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Single Idea 17789

[filed under theme 5. Theory of Logic / K. Features of Logics / 6. Compactness ]

Full Idea

No logic which can axiomatise arithmetic can be compact or complete.

Gist of Idea

No logic which can axiomatise arithmetic can be compact or complete

Source

John Mayberry (What Required for Foundation for Maths? [1994], p.412-1)

Book Ref

'Philosophy of Mathematics: anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.412

A Reaction

I take this to be because there are new truths in the transfinite level (as well as the problem of incompleteness).

The 17 ideas with the same theme [satisfaction by satisfying the finite subsets]:

9995 Proof in finite subsets is sufficient for proof in an infinite set [Enderton]
10771 Compactness is important for major theories which have infinitely many axioms [Tharp]
10772 Compactness blocks infinite expansion, and admits non-standard models [Tharp]
13544 Inconsistency or entailment just from functors and quantifiers is finitely based, if compact [Bostock]
13618 Compactness means an infinity of sequents on the left will add nothing new [Bostock]
13841 Why should compactness be definitive of logic? [Boolos, by Hacking]
10287 If a first-order theory entails a sentence, there is a finite subset of the theory which entails it [Hodges,W]
13496 First-order logic is 'compact': consequences of a set are consequences of a finite subset [Hart,WD]
17789 No logic which can axiomatise arithmetic can be compact or complete [Mayberry]
13630 Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro]
13646 Compactness is derived from soundness and completeness [Shapiro]
13699 Compactness surprisingly says that no contradictions can emerge when the set goes infinite [Sider]
10977 Compactness blocks the proof of 'for every n, A(n)' (as the proof would be infinite) [Read]
10976 Compactness makes consequence manageable, but restricts expressive power [Read]
10974 Compactness is when any consequence of infinite propositions is the consequence of a finite subset [Read]
10975 Compactness does not deny that an inference can have infinitely many premisses [Read]
17867 If a concept is not compact, it will not be presentable to finite minds [Almog]