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I would request for some more clarification regarding the proof of diagonal lemma in Enderton book on Mathematical Logic (page 235)

Suppose the numeral for the godel number for the formula theta( v1,v1,v3)is q.

Then theta(q,q,v3) is that " v3 is the numeral for the godel number of theta(q,q,v3)".

The clarification I request for is :

Taking v3 as a free variable , substituting it by a numeral should give a closed sentence.

So if we substitute S0 for v3, do we get

Theta(q,q,S0) is " S0 is the numeral for the godel number of theta(q,q,S0)"?

Or is it

Theta(q,q,S0) is "S0 is the numeral for the godel number of theta(q,q,v3)"?

Secondly,

If p is the sentence

"for all y[ theta ( q,q,y) -->beta y ]"

and we claim "p <-->beta (numeral code p)"

Is it necessary first to establish that there is at least one y for which theta (q,q,y) is true?

Sudhir
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  • Only three comments : (a) $q$ is not the g-number of $\theta$ but of formula (1) pag.235, i.e. of $\forall v_3[\theta \rightarrow \beta]$. (b) $S(0)$ cannot be the g-number of the formula (see Table at pag.225) : we have that $S(0)=1=\sharp('(')$. (c) The formula $\theta$ represents a function $Sb_1$ and we have that $\theta(\sharp \alpha, \sharp t, \sharp \alpha(t/x))$ is true iff $Sb_1(\sharp \alpha, \sharp t) = \sharp \alpha(t/x)$ so it is already established that, when $\theta$ is true, there is exactly one y for which $\theta$ is true. – Mauro ALLEGRANZA Jan 03 '14 at 16:35
  • a)yes, but my question is in relation to the formula theta.b)I had used S0 only as an example.I wanted to know which of the two interpretations I mentioned is the correct one.c)My mistake: the q here is the number of "for all y[ theta (x,x,y) -->beta y]" – Sudhir Jan 03 '14 at 17:48

1 Answers1

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I will try to answer your questions with a "line-by-line" comment.

First, some background definitions.

a) see Enderton, pag.212 :

We say that a formula $\alpha$ of formal number theory $T$ functionally represents a function (say, for simplicity, with two argument places) $f : \mathbb N$ x $\mathbb N \rightarrow \mathbb N$ when, for every $a, b \in \mathbb N$ , if $c=f(a,b)$, then

$T \vdash \forall v_3[ \alpha (S^a(0), S^b(0), v_3) \leftrightarrow v_3 = S^c(0)]$.

b) see Enderton's definition (pag.235) :

Let $\theta (v_1,v_2,v_3)$ functionally represents a function whose value at <# $\alpha, n$> is #($\alpha (S^n0)$).

Now, your questions :

Suppose the numeral for the godel number for the formula theta( v1,v1,v3)is q.

I will rephrase it as :

Suppose that the g-number of the formula $\theta (v_1,v_1,v_3)$ is $q$ [i.e. $q$ = # $\theta (…)$ ; the corresponding numeral will be : $S^q(0)$ ].

Then theta(q,q,v3) is that " v3 is the numeral for the godel number of theta(q,q,v3)".

Then the correct interpretation of $\theta (S^q(0), S^q(0), S^n(0))$ will NOT be " $n$ is the godel number of $\theta (S^q(0), S^q(0), v_3)$ ", because $\theta$ functionally represents a function whose value at … where $\alpha$ must have only one free variable, and $\theta (v_1,v_1,v_3)$ has two free variables.

So, you must start with $\forall v_3 [\theta (v_1,v_1,v_3) \rightarrow \beta (v_3)]$; now you have a formula of “type” $\alpha (x)$ (with only one free variable $x$).

Let $q$ be its g-number (i.e. $q$ = # ($\forall v_3 [\theta (v_1,v_1,v_3) \rightarrow \beta(v_3)] )$ ); then, substituting $S^q(0)$ into its (free) argument place you will obtain the new formula $\forall v_3[\theta (S^q(0), S^q(0), v_3) \rightarrow \beta (v_3)]$; call it $\sigma$ with a new g-number : $s$ = #$\sigma$.

Because $\theta$ functionally represents a function whose value at is #$\sigma$, we have that : $\forall v_3 [\theta(S^q(0), S^q(0),v_3) \leftrightarrow v_3 = S^s(0)]$.

Taking v3 as a free variable , substituting it by a numeral should give a closed sentence.

So if we substitute S0 for v3, do we get

Theta(q,q,S0) is " S0 is the numeral for the godel number of theta(q,q,S0)"?

Or is it

Theta(q,q,S0) is "S0 is the numeral for the godel number of theta(q,q,v3)"?

In general, $\theta (S^q(0), S^q(0), S^n(0))$ will be interpreted as " $n$ is the g-number of the result of the substitution into the formula with g-number $q$ [i.e.$\forall v_3 \theta(v_1,v_1,v_3)$ ] with the numeral $S^q(0)$, i.e. $n$ = # $\forall v_3 [\theta... ](v_1/S^q(0))$.

Mauro ALLEGRANZA
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