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I listed common failures appearing in definitions of large numbers. See also the list of common mistakes appearing in googology.


Contents

Careless Mistake

The start point to avoid common failures is to check what you are trying to post before you post it. Then you can find some of the errors. ...Right, not all errors.


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.


Regretful Title

You want to make your blog posts very famous, right? Then it is good to give an attrative title which appropriately describes the contents! But please be careful not to make a typo in the title, as it is a little tiresome to change the title than the body of the blog post. Even if you are certain that there is no typo, you might regret the title later because it contains a theoretic mistake, a wrong information, or an unreasonable conjecture. For example, there seems to be a jynx on titles like the following:

When the title includes such a misfortune, the author's trust might be decreasing. Then your other blog posts might not be seriously taken, or even not be read. Therefore in order to make your blog posts to be famous, it is good to be careful not to embed an information which is not justified by a theoretic reasoning as if it were an obvious fact or a reasonable conjecture into the title (and also the body).


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.


Use of Ambiguous Ellipses

"I define a positive integer \((a,b,c,d)\) for any positive integers \(a\), \(b\), \(c\), and \(d\) in the following recursive way: If \(d = 1\), then \((a,b,c,d) = a \uparrow^c b\). If \(d > 1\), then \((a,b,c,d) = (a,b,(a,b,\cdots,(a,b,c,d-1),\cdots,d-1),d-1)\). I put \(N = (10,10,10,10)\)."

The sentence does not define \(N\) unless you specify the precise meaning of the ellipses. If \(d = 2\), then what do \(\ldots\)'s mean? In general, how many times do you intend to repeat the expression? In order to avoid the ambiguity, it is better to use recursion instead of ellipses. This failure frequently occurs when a beginner who does not know recursion tries to imitate recursive notations. See also #Intuitive Pattern Matching and User blog:p進大好きbot/How to Create a Recursive Notation#General Setting.


Lack of the Clarification of the Domain

"I define my bracket notation in the following recursive way: \((a,b) = a^b\), \((a,b,0) = (a,b)\), and \((a,b,c+1) = (a,(a,b,c),c)\). I put \(N = (0,(0,(0,0),0),0)\)."

The sentence does not define \(N\) unless you add the precise definition of the domain, i.e. what tuple of values can be substitued. First of all, a notation is a function using a term rewriting process. A function is defined as a pair of the domain and the rule of the assignment. Therefore a function without a fixed domain is as ill-defined as a function without a fixed rule of the assignment. That is why the notation is ill-defined.

In order to specify the domain of a (partial) function or a function formula, we need to quantify all the variables expressing inputs and to clarify the precise conditions which are assumed on the tuple of inputs. For example, I list standard ways to clarify the domain, which should be written before you specify the rule of the assignment:

  1. I define my bracket notation \((x)\) for any sequence \(x\) of positive integers in the following recursive way:
  2. I define my bracket notation \((x)\) for any sequence \(x\) of non-negative integers in the following recursive way:
  3. I define my bracket notation \((x)\) for any sequence \(x\) of integers in the following recursive way:
  4. I define my bracket notation \((x)\) for any sequence \(x\) of real numbers in the following recursive way:
  5. I define my bracket notation \((x)\) for any formal string \(x\) in the following recursive way:
  6. I define my bracket notation \((x)\) for any non-empty sequence \(x\) of positive integers in the following recursive way:
  7. I define my bracket notation \((x)\) for any non-empty sequence \(x\) of non-negative integers in the following recursive way:
  8. I define my bracket notation \((x)\) for any non-empty sequence \(x\) of integers in the following recursive way:
  9. I define my bracket notation \((x)\) for any non-empty sequence \(x\) of real numbers in the following recursive way:
  10. I define my bracket notation \((x)\) for any non-empty formal string \(x\) in the following recursive way:
  11. I define my bracket notation \((x)\) for any non-empty sequence \(x\) of positive integers of length \(\leq 3\) in the following recursive way:
  12. I define my bracket notation \((x)\) for any non-empty sequence \(x\) of non-negative integers of length \(\leq 3\) in the following recursive way:
  13. I define my bracket notation \((x)\) for any non-empty sequence \(x\) of integers of length \(\leq 3\) in the following recursive way:
  14. I define my bracket notation \((x)\) for any non-empty sequence \(x\) of real numbers of length \(\leq 3\) in the following recursive way:
  15. I define my bracket notation \((x)\) for any non-empty formal string \(x\) of length \(5\) in the following recursive way:
  16. I define my bracket notation \((x)\) for any non-empty formal string \(x\) consisting of positive integers and commas of length \(5\) in the following recursive way:
  17. I define my bracket notation \((x)\) for any non-empty formal string \(x\) consisting of positive integers, commas, and parenthesis \((\) and \()\) of length \(15\) in the following recursive way:

After reading specific examples of the definition of the domain, you now understand how ambiguous the original description was.

Putting aside the ill-definedness, I explain how omiting the definition of the domain can cause ambiguity. Without the clarification of valid expressions, nobody knows whether the nested expression \((0,(1,2),3)\) is a valid expression of length \(11\), or the expression \((0,n,3)\) for the result \(n\) of \((1,2)\). The solution of such a nested expression can be quite ambiguous, because there are so many notations which deals with a nested array itself is a valid expression. Clarifying the domain roughly means the clarifying the precise condition that a given expression is "valid" in your notation. A beginner frequently guessed that computation rules characterise the domain, but it is wrong. If the domain has not been clarified, then even the way to apply the computation rules is not necessarily clear.

A similar problem on the lack of the clarification of the domain occurs when you deal with an infix notation like \(3 \triangleleft 2 \triangleleft 4\) without the clarification of an associativity law. Nobody knows whether \(3 \triangleleft 2 \triangleleft 4\) means a valid string of length \(5\), the expression \(n \triangleleft 4\) for the result \(n\) of \(3 \triangleleft 2\), or the expression \(3 \triangleleft m\) for the result \(m\) of \(2 \triangleleft 4\).

Please keep in mind that solving each bug pointed out by others is insufficient unless you fix the domain. When I point out the issue of the lack of the domain, beginners tend to ask what problems occur by the issue. When I point out specific examples, then they tend to consider that there are no other errors than those pointed outby me, and hence just try to fix them without fixing the domain. Examples of errors are just examples, and even when they remove them, there will be other errors unless you fix the issue of the lack of the domain. The point is that unless the notation has a specific domain, it is ill-defined even when you are unable to find any error.


Of course, the problems on the ambiguity might remain even after they fix the domain. In order to avoid such ambiguity, you need to universally quantify all the variables. If you do not know the quantification, see the section on the issue on the lack of the quantification. I note that quantifying each variable is not characterise the domain of your function, because you might assume a complicated restriction on the tuple of variable instead of each variable.

For example, if you want to define a function on a tuple \((a,b,s)\) of natural numbers \(a\) and \(b\) and a string \(s\) such that \(a+b\) is greater than the length of \(s\), it is obviously insufficient to quantify each variable by saying "For any natural numbers \(a\) and \(b\) and a string \(s\), \(a(s)b = \)...", because you dropped the intended condition on the tuple \((a,b,s)\).

Instead, you can simply declare "For any natural numbers \(a\) and \(b\) and a string \(s\) such that \(a+b\) is greater than the length of \(s\), I define a natural number \(a(s)b\) as ..." or "Let \(a\) and \(b\) be natural numbers and \(s\) a string such that \(a+b\) is greater than the length of \(s\). I define a natural number \(a(s)b\) in the following way:". Writing down all the precise condition on the tuple of inputs, you can complete the clarification of the domain. It helps others to point out an actual problem by saying "You are considering \(a(s)b-1\) in the rule 4, but the tuple \((a,b-1,s)\) does not necessarily belong to the domain of your notation."

You can skip the additional conditions after "such that" as long as you clarify that you are defining a partial function. Even in that case, you should clarify the whole space including the actual domain by declaring "I define a partial function \(a(s)b\) on a tuple \((a,b,s)\) of natural numbers \(a\) and \(b\) and a string \(s\)." If you are not familiar with creating a new notation, it is good to clarify the domain instead of defining it as a partial function, because the clarification of the domain helps you and others to find errors.

This ambiguity frequently occurs when beginners try to omit the precise description by writing something like "all symbols are positive integers" in front of the definition. They might say "As I said, all expression should be positive! Therefore the occurrence of the expression \(a-b\) automatically means that I trivially assume the obvious condition that \(a\) is greater than \(b\)!" but it is not reasonable. I would like to recall that the notion of a function in mathematics is defined as the data of the domain and the correspondence, and hence a function without the definition of the domain is as ill-defined as a function without the definition of the correspondence.


Lack of the Quantification of Variables

"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 fully quantify variables \(n\) and \(s\). 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 \((s)\)?

You need to quantify all variables in each expansion rule. Say, "\((n) = n+1\)" can be quantified as "For any natural number \(n\), \((n) = n+1\).", "For any positive integer \(n\) divisible by \(3\) and smaller than \(100\), \((n) = n+1\).", "Let \(n\) be a natural number such that the last digit of \(n^2+1\) is \(7\). I define \((n)\) as \(n+1\)", or something like that. Depending on the condition which gives the bound of the qunatification, the extent to which the rule is applicable to an expression will change, and might cause an incomplete case classification or an overlapping case classification. Since the actual problem heavily depends on the unwritten intended meanings of the quantification, the lack of the quantification makes it quite difficult for others to point out the issues.

This ambiguity frequently occurs because it is not easy to notice the importance of the precise range through which a variable runs. You should quantify all undefined objects when they appear first. See also the common failure due to an undefined substitution and the guideline to define a new object.

Moreovr, there are two distinct quantifiers: \(\forall\) and \(\exists\). If you completely omit them, then it can lead a serious ambiguity. For example, a sentence like "For any \(x \in T\), if \(x > g(s)\) (\(s \in S\)), then \(f(x) = 0\)." is inappropriate. Does "(\(s \in S\))" mean "for some \(s \in S\)" or "for any \(s \in S\)"? Whether the definition appropriately works or not heavily depends on whr\ether \(s\) is quantified by \(\forall\) or \(\exists\).

In addition, when you consider a relation among two or more variables such as an input of a multivariable function, then it is not simply sufficient to quantify the variables, because "the range of each variable" does not characterise "the range of each tuple of variables". For example, if you intend that \((n,m)\) is a pair of natural numbers such that \(n+m > 3\), then quantifying \(n\) and \(m\) by saying "all variables are natural numbers" is insufficient. See also the clarification of the domain for more details.


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 sum of the results of formal strings with respect to given expansion rules and a formal symbol in an expression of a new notation. For more details, see also the issue on overloaded equation and the issue on cofusion of operator symbols and operators.


Overloaded Equation

"I define a notation \(T\) as …. I define 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 equation "\(=\)" play distinct roles, i.e. the strict equation as strings, the overloaded equation, and the term rewriting. See also the issue on multiple use of the same symbol.

It is quite ambiguous to overload equation using the same symbol "\(=\)", because others cannot distinguish the overloaded one from the usual mathematical equation. An overloaded equality should be written by another symbol such as \(\equiv\) or \(==\). Of course, if you want to use an overloaded equality, then you should clearly define it. For example, even if you define some relation "<" which is intended to work as an order "with respect to some overloaded equation", it does not automatically give a definition of an overloaded equation. The lack of the precise definition of an overloaded equation is a serious problem, because it is just a use of undefined identification. See also the issue on undefined comparison and #Circular Reference of Comparison and Correspondence.

Also, it is quite ambiguous to denote a term rewriting using the same symbol "\(=\)". A term rewriting should be written by another symbol such as \(\mapsto\) or \(\to\). At least, it is good to follow the tradition that if you write a term rewriting "\(A = B\)", then it means "Convert \(A\) into \(B\)" because fixing the direction of conversion in a common way helps readers to guess the intended behaviour. Although it is rare to forget to define a term rewriting, it frequently occurs that one forgets to clarify when the term rewriting should be applied. For example, when you define both a term rewriting and an expansion rule and intend that a term rewriting should be applied before the application of the expansion rule, then you should clarify so. If you clarify so, then it is not allowed to apply a term rewriting at any other time, e.g. an arbitrary intermediate process to compute the expansion. If you want to apply a term rewriting to a substring of a given string, then you should clarify how to apply it. Application of a term rewriting to a substring frequently causes an error, because there can be two or more ways to apply a term rewriting to a single expression. In that case, the result of a term rewriting might not be unique, and hence the notation system might be ill-defined.

One typical failure related to overloaded undefined equalities is the confusion of the formal string "\(X+0\)" of length \(3\) and the formal string "\(X\)" of length \(1\). Although an average googologist might not confound a formal string of length \(3\) and a formal string of length \(1\), the confusion frequently occurs when a beginner try to imitate an ordinal notation without understanding the difference between an equality of formal strings and an overloaded undefined equality of formal strings. Indeed, there are so many ill-defined notations based on this misconception. Although it looks a quite tiny problem, it causes an actual problem such as circular logic, as explained in this example. See also the issue on confusion of operator symbols and operators.


Confusion of Operator Symbols and Operators

I define a set \(T\) in the following way: [Rule 1] \(\{\} \in T\). [Rule 2] For any \(a,b \in T\), \(a+b, a \times b \in T\). I set the priority order of the application as \(+ < \times\) so that \(\{\} + \{\} + \{\} \times \{\}\) means \(\{\} + \{\} + (\{\} \times \{\})\) intead of \((\{\} + \{\} + \{\}) \times \{\}\).


These sentences do not define \(T\) as you desire. Why? First, there are possible several interpretations of \(T\):

  1. \(T\) is some set, and \(+\) and \(\times\) are operators.
  2. \(T\) is a set of trees, and \(+\) and \(\times\) are operator symbols.
  3. \(T\) is a set of formal strings, and \(+\) and \(\times\) are operator symbols.

The first interpretation does not work, because operators, which are maps, should be defined before they are refered to. In particular, the domains of them should be defined before you define \(T\). If you insist that the domains are \(T^2\), then it causes a circular logic, because the definition of \(T\) refers to the function \(+\) and the definition of the function \(+\) refers to the set \(T^2\). For more details, see the issue on the circular reference of domains and functions. Also, you need to define the values \(a+b\) and \(a \times b\) for any \(a,b \in T\) without circular logic.

The second interpretation does not work, either, because \(\{\} + \{\} + (\{\} \times \{\})\) is quite ambiguous. It can be either \((\{\} + \{\}) + (\{\} \times \{\})\) or \(\{\} + (\{\} + (\{\} \times \{\}))\). You might insist that \(+\) is associative, but it has an obvious problem. Since you have not clarified the way to encode the expression \(a+b\) into a tree, one of the most natural interpretation is the tree whose entries are \(+\), \(a\), and \(b\), for which the associativity does not hold. Therefore you need to specify the way to encode the expression \(a+b\) into a tree so that the associativity actually holds. Instead, you can set a left (resp. right) associative convention. It means that \(a + b + c\) is just an abbreviation of \((a + b) + c\) (resp. \(a + (b + c)\)). However, it does not imply the associative law \((a + b) + c = a + (b + c)\), and hence does not work if you actually require the associativity law. As the examples show, trees are much more difficult than formal strings. Therefore if you are not familiar to trees, then it is better not to use it because you might not even understand errors pointed by others.

The third interpretation does not work, either, because "the priolity order of the application" does not make sense. Two expressions \(\{\} + \{\} + X\) with \(X = \{\} \times \{\}\) and \(Y \times \{\}\) with \(Y = \{\} + \{\} + \{\}\) precisely coincide with each other as formal strings, and hence you cannot distinguish them.

The biggest failure is the ambiguity of the meanings of \(T\) and the symbols \(+\) and \(\times\). If you clarified from the beginning that \(T\) is a set of formal strings or something like that, and that \(+\) and \(\times\) are just operator symbols or something like that, then you might notice this issue by yourself. Moreover, since you omit those clarifications, it is hard for others to point out the issue, because they should mention all the possibilities. If you want not to bother others who give you feedbacks, it is much better to remove the ambiguity so that you can notice problems by yourself and other can easily point out errors.

One typical failure related to the confusion of an operator symbol and an actual operator is the confusion of the formal string "\(1+1\)" of length \(3\) and the formal string "\(2\)" of length \(1\). Although an average googologist might not confound a formal string of length \(3\) and a formal string of length \(1\), the confusion frequently occurs when a beginner try to imitate an ordinal notation without understanding the difference between an equality of formal strings and an overloaded undefined equality of formal strings. Indeed, there are so many ill-defined notations based on this misconception. See also the issue on overloaded equalities.


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. For details, see the section of circular logics.


Self Loop

"I define a function \(f(x)\) as \(f(x)^2\)."

This sentence does not define \(f\), because the computation of \(f(1)\) refers to \(f(1)\) itself. This ill-definedness sometimes occurs when a googologist simply forgets to check the behaviour for the easiest cases. It is the simplest example of a circular logic.


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.


Infinite Searching

"I define \(<\) as the binary relation on the notation system \(T\) defined above characterised by the following: (1) For any \(s \in T\), \(s < s\) is false. (2) For any \(s,t \in T\), if \(s < t\), then \(t < s\) is false. (3) For any \(s,t,u \in T\), \(s < t\) and \(t < u\) implies \(s < u\). (4) For any \(s,t \in T\), \(s < s+t\). (5)…"

This sentence does not give an explicit algorithm to compute \(<\), and hence might not be used in computable googology. Indeed, if you directly try to interprete those restrictions into algorithms, the transitivity (3) gives you an infinite searching: In order to check that \(s < u\) does not hold, you need to check that \(s < t\) and \(t < u\) does not simultaneously hold for any \(t \in T\). (If you forget to avoid the case \(t = u\), then your algorithm will include a self loop. Even if you reject the case \(t = u\), then you need to search infinitely many \(t\).)

Similarly, use of an unbounded quantifier in a case classification causes the same issue. For example, if you try to define a function using "If \(\exists i \in \mathbb{N}\) such that...", then the direct implementation of the case classification yields an infinite loop if there is no such an \(i\), and hence does not actually give an implementation of the characterising formula.

Of course, the use of characterising formulae itself is ok as long as your regulation allowed a computable function without an explicitly written algorithm to compute it. However, then your regulation would allow "the constant function which outputs Rayo's number", because it is computable. Therefore the regulation is not sound. If you are working in computable googology, which should require an algorithm to compute all relations and functions to define a large number, then the use of a characterising formula without an explicitly written algorithm will be problematic.


Lack of the Well-foundedness

"I define \(<\) as the binary relation on the notation system \(T\) defined above in the following recursive way. Then the system is terminating, because for any \(A \in T\) and \(n \in \mathbb{N}\), we have \(A[n] < A\) as long as \(A \neq 0\)."

This argument is incorrect, because you have never show the well-foundedness of \(<\). The most difficult part in a proof of the termination of a notation system is to show the well-foundedness of an order compatible with the expansion rule, and hence omitting the well-foundedness does not solve the issue. If \(<\) is not well-founded, then the notation system might include an infinite loop, even if the expansion rule is compatible with the order.


Non-Existing Reference

When you use a new term \(t\), then you need to do either one of the following:

  1. Define \(t\) only using other existing terms, i.e. 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 references to non-existing objects.


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. Also, beginners tend to believe that they can omit "obvious" rules by saying something like "why should I write such a trivial rule? It is obvious that \(g(x,0,1) = g(x,1)\) is assumed for any positive integer \(x\) !" when we point out that there is no rule applicable to \(g(x,0,1)\) for their own function \(g\). Unfortunately, such an excuse does not make sense, because no rule which exists only in their mind is assumed. We can only assume the written rules.


Incomplete Partial Specialisation

"Using the function \(g(x)\) defined above, I define another function \(f(x)\) on a natural number \(x\) by \(f(0) = 1\) and \(f(g(x)+g(s)) = x^{x+s+1}\)."

This sentence does not define a function \(f\), unless \(s\) is already defined (or appropriately quantified) and satisfies that every \(x \neq 0\) in the domain of \(f\) admits precisely one \((y,s)\) such that \(x = g(y) + g(s)\). This ill-definedness sometimes occurs because terms appearing in an expression of the input look similar to indeterminates. See also the issue on overlapping partial specialisation and the guideline to avoid partial specialisation.


Ill-defined Value

"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 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. You are not allowed to use an undefined and unquantified object in the definition of a new object. For example, we frequently observe the following failure:

  1. Defining a number as a result of an expression including undefined and unquantified objects.
  2. Defining a function by case classifications whose conditions include undefined and unquantified objects.

This ill-definedness sometimes occurs by forgetting to write down the definition which is regarded as a "trivial" one by the author.


Ill-formed Expression

"I define a non-negative integer \(f(x,y)\) for any non-negative integers \(x\) and \(y\) in the forllowing recursive way: If \((x,y) = (0,0)\), then \(f(x,y) = 1\). Otherwise, \(f(x,y) = f(f(x,y-1),y-1)\)."

This sentence does not define \(f\) unless you add a case classification for the case \(x > 0\) and \(y = 0\), because the current rule refers to the ill-formed expression \(f(x,-1)\). You are not allowed to input to a function arguments which does not belong to the domain. For example, we frequently observe the following failure:

  1. Inputting a tuple of numbers including \(-1\) (resp. \(0\)) to a function whose domain is the set of non-negative (resp. positive) integers.
  2. Inputting an ordinal greater than \(\omega\) such as \(I\) to the subscript of Buchholz's function rather than that of Extended Buchholz's function. (See also #Buchholz's function#Common misconceptions.)
  3. Inputting an ordinal greater than or equal to \(\Gamma_M\) to the arguments of Rathjen's psi. (See also [[User blog:p進大好きbot/List of common misconceptions about Rathjen's psi#Does the countable limit coincide with \(\psi_{\Omega\}(\Gamma_{M+1})?]])
  4. Inputting a transfinite ordinal to hyper operators without fixing a definition. (See also #Undefined Transfinite Extension.)

This ill-definedness sometimes occurs when googologist miss the domains of their own functions or use existing functions whose definitions they do not know.


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.

Another typical example of an undefined comparison is the use of an undefined \(\leq\) when \(<\) is defined. The relation \(x \leq y\) can be defined in several ways, e.g. "\(x < y\) or \(x = y\)", "\(x < y\) or \(x \equiv y\)" using some non-trivial equivalence relation \(\equiv\), and "the negation of \(y < x\)", and hence the abbreviation of the definition of \(\leq\) is quite ambiguous. See also the issue on overloaded equation and the issue on circular reference of comparison and correspondence to ordinals.


Undefined Operation

"I define a notation \(T\) as the set of formal strings given as …. I define a recursive system of fundamental sequences. For any \((s,t) \in T^2\), if \(s+(t-1)\) is greater than…."

This sentence does not define a recursive system of fundamental sequences unless the formal string "s + (t - 1)" is a valid expression in the notation or you explicitly define the operation \(+\) and \(-1\) as maps whose domains are \(T^2\) and \(T\) respectively. 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.


Undefined Transfinite Extension

I define my ordinal \(\alpha\) as \(f_{\Omega \uparrow^{\Omega} \underbrace{\Omega_{\Omega_{\cdot_{\cdot_{\cdot_{\Omega}}}}}}_{\Omega}}(\Omega)\).

This sentence does not define an ordinal \(\alpha\) unless the transfinite hyperoperator and the transfinite FGH are precisely defined. Since there are so many candidates which have distinct reasonabilities, we do not have a single common agreed-uopon choice.

Further, when people use them and others point out this issue, they tend to explain that \(a \uparrow^b c\), \(\underbrace{\Omega_{\Omega_{\cdot_{\cdot_{\cdot_{\Omega}}}}}}_{a}\), and \(f_a(b)\) are simply defined as the same rule as the "usual" ones. But the point is that the transfinite iteration of a function also has so many candidates, and that is why the resulting functions have so many candidates. Therefore such an explanation is meaningless. They should precisely define the functions instead of saying "\(\omega \uparrow \uparrow (\omega+1)\) is \(\omega^{\varepsilon_0+1}\) by definition" or "\(f_1(\omega) = 2 \omega\) by definition" without fixing precise definitions. Although those equations do not hold for many natural formulations, they somewhy believe that they do not have to write down the precise definitions in their mind.

This ill-definedness frequently occurs when beginners do not understand what the domain of a function means. Given a function whose domain is originally given as \(\mathbb{N}\), we are not allowed to use it as a function whose domain is the class of all ordinals unless we fix a precise definition of a transfinite extension, because the behaviour of the original function does not characterise its extension.

See also the issue on lack of the declaration of the domain, the issue on undefined substitutions, and the guideline on how to define a new object.


Undefined Substitution

"For positive integers \(a\) and \(b\), I define \(f(a,b) = a^{f(a,b-1)}\). I put \(N = f(100,100)\)."

This sentence does not define \(N\) because \(N\) refers to \(f(100,1)\), which refers to the expression \(f(100,0)\) with undefined substitution. More awfully, when people define \(N\) in this way and others point out the error, they sometimes say "\(f(a,0) = a\) by definition" or something like that. Unfortunately, we are not talking about unwritten rules which exist only in their minds, but are talking about the written rules.

This ill-definedness frquently 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, the issue on undefined transfinite extensions, 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.


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.


Non-Unique 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.


Overlapping Partial Specialisation

"I define a set \(T\) of formal strings in the following recursive way: \(0 \in T\), \(\omega^a \in T\) for any \(a \in T\), and \(a+b \in T\) for any \(a,b \in T\). I define a fundamental sequence \([] \colon T \times \mathbb{N} \to \mathbb{N}\) by \(0[n] = 0\), \(\omega^0[n] = 0\), \(\omega^a[n] = \omega^{a[n+1]}\) for any \(a \in T \setminus \{0\}\), \(a + 0[n] = a[n+1]\) for any \(a \in T\), and \((a+b)[n] = a+(b[n+1))\) for any \(a \in T\) and \(b \in T \setminus \{0\}\)."

This sentence does not define a function \([]\), because the partial specialisation \((a+b)[n] = a+(b[n])\) is not solvable in a unique way. For example, when you compute \(\omega^{0}+\omega^{0}+\omega^{0}[n]\), the rule is applicable to it in two distinct ways, i.e. \((a,b) = (\omega^{0}+\omega^{0},\omega^{0})\) and \((a,b) = (\omega^{0},\omega^{0}+\omega^{0})\). This ill-definedness frequently occurs when beginners heavily use partial specialisations without understanding what a partial specialisation means.

As a related topic, a sentence like "I define \(h(x)\) by \(h(g(x)) =\) (an expression including \(g(x)\)) for any function \(g(x)\)" is also a combination of partial specialisation. It can be justified only when the right hand side does not include an undefined term other than \(g(x)\). In that case, it is better to define \(h(x)\) by the expression replacing \(g(x)\) by x without partial specialisation. If the right hand side includes an undefined term \(t\) such as \(g(x+1)\), then it is just ill-defined because \(g(x)\) does not characerise other values of \(g\). See also the issue on incomplete partial specialisation and the guideline to avoid partial specialisation.


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 \((\#_1,\#_2)\). Basically, division into substrings including two or more wildcards frequently cause this problem.

One simplest example of a non-unique division into substrings is the decomposition into an "addition expression". For example, consider the following rule to compute \(s[n]\) for a formal string \(s\) and a natural number \(n\): "If \(s\) can be expressed as \(A+B\) using substrings \(A\) and \(B\), then \(s[n]\) is defined as \(A+(B[n^2])\)." It does not work, because this rule uses two wildcards \(A\) and \(B\). For example, this rule is applicable to \(s = X+X+X\) in two distinct ways. One is the application with respect to the division by \(A = X\) and \(B = X+X\), and the other one is the application with respect to the division by \(A = X+X\) and \(B = X\). Since the result of the computation of \(s[n]\) heavily depends on the way to apply the rule to it, \(s[n]\) is ill-defined.

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. For example, googologists who the ordinal notation associated to Cantor normal forms but actually do not know the definition of an ordinal notation tend to make this kind of mistakes.


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 or a new fundamental sequence. See also the issue on intuitive normal forms.


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 \(T\) 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 \(\{0,T\}\) with respect to …."

This sentence does not define \(\psi\) unless you explicitly fix \(T\). Stating something like "\(T\) 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.


Unspecified OCF

"I define a natural number \(N\) as \(f_{\psi(\Omega_{I_{M_{K+1}+1}+1})(10)}\)."

This sentence does not define \(N\) unless you specify the \(\psi\). Beginners tend to use an ill-defined OCF like UNOCF without understanding the issue, or an unspecified OCF without understanding the ambiguity. When we ask what OCF they are using, then they frequently answer "normal OCF". When we ask the definition of a "normal OCF", then they answer that they do not know the definition, or they answer that it is Buchholz's OCF or Rathjen's OCF, even though those OCFs behave in a completely different way. Readers should be careful that beginners tend to refer to Buchholz's OCF and Rathjen's OCF without knowing their definitions and spread wrong information about them.

A: "What is the definition of the \(\psi\) which you are using in your analysis?"
B: "The normal OCF."
A: "Sorry, but I am asking the definition rather than the name which you use when you call the OCF. What is the definition of the normal OCF?"
B: "It is Buchholz's \(\psi\) or Rathjen's \(\psi\). I do not have to recall its definition, because it is a common knowledge for experts in googology."
A: "Or? Why don't you know which one you are using? If you cannot specify the actual definition of the OCF which you are using, then your analysis is highly doubtful because it is not based on the actual definition."
B: "OK. Then it is Buchholz's \(\psi\), in fact. Of course, I know its definition very well."
A: "But the expression \(\psi_M\), which you used in your analysis, is ill-defined for Buchholz's \(\psi\)."
B: "Alright. To tell the truth, I intended to use Rathjen's \(\psi\), whose definition I professionally understand. I just tested you. I admit that you seem not to be such a beginner as I first guessed."
A: "But \(\psi_M\) is ill-defined for Rathjen's \(\psi\), too."
B: "Hey, stop it. It is obvious that I used the extented Madore's \(\psi\). Why couldn't you understand such a trivial thing?"
A: "I am asking the definition of the OCF, but not the name which you use when you call the OCF. Where is the definition?"
B: "I said that it is the extended Madore's \(\psi\). I do not have to explain the definition by myself, because it is a common knowledge for experts in googology."
A: "But there is no actual definition of what you call the extended Madore's \(\psi\)..."

For fakes on OCFs which people yielded in googology without referring to the actual definitions, see also Buchholz's function#Common misconceptions, User blog:p進大好きbot/List of common misconceptions about Rathjen's psi and User blog:p進大好きbot/Historical Background of the Ill-definedness of UNOCF. Anyway, the magical word "normal OCF" is meaningless, because there is no actual definition. When a googologist refers to a "normal OCF" without specifying the definition, then it is good to suspect that it is ill-defined.


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 instead of listing finitely many examples only for which you can prcisely specify how it is intended to work. Generally speaking, the ambiguous words like "Pattern repeats" and "and so on" are not almighty power which solve whatever you do not have an idea to clarify the definition of. Please keep in mind that if you do not know how to precisely define a notion in your mind, then you should not use it. 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. This failure is mainly due to the wrong belief that finitely many examples uniquely characterise a function, which is based on the lack of the knowledge of the precise definition of the notion of a function or a map. See also #Use of Ambiguous Ellipses, #Lack of the Clarification of the Domain and #Incomplete Case Classification.


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 and the issue on the intuitive use of normal forms.


Intuitive Normal Forms

"I define the normal form of an ordinal \(\alpha\) in the following way: If \(\alpha = 0\), then its normal form is \(0\). If \(\alpha = 1\), its normal form is \(\varphi_0(0)\). ..."

This sentence does not define a correct notion of a normal form as intended. For example, since \(\omega^{0+0} = \varphi_0(0)\). \(\omega^{0+0}\) is the "normal form" of \(1\) with respect to the inappropriate formulation of the normal form above. In order to design the notion of a normal form in an appropriate way, we need to formalise a predicate like \(\alpha =_{\textrm{NF}} \varphi_{\beta}(\gamma)\) on three ordinals \(\alpha\), \(\beta\), and \(\gamma\) with formal separators \(=_{\textrm{NF}}\) and \(\varphi\), which should be distinguished from the actual function \(\varphi\).

This failure frequently occurs when googologists who do not understand the difference between ordinals and their expressions try to imitate the definition of normal forms following wrong explanation by other googologists. See also the issue on non-unique expressions and the issue on the intuitive use of expressions.


Intuitive Conversion of Growth Rates and Ordinals

"For each function \(F \colon \mathbb{N} \to \mathbb{N}\), I define a stronger function \(F^{\uparrow} \colon \mathbb{N} \to \mathbb{N}\) as the function with the growth rate \(\omega^{\alpha+1}\), where \(\alpha\) is the orginal given as the growth rate of \(F\)."

The sentence does not define the correspondence \(F \mapsto F^{\uparrow}\), because of the following two typical problems:

  1. The growth rate does not characterise a unique function, because there are two distinct functions with that growth rate.
  2. "The ordinal corresponding to (the growth rate) a function" is ill-defined, because it can be just used in analyses in a context where a system of fundamental sequences below a fixed countable ordinal is explicitly fixed and the corresponding FGH actually gives a linear hierarchy.
  3. The FGH \(f_{\omega^{\alpha+1}}\) itself is ill-defined because no specific choice of a system of fundamental sequences up to \(\omega^{\alpha+1}\) is fixed.

The problem 1 occurs when beginners talk about growth rates without understanding the definition of (eventual) domination. See also the issue on the lack of the unique existence.

The problem 2 and the problem 3 occur when beginners talk about ordinals without understanding the definition of FGH and regard it as a magical tool to scale all functions using countable ordinals. See also the issue on unspecified fundamental sequences and the issue that a larger ordinal does not necessarily correspond to a greater function.


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.


Confusion of Distinct Notions

Googologists tend to confound two distinct notions. The failure frequently occurs when they talk about what they do not know the precise definitions of.


Confusion of Ordinals and Expression

Ordinals are not formal strings. See #Intuitive Use of Expressions and User blog:P進大好きbot/List of common mistakes in googology#Is a cardinal just a formal string?.


Confusion of Ordinals and Functions

Ordinals are not functions. See #Intuitive Conversion of Growth Rates and Ordinals.


Confusion of Ordinals and Growth Rate

Ordinals are not growth rates. See #Intuitive Conversion of Growth Rates and Ordinals.


Confusion of Distinct OCFs

Two OCFs are not necessarily the same. See also #Unspecified OCF and Buchholz's function#Common misconceptions.


Confusion of Ordinal Functions and Ordinal Notations

Ordinal functions such as OCFs are not ordinal notations. See also User blog:P進大好きbot/Relation between an OCF and an Ordinal Notation.


Confusion of Ways to Write Ordinals and Ordinal Notations

An ordinal notation does not mean a way to write ordinals. See also User blog:P進大好きbot/Relation between an OCF and an Ordinal Notation.


Confusion of Expansion Rules and Ordinal Notations

An ordinal notation does not mean a notation with an expansion rule. See also User blog:P進大好きbot/Relation between an OCF and an Ordinal Notation.


Intuition-Based Interpretation

When you use notions in formal logic, you should follow the precise definition of them, or should precisely redefine them by yourself. However, googologists tend to use "their own intuition-nased interpretations" of formal logic.

Here are examples of the intuition-based interpretation.


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 Inconsistency

"For a natural number \(n\), I define \(f(n)\) the greatest natural number definable by less than or equal to \(n\) symbols in \(\textrm{ZFC}\) set theory which does not cause contradiction or Berry's paradox."

This sentence is meaningless, because the intuitive predicate like "its existence does not cause a contradiction" or "its definition does not cause Berry's paradox" is not actually a formalised predicate on a natural number. Namely, the predicate itself is ill-defined in \(\textrm{ZFC}\) set theory. (If a statement like "It avoids contradiction" actually avoids contradiction, then everybody would write so without making an effort to create a consistent notion.)

The same holds even when you replace "\(\textrm{ZFC}\) set theory" by a natural language, because the corresponding definability is unformalisable. For example, consider the sentence "one plus the greatest natural number definable by less than or equal to one hundred symbols in English which does not cause contradiction or Berry's paradox". It "defines" a natural number \(N\) in your convention, which satisfies both \(N \leq f(100)\) and \(N > f(100)\), i.e. you are contradicting. A related concept is "the least natural number \(N\) greater than any ill-defined natural numbers". If \(N\) is ill-defined, then it satisfies \(N > N\) and hence causes contradiction. Does it mean that \(N\) is well-defined? No. It is just ill-defined because it is unformalisable. It does not cause contradiction in any formal theory, and just you are contradicting if you allow such an intuition-based description.

Similarly, defining a computable large function \(f(n)\) as the Busy Beaver function except for avoiding Turing machines with undecidable problems is invalid, because the predicate referring to the undecidability itself is ill-defined in the realm of computable functions, i.e. uncomputable.

This ill-definedness frequently occurs when a googologist tries to avoid a diagonalisation problem such as Berry's paradox and the halting problem without understanding the actual issue.


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 by formulae of length \(< 10^{100}\)."

This sentence does not define \(N\) unless a formal language \(L\) with precise definitions of the syntax and the semantics is explicitly written 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 or even what formalisation means tries to imitate Rayo's number or variables of transcendental integers using a programming language.

It does not matter if you replace the definablity by similar words like namability, nameability, describability, expressibility, mentionability, writability, specifiability, characterisability, coinability, or whatevery you want to use to intuitively refer to the intuitive status of numbers without formalisation, and the language \(L\) by words like English, people's actual conversation, people's imagination, all possible intelligences, or whatever you want to use to intuitively describe numbers. As long as you drop the formalisability, the resulting number is as ill-defined as a number causing Berry's paradox, even if you yourself are unable to find any error.

By the way, when beginners fall into this trap, then it is certain that they have no knowledge of formalisability and truth predicates. However, they tend to insist something like "My number is larger than Rayo's number and its variants, and is the greatest in the world!", although the precise definitions of them requires the knowledge of how to formalise truth predicates. Therefore they are talking about what they do not understand the precise definitions of. This failure is closely related to the cheating explained in the list of common mistakes in googogology.


Incomplete Original Logic

We can abbreviate the formal theory in which we work, only when we are assumed to be working in the traditional formal theory, i.e. \(\textrm{ZFC}\) set theory. Therefore when you use another formal theory especially in uncomputable googology, you need to declare what axioms you employ. Otherwise, even the notion of a natural number is ill-defined. See also the explanation of the necessity of an axiom to define a natural number.

Here are examples of the use of intuition-based original logic.


Lack of Base Theory

"I define a new set theory \(T\) stronger than \(\textrm{ZFC}\) as …. I defined \(N\) as the least natural number greater than any natural numbers definable by formulae in \(T\) of length \(< 10^{100}\)."

This sentence does not define \(N\), because you are refering to the intuitive definability of a natural number in the unspecified base theory by formulae in \(T\). You need to fix the base theory, in which \(T\) and the definability of a natural number by formulae in \(T\) are provable. This ill-definedness sometimes occurs when a googologist without the knowledge of formal logic imitates Utter Oblivion and confound the base theory and a coded theory. See also the issue on intuitive truth predicate and the issue on intuitive definability.


Lack of Language

"I work in a new set theory \(T\) defined by the following axioms: …. 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 the base theory \(T\) or \(N\), because the language of \(T\) is undefined. Without fixing the language, axiom schema do not make sense, and the notion of a formula is ambiguous. This ill-definedness occurs when a googologist without the knowledge of formal logic tries to imitate Utter Oblivion and confound a theory with a collection of formulae (intended to be axioms) without fixing a formal language.


Lack of Axioms

"I work in a new set theory \(T\) 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 the base theory \(T\) 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 and confound a formal language with a theory.


Incomplete Higher Order Coded 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 set. 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.


Incomplete Higher Order 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.


Incomplete New Foundation

"I define a new foundation of mathematics because higher order logics are insufficient for me. I call it System \(X\). It consists of terms, functions, truth predicates indexed by terms, quantifiers indexed by terms, and logical connectives. A truth predicate indexed by a term \(t\) returns the truth of a formula at "the \(t\)-th universe", and a quantifier indexed by a term \(t\) works as a "\(t\)-th order quantifier". For a natural number \(n\), I define \(f(n)\) as the least natural number greater than any natural number definable by a formula of length \(n\) or less in a formal theory \(T\) in System \(X\) consisting of a formal language with \(n\) or less symbols and \(n\) or less axioms with length \(n\) or less.

This sentence does not define a function \(f\), because notions such as natural numbers, functions, predicates, truths, logical connectives, universes, quantifiers, axioms, definability, formulae, formal languages, and so on are just defined for predicative logic in the usual mathematics and equivalent formal systems, and hence you need to define it for your own system, i.e. System \(X\).

This ill-definedness frequently occurs when a googologist who does not know that even such foundational notions have both syntax and sematincs theoretic precise definitions tries to formalise Utter Oblivion. Defining a a new foundation in a precise manner so that the resulting number is actually significantly larger than Rayo's number is quite difficult, and that is why Utter Oblivion has never been completely defined. Therefore skipping the precise definition of a new foundation is non-sense if you want to try to complete the definition of Utter Oblivion.

Also, you need to fix the base theory. If you want to define the base theory as \(\textrm{ZFC}\) set theory and refer to the definability in a new foundation within the base theory, then it does not work as intended, because the resulting number can be definable in the base theory, which is just \(\textrm{ZFC}\) set theory. In other words, it is not significantly larger than Rayo's number. On the other hand, if you want to define the base theory in terms of a new foundation, you need to fix the meta theory and define the new foundation in the meta theory. In that case, you need to define notions in the base theory (using the new foundation instead of first order logic), e.g. a natural number, the definability of a natural number in a theory coded by using the new foundation, and so on. In other words, you should reinvent complicated mathematical notions using the new foundation by yourself. However, googologists unintentionally tend to skip the definition of the meta theory and the definition of notions in the base theory, because people who try to formalise Utter Oblivion tend not to know the usual formal logic.


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. For details, see the section of infinite loops.


Confusion of Conclusions and Assumptions

"I define a binary relation \(a \triangleleft b\) on \(\mathbb{N}\). If \(a - b \equiv 0\) (mod \(2\)) and \(a < b\), then \(a \triangleleft b\). If \(a\) is an even number and \(b\) is an odd number, then \(a \triangleleft b\). Then \(\triangleleft\) is a strict well-ordering on \(\mathbb{N}\)."

This sentences do not define \(\triangleleft\), unless you further assume that \(\triangleleft\) is a strict well-ordering on \(\mathbb{N}\). Indeed, there is no rule to determine whether \(1 \triangleleft 1\) and \(1 \triangleleft 2\) hold or not. You might say that \(1 \triangleleft 1\) does not hold because \(1 = 1\) and \(1 \triangleleft 2\) does not hold because the rule implies \(2 \triangleleft 1\), but it is a wrong deduction, as you used the implicit assumption that \(\triangleleft\) is a strict well-ordering on \(\mathbb{N}\). You can only use what you actually wrote. Using the unwritten assumption that \(\triangleleft\) is a strict well-ordering on \(\mathbb{N}\) is not allowed, and also causes a circular logic because you concluded "Then \(\triangleleft\) is a strict well-ordering on \(\mathbb{N}\)" as if it were a proprty which can be verified from what you wrote.

This failure frequently occurs when googologists try to define an ordinal notation under a belief that what they wrote obviously give valid definitions of well-orderings.


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 tries to immetate a mutual recursion of an ordinal collapsing function \(\psi\) and a constructible set function \(C\). Typically, \(\psi(\alpha)\) refers to \(C(\alpha)\) or something like that, which refers to some other function \(\Xi(\alpha)\), which refers to \(C(\alpha)\) or \(\Xi(\alpha)\) itself.


Circular Reference of Domain and Function

"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 Reference of Functions

"I define an ordinal function \(f \colon \Omega \to \Omega, \ \alpha \mapsto f(\alpha)\) in the following way: If \(\alpha = 0\), then \(f(\alpha) = \omega\). If \(\alpha = \beta+1\) for some \(\beta \in \Omega\), then \(f(\alpha)[n] = (f(\beta)+1) \uparrow \uparrow n\). If \(\alpha\) is a non-zero limit ordinal, then \(f(\alpha)[n] = f(\alpha[n])\)."

This sentence does not define \(f\), because the system \([]\) of fundamental sequence is ill-defined. This ill-definedness occurs when a googologist who believes that a fundamental sequence is unique, or at least that there should be one "natural" choice, tries to define an ordinal function using fundamental sequences.

If you assume that a system of fundamental sequence is fixed from the beginning, then \(f(\alpha)[n] = f(\beta) \uparrow \uparrow n\) is a multiple definition of fundamental sequences. If you insist that you are simultaneously defining a system of fundamental sequences, then it is an incomplete partial specialisation because you are not defining for all limit ordinals in the domain of \(f\) and also might be an overlappring partial specialisation because you are defining \(\alpha[n]\) using an expression of \(\alpha\). See also the issue on the non-unique exression.

One possible way to fix the issue is to restrict the domain of \(f\), and simultaneously define a system of fundamental sequences for all limit ordinals in the domain of \(f\). If you want to define a system of fundamental sequences using expressions by \(f\), you need to fix a unique form expressing every limit ordinals in the domain of \(f\), and explicitly write down the definition of the fundamental sequence for any limit ordinal in the domain of \(f\) using the unique form. Otherwise, the issue on the incomplete partial specialisation is still serious.

Also, you need to be careful that you cannot define the domain of \([]\) as "the set of ordinals which can be expressed by \(0\), \(1\), \(\omega\), \(+\), and \(f\)" or something like that. It is simply a circular reference of the domain of \([]\) and \(f\), because the definition of \(f\) refers to the definition of \([]\), which refers to the definition of the domain of \([]\), which refers to the definition of \(f\). See also the issue on a circular reference of the domain and a function.


Circular Reference of Comparison and Correspondence

"I define a set \(T\) of formal strings as …. I define an expansion rule \([] \colon T \times \mathbb{N} \to T\) as …. Here, I need to add the definition of the comparison \(<\) and the equivalence relation \(\equiv\) used in the definition of \([]\). It is quite simple. The comparison \(<\) means the inequality of the corresponding ordinals, and the equivalence relation \(\equiv\) means the equality of the corresponding ordinals."

The sentence does not defined \(<\) or \(\equiv\), because it is based on a circular logic. The correspondence to ordinals makes sense only when you have defined some structures such as an expansion rule \([]\) or a well-founded ordering \(<\). However, you used the correspondence in the definitions of \([]\) and \(<\), and hence the correspondence is ill-defined.

Please be careful that even if two distinct expressions are intended by the creator to "express" the same ordinal or the same structure, the identification of them is valid only when an appropriate equivalence relation \(\equiv\) is completely defined. Also, such an identification based on an equivalence relation is not an actual equality. This ill-definedness frequentlys occurs when one tries to interprete ordinals into strings and to identify distinct expressions with each other, because the difference of the three roles of an equation is not easy to realise before pointed out. See also the issue on overloaded equation.


Circular Quantification

"I define a map \(f(\alpha)\) on an ordinal \(\alpha\) in the following way: If \(\alpha = 0\), then \(f(\alpha) = 1\). If \(\alpha = \beta + f(\gamma)\) for some ordinals \(\beta\) and \(\gamma\), then ... ."

This sentence does not define \(f\) because the definition of \(f(\alpha)\) refers to the value of \(f(\gamma)\) for each ordinal \(\gamma\), and especially to the value of \(f(\alpha)\) itself. This ill-definedness frequently occurs when a googologist who comfounds a function with a function symbol.


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.

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