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proof of compactness theorem for first
order logic∗
CWoo†
2013-03-21 14:17:06
The theorem states that if a set of sentences of a first-order language L is
inconsistent, then some finite subset of it is inconsistent. Suppose ∆ ⊆ L is
inconsistent. Then by definition ∆ `⊥, i.e. there is a formal proof of “false”
using only assumptions from ∆. Formal proofs are finite objects, so let Γ collect
all the formulas of ∆ that are used in the proof.
∗ hProofOfCompactnessTheoremForFirstOrderLogici created: h2013-03-21i by: hCWooi
version: h33034i Privacy setting: h1i hProofi h03B10i h03C07i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
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