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The definition of $\text{Sat}([φ],s)$ can be found here.

All I want is an explanation of what each line in this definition means and how $\text{Sat}([φ],s)$ works. The only relevant thing that I have found is the following comment ( source ):

He is referring to truth in the set-theoretic universe (not the natural numbers), and the class R is a truth function, for the clauses in the definition assert that R obeys the Tarskian recursion, and the final clause asserts that phi is true of s, according to the truth predicate R.

lyrically wicked
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1 Answers1

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That is the satisfaction predicate. It takes a formula and an assignment to the variables of the formula, and tells you whether or not the statement is true or false.

Rayo's number is really just a complicated way of saying that it is the smallest number which larger than all those numbers definable in the set theoretic universe with a formula of at most googol symbols.

The catch, of course, is that almost always we talk about the natural numbers as their own entity, so naming large numbers is taken by talking about some function like $\rm TREE$, or Busy Beaver, or so on. But the mathematical universe is so much larger. The set theoretic universe gives us tools much larger than just the natural numbers.

It also opens the door to the ills of independence, at least when playing this game of naming the largest number (e.g. $n$ is the least integer such that $2^{\aleph_n}>\aleph_{n+1}$, or $0$ otherwise). Which is why we need to refer to a fixed universe, and to the satisfaction relation of that universe.

Asaf Karagila
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  • Regarding the quote “Which is why we need to refer to a fixed universe, and to the satisfaction relation of that universe_” — does the definition imply that we are allowed to choose _any universe? If no, then why doesn’t the definition explicitly state what set-theoretic universe should be chosen? And what is the meaning of each of the following three lines? 1) ([ψ] = `x_i ∈ x_j' ∧ t(x_1) ∈ t(x_j)); 2) ([ψ] = `(θ∧ξ)' ∧ R([θ],t) ∧ R([ξ],t)); 3) ([ψ] = `∃x_i (&theta)' and, for some an x_i-variant t' of t, R([θ],t')) – lyrically wicked Oct 20 '18 at 12:07
  • No, the definition is always with respect to a single universe which is implicitly prefixed. Which might as well be the mathematical universe, as a whole, if you take set theoretic foundations of mathematics along for a ride. – Asaf Karagila Oct 20 '18 at 12:24
  • The three lines you define literally state the definition of truth in the set theoretic universe. Do you know Tarski's definition of the truth in a structure? – Asaf Karagila Oct 20 '18 at 12:26
  • I have read Tarski's Truth Definitions here, but haven't seen any definition that resembles the definition of $\text{Sat}$. Is it possible to translate these three lines to a more readable text? And where can I read more about the definition of truth in the set theoretic universe? – lyrically wicked Oct 20 '18 at 12:44
  • Well. The definition of truth start with atomic formulas—in the set theoretic context that is just $x=y$ or $x\in y$—and proceeds to connectives like $\land$ and negation, and finally quantifiers. That's all that there is to those three lines. And the truth definition in the set theoretic universe is no different. The set theoretic universe, after all, is just a structure, whose truth is given by the same Tarskian definition. – Asaf Karagila Oct 20 '18 at 12:46
  • It seems to me that I can imagine the meaning of all lines, but I still cannot obtain the meaning of ([ψ] = `∃x_i (&theta)' and, for some an x_i-variant t' of t, R([θ],t')) – lyrically wicked Oct 20 '18 at 12:54
  • Do you know what it means that $M\models_t\exists x\theta(x)$? – Asaf Karagila Oct 20 '18 at 13:03
  • In this formula, I am not sure I understand the meaning of subscript $t$. Also, I am not sure I can understand the meaning of $\theta(x)$. – lyrically wicked Oct 20 '18 at 13:35
  • The meaning is that $t$ is an assignment, and $\theta(x)$ is a formula whose free variable is $x$ (if it even has a free variable). Perhaps it would be a good choice to start with learning about predicate logic and first-order logic before trying to jump as far as the truth in the set theoretic universe and the satisfaction predicate. – Asaf Karagila Oct 20 '18 at 13:38
  • I knew that $\phi(x)$ might imply a formula whose free variable is $x$, but then what does ∃x_i (&theta) mean? – lyrically wicked Oct 20 '18 at 13:55
  • It is a formula that has the form $\exists x_i\theta$, where $x_i$ is the $i$th free variable in the language (because the language has infinitely many free variables). And $\theta$ is some other formula. – Asaf Karagila Oct 20 '18 at 13:57
  • The last claification: can I read the excerpt for some an x_i-variant t' of t as [for some $x_i$-variant $t'$ of $t$]? – lyrically wicked Oct 20 '18 at 14:11
  • It means that you take $t$, and only change its assign of $x_i$. In other words, $\exists x\theta$ is true if and only if there is some object $a$ such that when assigned $x$ with $a$, $\theta$ is true. – Asaf Karagila Oct 20 '18 at 14:12
  • Thank you! By the way, I am interested what is the meaning of the formula $M \models_t \exists x \theta(x)$. If $t$ is a variable assignment, then...? – lyrically wicked Oct 20 '18 at 14:15
  • I think that [it is the smallest number which is not definable in the set theoretic universe with a formula of at most googol symbols] in the current answer should be edited to [it is the smallest number bigger than every finite number $N_i$, assuming that $N_i$ is definable in the set theoretic universe with a formula of at most googol symbols]. The difference is very important. – lyrically wicked Oct 27 '18 at 10:13
  • @lyricallywicked: I don't understand your comment. What is the distinction? – Asaf Karagila Oct 27 '18 at 10:20
  • 1/2: Maybe I am wrong, but here's how I understand it. Assuming that the set-theoretic universe is fixed and $y \le 10^{100}$, let $X = {X_1, X_2, \ldots, X_{i-1}, X_i}$ denote a finite set of all numbers that are definable by some formula of $y$ symbols. Then let $T = {T_1, T_2, T_3, \ldots}$ denote an infinite set of all numbers that are not definable (in the same set-theoretic universe) by any formula of $y$ symbols. – lyrically wicked Oct 29 '18 at 07:43
  • 2/2: Then consider a finite set $Y = {0, 1, \ldots, X_i-2, X_i-1}$. It will contain unimaginably many elements that intersect with $T$. All these numbers are less than $X_i$, so they should be ignored. This is why Rayo's number $N$ was defined as $N = X_i + 1$, not as $N = T_1$. – lyrically wicked Oct 29 '18 at 07:44
  • Perhaps for future reference, but here's a simpler way to make the same point: "the smallest number not in ${0,5,42}$ is not the same as the least number larger than all members of ${0,5,42}$". – Asaf Karagila Oct 29 '18 at 07:59