An Application of 1-Genericity in the Enumeration Degrees

Using results from the local structure of the enumeration degrees we show the existence of prime ideals of

enumeration degrees. We begin by showing that there exists a 1-generic enumeration degree

which is noncuppable—and so properly downwards

$\Sigma^0_2$

—and low

2

. The notion of

enumeration

1

-genericity

appropriate to positive reducibilities is introduced and a set

A

is defined to be

symmetric enumeration

1

-generic

if both

A

and

$\ensuremath{\overline{A}} $

are enumeration 1-generic. We show that, if a set is 1-generic then it is symmetric enumeration 1-generic, and we prove that for any

enumeration 1-generic set

B

the class

$\{\, X \,\mid \, \;\ensuremath{\negmedspace\leq_{\ensuremath{\mathrm{e}} }\negmedspace}\; B \,\}$

is uniform

. Thus, picking 1-generic

(from above) and defining

it follows that every

only contains

sets. Since

is properly

$\Sigma^0_2$

we deduce that

contains no

$\Delta^0_2$

sets and so is itself properly

.