whose termination is strongly believed.

= Unspecified Axiom =

When you abbreviate the formal theory in which we work, then you are assumed to be employing the traditional formal theory called \(\textrm{ZFC}\) set theory. Therefore when you use another formal theory especially in uncomputable googology, you need to declare what axioms you employ.

Here are examples of unspecified axioms.

Original Formalised Theory
"I use a new set theory. A sset is a collection of sets, which is not necessarily a set. A ssset is a collection of ssets, which is not necessarily a sset. Similarly, a ss…set is a collection of s…sets, which is not necessarily a s…set. I denote by \(N\) the least natural number greater than any natural number definable by formuae in this theory of length \(10^{100}\)."

This sentence does not define \(N\), because the notion of a "collection", the full collection of the axioms, and the formalisation of natural numbers in the base theory into terms in the formalised theory are not defined. This ill-definedness occurs when a googologist without the knowledge of formal logic tries to imitate higher order theories.

Original Base Theory
"I use a new set theory. A clas is a set. A class is a collection of clas'es, which is not necessarily a clas. Similarly, a clas…ss is a collection of clas…s'es, which is not necessarily a clas…s. I denote by \(N\) the least natural number in this theory greater than any natural number \(n\) admitting formulae \(f(x)\) of length \(< 10^{100}\) satisfying \(f(n) \land \forall x(f(x) \to (x=n))\)."

This sentence does not define \(N\), because the notion of a "collection", the full collection of the axioms, and the truth predicate used to formulate the satisfaction are not defined. This ill-definedness occurs when a googologist without the knowledge of formal logic tries to imitate higher order theories.

Comfounding with Formal Languages
"I use a new set theory defined as \(\textrm{FOST}\) augmented by all symbols corresponding to all ordinals and all functions. I define \(N\) as the least natural number in the theory greater than any natural numbers definable by formulae of length \(< 10^{100}\)."

This sentence does not define a new set theory or \(N\), because adding symbols to the formal language of a given formalised theory does not yield a formal theory unless you specify the full collection of axioms.This ill-definedness occurs when a googologist without the knowledge of formal logic tries to imitate Rayo's number.