Content deleted Content added
mNo edit summary |
Got started...I'll come back and leave it less half-finished later |
||
Line 1:
'''Forcing''' in [[recursion theory]] is a modification of [[Paul Cohen (mathematician)|Paul Cohen's]] original [[set theory|set theoretic]] technique of [[forcing (set theory)|forcing]] to deal with the effective concerns in [[recursion theory]]. Conceptually the two techniques are quite similar, in both one attempts to build [[generic set|generic]] objects (intuitively objects that are somehow 'typical') by meeting dense sets. Also both techniques are elegantly described as a relation (customarily denoted <math>\Vdash</math>) between 'conditions' and sentences. However, where set theoretic forcing is usually interested in creating objects that meet every dense set of conditions in the ground model, recursion theoretic forcing only aims to meet dense sets that are arithmetically or hyperarithmetically definable. Therefore some of the more difficult machinery used in set theoretic forcing can be eliminate or substantially simplified when defining forcing in recursion theory. But while the machinery may be somewhat different recursion theoretic and set theoretic forcing are properly regarded as an application of the same technique to different classes of formulas.
== Terminology and General Remarks ==
In this article we use the following terminology (standard in the field). A '''real'''is an element of <math>2^\omega</math>. In other words a function that maps each integer to either 0 or 1. A
'''string''' is an element of <math>2^{<\omega}</math>. In other words a finite approximation to a real.
== Definition of Forcing ==
We now define the basic notions
; notion of forcing : A notion of forcing is a set <math>P</math> and a [[partial order]] on <math>P</math>, <math>\succ_{P}</math> with a ''greatest element'' <math>0_{P}</math>.
; <math>\sigma</math> is stronger than <math>\tau</math> : Given <math>\sigma,\tau \in P</math> for a notion of forcing <math>P</math> we say that <math>\sigma</math> is stronger than <math>\tau</math> when <math>\tau \succ_P \sigma</math>. Note that this means that <math>\succ_{P}</math> will usually turn out to be the '''reverse''' of the containment relation. '''Be careful some recursion theorists reverse the direction of <math>\succ_P</math>''' which seems more natural in many recursion theoretic contexts but is at odds with the notation used in set theory.
Technically as in set theoretic forcing the result of a forcing construction will be an [[ultrafilter]] on <math>\mathbb{P}</math>. That is we will produce a set of conditions <math>F\subset P</math> so that if <math>p,q \in P</math> then there is an <math>r\in P</math> so that
{{mathlogic-stub}}
| |||