Today I Learned

Some of the things I've learned every day since Oct 10, 2016

151: Henkin Witness (Logic)

Let T be a theory and \exists x \phi (x) be a sentence in T. A witness of this sentence is a constant c such that T \vdash \phi (c).

Similarly, a theory T is said to have the witness property if wherever \phi(x) is a formula with 1 free variable x there is a constant c such that

T \vdash ((\exists x \phi(x)) \rightarrow \phi(c)),

so that if T \vdash (\exists x \phi(x)) then T \vdash \phi(c).

Witnesses and the witness property play an essential role in certain proofs of the Completeness and Compactness theorems.

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One response to “151: Henkin Witness (Logic)

  1. Pingback: 165: The Compactness Theorem (Henkin Construction) | Today I Learned

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