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.

Typo

It is a common enemy to googologists. When you submit your blog post, then you will find many typos. When you extend your system, then you will find new typos. When you fix typos, then you will find that your correction generated other typos. The least weakly typo cardinal can be used to diagonalise higher inaccessible typos. It can be replaced by the least recursively typo ordinal so that it works in \(\textrm{ZFC}\) set theory.

Hear ar exanples of tyops.

Ambiguity

Definitions should be described in clear sentences without much ambiguity. Of course, sentences in a natural language usually include ambiguous phrases to some extent. However, if you write descriptions which are too ambiguous to give us unique interpretations, then they do not allow us to point out exact errors. For example, consider a description D can be interpreted in to three definitions A, B, and C. Suppose that A has an error X, B has an error Y, and C has an error Z. When we point out "D has an error X," then you might answer "No, D does not mean A." Next, when we point out "Oh, then D has an error Y," then you might answer "No, D does not mean B." Finally, when we point out "Ok, then D has an error Z", then you might answer "Right! Thank you!" It is not good way to spend time. If you want other to understand correctly and to point out exact errors, then it is better for you to describe a definition as clear as possible.

Here are examples of ambiguous descriptions.

Lack of the Declaration

"I define term rewritings in the following recursive way: \((n) = n+1, (0,s) = ((s)), (n,s) = (n-1,(n-1,s)), (0,0,s) = ((s),(s))\). I put \(N = (0,0,0,0,0)\)."

This sentence does not define \(N\), because you do not declare the domains of variables \(n\) and \(s\), and the domain of valid expressions. Is \(n\) supposed to be any natural number? Or a positive integer? Is \(s\) supposed to be any natural number? Or any string? Or any array of natural numbers? Does \(((s))\) mean the string "\(((s))\)" itself? Or the string "\((n)\)", where \(n\) is the result of \((s)\)?

In order to avoid such ambiguity, it is better to qunatify the variables by writing something like "For any natural number \(n\)" or "Let \(n\) be a natural number", which clarifies that \(n\) can be chosen to be an arbitrary natural number. This ambiguity occurs because it is not easy to notice the importance of the precise range which a variable runs through. You should quantify all undefined objects when they appear first. For mor details, see the guideline on how to define a new object.

Multiple Use of the Same Symbol

"Let \(n\) denote any natural number. I define term rewritings in the following recursive way: \((0,0) = 1, (n+1,0) = (n,(n,0)), (n,n+1) = (n-1,(n,n))\). I put \(N = (10,10)\)."

This sentence does not defined \(N\), because the rule set for \((n,n+1)\) is not applicable to \((0,2)\). The same symbol appearing in a single equality is automatically supposed to be the same value, and hence "\((n,n+1)\)" is not supposed to mean "\((n_1,n_2+1)\) for some natural numbers \(n_1\) and \(n_2\)". If such a convention is allowed, then it is impossible for others to determine whether the occurrence of \(n\) in the right hand side means the first or the second \(n\) in the left hand side. This ambiguity rarely occurs for natural numbers, but often occurs for symbols for strings.

This is not only the case for constants or variables. For example, if you use "=", "+", or other symbols with two or more distinct meanings, then it causes serious ambiguity. This ill-definedness frequently occurs when googologists defines their own notation. They use "=" for the strict identity of formal strings, undefined relation for a certain "identification" of two distinct formal strings, and a term-rewriting law. They use "+" for the usual addition of the result of the computable functions corresponding to formal strings with respect to given expansion rules and a formal symbol in an expression of a new notation.

Lack of the Declaration of the Domain

"I define \(f\) as the function \(\alpha \mapsto \alpha + 1\)."

This sentence doed not define \(f\), unless you specify the domain of \(f\). This ill-definedness sometimes occurs when a googologist applied unspecified hyper operators to ordinals. See also the common failure due to an undefined substitution and the guideline to define a new object.

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 \(f(x) = \sum_{i=0}^{x} f(2^i)\). Put \(N = f(10^{100}).\)"

This sentence does not define \(f\) or \(N\), because the computation of \(f(1)\) includes an actual loop. 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.

Diverging Loop

"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: \(f() = 1, f(s,0) = f(s)+1, f(s,x+1) = f(\underbrace{s,x,\ldots,s,x}_{x+2})\). 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:

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.

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 \(t(s)\).

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 or 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 or 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. Define \(f(x,y)\) in the following way: \(f(0,y) = 2^y\), \(f(x+1,0) = f(x,x)\), and \(f(x,2y) = 2^{f(x,y)}\). 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 \(x \neq 0\), \(y \neq 0\), and \(y\) is an odd number. 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 Partial Specialisation

"Let \(x\) be a natural number. Put \(f(x+s) = x^{s+1}\)."

This sentence does not define a function \(f\) unless \(s\) is a term which has already been defined or quantified or is a free variable in the current scope. This ill-definedness sometimes occurs because terms appearing in an expression of the input look similar to indeterminates.

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. See also the issue on multiple use of the same symbol.

Undefined Operation

"I define a notation \(T\) as …. I define a recursive system of fundamental sequences. For any \((s,t) \in T^2\), if \(s+t\) is greater than…."

This sentence does not define a recursive system of fundamental sequences unless the concatenation of s, "+", and t is a valid expression in the notation or you explicitly define the operation \(+\) as a map whose domain is \(T^2\). This ill-definedness sometimes occurs when a googologist tries to imitate an ordinal arithmetic without understanding that an extension of arithmetic operators to formal strings is not unique. See also the issue on multiple use of the same symbol.

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 in order to complete the definition. 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. If you want to substitute formal strings 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 or formal strings. See also the issue on lack of the declaration of the domain and the guideline on how to define a new object.

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, because you have two candidates \(()\) and \(((),())\) of \(t\). 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, because you have two candidates \(( \ \ (() \ \ , \ \ (),()) \ \ )\) and \(( \ \ (),(() \ \ , \ \ () \ \ )\) of \((\#_1,\#_2)\). 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.

Multiply Defined 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 \(f(2) = \psi_{\chi_0(0)}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0))\). Repeat this pattern. Put \(\alpha = f^{100}(100)\)."

This sentence does not define \(f\) or \(\alpha\) 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 Use of Expressions

"I define \(f(\alpha)\) for any ordinal \(\alpha\) in the following way: If \(\alpha\) is written as \(\beta + \gamma\), then \(f(\alpha) = \varphi_{\beta}(\gamma)\). If \(\alpha\) is written as \(\varphi_{\beta}(\gamma)\), then \(f(\alpha) = \beta + \gamma\)."

This sentence does not define \(f\) because the predicate "written as" refers to a property on an expression rather than an ordinal itself. This ill-definedness frequently occurs when a googoligist who does not understand the definition of the notion of an ordinal and comfound an ordinal with an expression try to define an ordinal function.

Even an "expression" does not make sense unless you specify a translation map from a set of formal strings to a set of ordinals. Even an epression is not unique for a fixed translation map unless it is injective. You can solve the issue to define the subset of "standard" expression to which the restriction of the translation map is injective. Also, you can avoid the use of an expression by defining the notion of a "normal form" as a predicate on a tuple of ordinals giving "normal forms". For example, the \(3\)-ary relation \(\alpha =_{\textrm{NF}} \varphi_{\beta}(\gamma)\) on a tuple \((\alpha,\beta,\gamma)\) of ordinals is frequently defined as \(\{\beta,\gamma\} \subset \alpha = \varphi_{\beta}(\gamma)\). See also the issue on a non-unique expression.

Intuitive Translation of Notations

"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 \(\textrm{ZFC}\) 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 formulae in \(\textrm{ZFC}\) set theory 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 in the base theory. Otherwise, \(t\) is ill-defined, because there could be unspecified multiple candidates or no candidate of \(t\).

Here are examples of lacks of the unique existence.

Unspecified Fundamental Sequences

"Let \(\alpha\) denote the countable ordinal defined as …. Put \(N = f_{\alpha}^{100}(10^{100})\)."

This sentence does not define \(N\) 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 \(\textrm{PTO}(\textrm{ZFC} + \textrm{I}0)\).

Unspecified Large Ordinals

"Let \(X\) denote a sufficiently large ordinal. For an ordinal \(\alpha\), I define \(\psi(\alpha)\) 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.

Non-Unique Expression

"I define a function \(f(\alpha)\) of any ordinal \(\alpha\) in the following way: If \(\alpha = \beta + \gamma\) for some ordinals \(\beta\) and \(\gamma), then \(f(\alpha) = f(\beta) \times f(\gamma)\). If \(\alpha = f(\beta)\) for some ordinal \(\beta\), then ..."

This sentence does not define \(f\) unless you add a condition to uniquely characterise \(\beta\) and \(\gamma\) using \(\alpha\), because a pair \((\beta,\gamma)\) satisfying \(\alpha = \beta + \gamma\) is not unique. Even if you add such a condition, the case classification can be overlapping if \(f(\beta)\) in the second case can be expressed as a sum satisfying the condition. Even if it is not an overlapping, the second case does not work unless you fix a well-ordering \(R\) on the class of all ordinals (such as \(\in\)) and add the condition \(\beta R \alpha\) because the reference of the value of \(f\) for any input in the definition of \(f(\alpha)\) refers to the value of \(f(\alpha)\) itself, and causes a circular reference. Such ill-definedness frequently occurs when a googologists who comfound an ordinal with its expression to create a new function.

Unprovability

"I work in \(\textrm{ZFC}\) set theory, and assume its consistency. I denote by \((M_n)_{n \in \mathbb{N}}\) the enumeration of Turing machines given by the universal Turing machine, and by \(N\) the least natural number such that there exists a system of fundamental sequences below \(\textrm{PTO}(\textrm{ZFC})\) for which \(M_N\) coincides with the FGH along \(\textrm{PTO}(\textrm{ZFC})\)."

This sentence does not define \(N\), because the existence of such an \(N\) is meta-theoretically provable to be unprovable in \(\textrm{ZFC}\) set theory. It is not an intuition-based argument such as "The existence is actually true, but just is not provable", but is a mathematical argument which ensures that there exists a model of \(\textrm{ZFC}\) set theory in which \(N\) does not exist. This ill-definedness sometimes occurs when a googologist who refers to provability, consistency, and \(\textrm{PTO}\) without understanding their precise definitions tries to imitate the least transcendental integer.

Contradiction

"I denote by \(\alpha\) the least ordinal such that for any (Goedel number of a) formula \(P\), \(P \leftrightarrow V_{\alpha} \models P\) holds."

This sentence does not define \(\alpha\), because the existence of such an \(\alpha\) contradicts \(\textrm{ZFC}\) set theory. I note that if you do not clarify the base theory, then you are traditionally supposed to work in \(\textrm{ZFC}\) set theory. Therefore statements like "No. I can weaken the set theory so that my definition works." are not reasonable unless you explicitly specify what axioms you employ from the beginning. This ill-definedness sometimes occurs when a googologist who uses notions without understanding their precise definitions of notions tries to imitate Rayo's number.

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 Reference

"I define \(X\) as the set of terms of the form \(0\), \(f(0)\) for an \(f \in S\) or \(f(x)\) for an \((f,x) \in S \times X\), where \(S\) is the set of functions satisfying …."

This sentence does not define \(X\) or \(S\) unless you specify the domain of each element of \(S\) without referring to \(X\). This ill-definedness occurs when a googologist who tends to omit the declaration of domains of functions tries a mutual recursion of a set of functions and the common domain. See also a issue on lack of the declaration of the domain.

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 expression \(t \in T\), 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 to 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.