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Single Idea 10588

[filed under theme 5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic ]

Full Idea

Early study of first-order logic revealed a number of important features. Gödel showed that there is a complete, sound and effective deductive system. It follows that it is Compact, and there are also the downward and upward Löwenheim-Skolem Theorems.

Gist of Idea

First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems

Source

Stewart Shapiro (Higher-Order Logic [2001], 2.1)

Book Ref

'Blackwell Guide to Philosophical Logic', ed/tr. Goble,Lou [Blackwell 2001], p.34

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The 12 ideas from 'Higher-Order Logic'

10588

First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro]

10290

Second-order variables also range over properties, sets, relations or functions [Shapiro]

10590

Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro]

10292

Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro]

10294

Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro]

10591

Logicians use 'property' and 'set' interchangeably, with little hanging on it [Shapiro]

10296

The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro]

10297

The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro]

10298

Some say that second-order logic is mathematics, not logic [Shapiro]

10299

If the aim of logic is to codify inferences, second-order logic is useless [Shapiro]

10300

Logical consequence can be defined in terms of the logical terminology [Shapiro]

10301

The axiom of choice is controversial, but it could be replaced [Shapiro]