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Single Idea 10292

[filed under theme 5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems ]

Full Idea

Downward Löwenheim-Skolem: a finite or denumerable set of first-order formulas that is satisfied by a model whose domain is infinite is satisfied in a model whose domain is the natural numbers

Gist of Idea

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

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]