I've been working my way through Godel's original paper of the incompleteness theorem in my spare time, and I'm stuck with something stupidly simple. I'm looking at the list of 45 definitions of concepts that are recursively defined. The definitions I'm concerned with here as as follows::
$$ 0Pr(x) \equiv 0 $$ $$(n+1)Pr(x) \equiv \epsilon y [ y \leq x \& Prim(y) \& x/y \& y > nPr(x)] $$
$$ l(x) \equiv \epsilon y [y \leq x \& yPr(x) > 0 \& (y+1)Pr(x) = 0]$$
Anyways, what I'm curious about is the fact that in definition 7, the length of the series of numbers assigned to x, the $y$th prime in $x$ appears to be less than the $y+1$st prime in $x$ if we wish for $l(x)$ to have a value. However, in definition $3$, we see that the $y+1$st prime in $x$ is always larger than the $yth$; $y > nPr(x)$. I think I have stared at this so long my eyes are starting to bleed. What am I missing? How is it possible that $l(x)$ is definable?
I assume that since $n Pr(x)$ needs to be defined for all $n$, then after you run out of primes in $x$, the primes after that are all defined as 0. But I can't see this interpretation coming out of definition 3.
A page or two earlier in the paper, I think you will find that Goedel defines $\epsilon x\, F(x)$ to mean the smallest number $x$ such that $F(x)$ holds or $0$ if there is no such number. So Definition 3 makes the $y+1$-st prime divisor of a number with only $y$ prime divisors equal to $0$.